Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2923 https://internationalpubls.com The Normalization of Complex Intuitionistic Fuzzy Matrices Jaikumar. S1, Ragavan. C1* 1, Department of Mathematics, Sri Vidya Mandir Arts & Science College (Autonomous), Uthangarai, Krishnagiri, Tamilnadu, India sjaisan18@gmail.com, ragavanshana@gmail.com Article History: Received: 14-01-2025 Revised: 15-03-2025 Accepted: 21-03-2025 Abstract: The concept of intuitionistic fuzzy sets (IFS) provides a comprehensive framework for dealing with uncertainty, incorporating both membership and non-membership functions. Normalization of intuitionistic fuzzy matrices is an essential process in decision-making and data analysis where the matrix entries are expressed in terms of intuitionistic fuzzy numbers (IFNs). This paper explores the methods and techniques for normalizing intuitionistic fuzzy matrices to ensure that the degree of membership and non-membership values are consistent, thus enhancing the reliability of the matrix in decision analysis problems. We propose an efficient approach to normalize intuitionistic fuzzy matrices, ensuring the transformed values retain their essential characteristics while adhering to the constraints of the fuzzy set theory. The study also addresses the application of normalized intuitionistic fuzzy matrices in multi-criteria decision-making (MCDM) and other relevant areas such as image processing, pattern recognition, and system modelling. Several illustrative examples are provided to demonstrate the effectiveness of the proposed normalization techniques Keywords: Complex Intuitionistic Fuzzy Sets, Fuzzy Matrices, Normalization Techniques, Membership and Non-membership Functions, Fuzzy Logic, Matrix Normalization, Intuitionistic Fuzzy Logic. 1. 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G[43], discussed “Study on bond selection under intuitionistic fuzzy conditions”in this year 2018.Zhang, F. W[44] , discussed “Generalized fuzzy additive operators on intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets and their application”in this year 2019.Zadeh, L.A [45] ,discussed “Fuzzy sets”in this year 1965.L.A.Zadeh[46]discussed Fuzzysets ,Information and Control in this year 1965. Objective: The objective of normalization of Intuitionistic Fuzzy Matrices (IFMs) is to transform the matrix elements, which represent uncertain or vague information, into a standardized form while preserving the relationships and inherent structure of the fuzzy data. The purpose of normalization is to ensure that the elements in the matrix are comparable and consistent, allowing for more accurate and reliable decision-making or analysis 2.Preliminary 2.1Definition of Fuzzy: Fuzzy refers to something that is unclear, imprecise, or ambiguous. It often describes situations where boundaries or definitions are not strictly defined, leading to a certain level of vagueness. In various fields, "fuzzy" can have different connotations, but the core idea revolves around uncertainty or a lack of exact precision. 1.Physical/Visual Context: When something is blurry, soft, or lacking sharp details, it is often described as fuzzy. For example, "The picture is fuzzy" means the image isn't clear. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2925 https://internationalpubls.com 2.Metaphorical/Conceptual Context: It can refer to something vague, ambiguous, or hard to understand. For example, "His explanation was fuzzy" means the explanation was unclear or imprecise. A fuzzy set can be represented in a matrix form, where the membership values of elements are arranged in rows and columns. Each element in the matrix corresponds to a specific item, and the values in the matrix represent the degree of membership of that item in the fuzzy set. 2.2 Examples 2.2.1. Temperature (Hot vs. Cold): Set Description: We can define a fuzzy set for temperature, where the degree of membership represents how hot or cold a temperature is, Example: Consider the fuzzy set "Hot temperatures" where: 0°C → Membership degree of 0 (completely cold) 20°C → Membership degree of 0.2 (slightly hot) 30°C → Membership degree of 0.5 (moderately hot) 40°C → Membership degree of 0.8 (very hot) 50°C → Membership degree of 1 (completely hot) 2.2.2. Height (Tall vs. Short): Set Description: A fuzzy set can be defined for the height of a person, with the degree of membership indicating how tall someone is, Example: Consider the fuzzy set "Tall people" where 150 cm → Membership degree of 0.2 (slightly tall) 160 cm → Membership degree of 0.4 (moderately tall) 170 cm → Membership degree of 0.7 (fairly tall) 180 cm → Membership degree of 1 (completely tall) 2.3 Application of Fuzzy 2.3.1. In mathematics and logic: In fuzzy logic, values are not limited to just true or false (1 or 0), but can take on any value between 0 and 1, reflecting degrees of truth. For example, instead of saying "a person is tall" as either true or false, fuzzy logic might say they are 0.7 tall (indicating they are somewhat tall but not extremely so). 2.3.2 In computing: A fuzzy search allows for approximate matches rather than exact ones. For example, searching for "appl" might also return "apple" or "applause" because the system is designed to tolerate small errors in spelling or typing. 2.3.3. Computer Science and Mathematics: In traditional binary logic, something is either true or false (1 or 0), but in fuzzy logic, truth values can be any number between 0 and 1, representing degrees of truth. Example: A thermostat that adjusts the temperature gradually based on a range, not just switching between "on" and "off." If the target temperature is 70°F, and the room temperature is 68°F, the thermostat may turn on at 50% power instead of 100%. 2.3.4. Technology: In searching algorithms, fuzzy search allows for matching terms that are similar but not identical. This can be used when users make spelling errors or variations. Example: Searching for "apple" could also return results for "apple," "appl," or "apples." 2.3.5. Fuzzy Concepts (Philosophy and Linguistics): A fuzzy concept refers to something that doesn’t have clear-cut boundaries or definitions. It’s often subjective. Example: The term "tall" is fuzzy Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2926 https://internationalpubls.com because what one person considers tall (e.g., 6 feet) might be different from another person’s perception (e.g., 5'10"). 2.3.6. Fuzzy Logic in Everyday Life: Using fuzzy reasoning in everyday decisions, where things aren’t just black and white. Example: Deciding whether to go outside when the weather is "a little rainy" is a fuzzy decision it's not a simple "yes" or "no" answer, but something in between based on personal comfort. 2.4 Fuzzy Matrix 2.4.1Definition of fuzzy matrix: A fuzzy matrix is a mathematical structure used to represent data where the relationships between elements are not precisely defined, but instead are characterized by degrees of membership. This concept stems from fuzzy set theory, which allows for partial membership in a set rather than a strict binary classification (where something is either in the set or not). In a fuzzy matrix, each element in the matrix represents a relationship or association between two entities, and the value in the matrix is a fuzzy number (typically ranging from 0 to 1) that indicates the degree of association. A value of 0 might mean "no association," while a value of 1 could represent "full association," and values in between indicate partial associations. Membership Degrees: Instead of having binary values (0 or 1), fuzzy matrices contain values between 0 and 1, representing the degree of membership. Symmetry: In some applications, the fuzzy matrix can be symmetric, meaning that if element A is related to element BBB to some degree, element BBB is related to element A to the same degree. Application: Fuzzy matrices are widely used in decision-making processes, pattern recognition, image processing, and systems where uncertainty or vagueness in data exists. 2.5 Examples of Fuzzy matrix: A fuzzy matrix is a matrix in which the elements are fuzzy values, typically represented by fuzzy numbers or membership functions. Fuzzy matrices are used in various fields, such as fuzzy logic, fuzzy systems, and decision-making processes, to handle imprecision and uncertainty. Let's consider a fuzzy matrix representing the relationship between three factors (A, B, C) and their membership values. Here, each element in the matrix could represent a degree of relationship or membership function value ranging from 0 to 1. Fuzzy Matrix:[ 0.8 0.6 0.4 0.5 0.7 0.9 0.3 0.6 0.8 ] In this matrix, the value 0.8 in the first row and first column indicates a high degree of membership or relationship between factor A and itself. The value 0.6 in the first row and second column represents a medium degree of membership between factor A and B. The value 0.4 in the first row and third column indicates a low degree of relationship between A and C. Each element in the matrix is a fuzzy value between 0 and 1, Where 0 means no membership or no relationship. 1 means full membership or a strong relationship. 2.5.1. Basic fuzzy Matrix (Binary Fuzzy Matrix): The matrix represents relationships between elements using values in [0,1] A=[ 0.8 0.4 0.6 0.3 1.0 0.7 0.5 0.2 0.9 ] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2927 https://internationalpubls.com where 𝑎𝑖𝑗 represents the degree of membership or confidence in a relationship between elements. 2.5.2 Fuzzy Adjacency Matrix: In a fuzzy graph , has three nodes with fuzzy edges weights ,its adjacency matrix may look like: M=[ 0 0.7 0.5 0.7 0 0.8 0.5 0.8 0 ] where 𝑀𝑖𝑗 represents the fuzzy weight of the edge between nodes i and j 2.5.3. Fuzzy Relation Matrix: A fuzzy matrix showing relationships between elements of two sets R=[ 0.9 0.6 0.3 0.5 0.8 0.7 0.4 0.2 0.1 ] where represents the fuzzy relationships strength between elements of two sets 2.6 Fuzzy matrix application: A fuzzy matrix is a generalization of a traditional matrix that deals with fuzzy relations and fuzzy logic. In a fuzzy matrix, elements represent degrees of membership to certain sets, usually expressed as values in the range [0, 1], where 0 means no membership, 1 means full membership, and intermediate values represent partial membership. This concept is widely applied in fields where uncertainty, vagueness, or imprecision is present. Below are some applications of fuzzy matrices: 2.6.1 Fuzzy Logic Systems: Fuzzy matrices are used in fuzzy logic systems for decision-making, where each element in the matrix represents the degree of truth for a given fuzzy relationship between variables. For example, in a fuzzy control system for a thermostat, the fuzzy matrix can represent rules and conditions like "temperature is somewhat high," "humidity is low," etc., and the matrix helps to decide the appropriate output (e.g., turn on the air conditioning). 2.6.2 Fuzzy Graph Theory: In fuzzy graphs, the adjacency matrix can have values between 0 and 1, indicating the degree of connection between two nodes. This approach is useful in networks where relationships between nodes are not binary, such as in social networks, recommendation systems, or transportation networks. It allows the representation of relationships that aren't just "connected" or "not connected" but instead have varying strengths. 2.6.3 Fuzzy Relational Databases: Fuzzy matrices can be used in fuzzy relational databases where database relationships (such as queries or connections between tables) are not precise but instead have degrees of certainty. For instance, in a database of products, a fuzzy matrix could represent the degree of similarity between different items, which can help with fuzzy searching or recommendation engines. 2.6.4 Fuzzy Control Systems: In fuzzy control, fuzzy matrices are often used to represent the relationship between input and output variables in fuzzy inference systems (FIS). For example, in controlling the speed of a fan based on temperature readings, a fuzzy matrix might define how the degree of "high temperature" should correlate to the fan's speed in a fuzzy way. 2.6.5 Pattern Recognition and Classification: Fuzzy matrices are employed in fuzzy pattern recognition to classify patterns when there is ambiguity or overlap between classes. A fuzzy matrix can represent the degree to which a certain input belongs to multiple classes simultaneously. This is useful in applications like handwriting recognition, image segmentation, and speech recognition, where categories are not perfectly distinct. 2.6. 6 Fuzzy Decision Making: In multi-criteria decision analysis (MCDA), fuzzy matrices can represent the degrees of importance or preference for various criteria under uncertain conditions. For example, in selecting a vendor for a project, decision-makers might use a fuzzy matrix to rank multiple Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2928 https://internationalpubls.com options based on subjective and vague criteria like "trustworthiness" or "reliability," which are not easily quantifiable. 2.6.7 Fuzzy Systems in AI: Fuzzy matrices can be used in AI-based decision support systems, where the relationships between inputs and outputs are not deterministic. They allow AI systems to handle uncertainty and vagueness, which is essential when processing real-world data, such as in natural language processing or autonomous driving. 2.6. 8 Fuzzy Optimization Problems: In optimization problems, fuzzy matrices can be used to represent uncertainty in constraints or objective functions. This is particularly useful in cases where the optimization criteria are vague or approximate, such as in supply chain management, where costs and demand are uncertain, and solutions need to be found that work well under a range of possible conditions 2.6.9 Image Processing: Fuzzy matrices are often applied in fuzzy image processing, where an image's pixel intensities are considered fuzzy rather than precise. This can help with tasks like image denoising, edge detection, or segmentation, where the boundaries between objects in an image are not perfectly clear. 2.6.10 Fuzzy Clustering: In fuzzy clustering algorithms (e.g., fuzzy c-means), fuzzy matrices represent the degree of membership of data points in different clusters. This allows for the classification of points that belong partially to multiple clusters, which is useful in situations like market segmentation, medical diagnosis, and customer behaviour analysis. 3.Intuitionistic Fuzzy Matrix 3.1Definition of intuitionistic fuzzy matrix: An intuitionistic fuzzy matrix A of order m × r is defined as: A=[(µ𝑖𝑗 , 𝜗𝑖𝑗)]𝑚×n where, µ𝑖𝑗is the membership degree of the element (i, j), representing the degree of belonging to a set. 𝜗𝑖𝑗 is the non - membership degree, representing the degree of not belonging to a set. For every elements (i ,j the sum of membership and non-membership degree satisfies: 0 ≤ µ𝑖𝑗 + 𝜗𝑖𝑗 ≤ 1𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖, The uncertainty degree (hesitancy)is given by( µ𝑖𝑗 + 𝜗𝑖𝑗) which quantifies the hesitation or lack of complete knowledge about the membership status. 3.2 Examples of intuitionistic fuzzy matrix: Intuitionistic fuzzy sets (IFS) extend traditional fuzzy sets by incorporating both membership and non- membership functions, as well as a degree of hesitation. In an intuitionistic fuzzy matrix, each element has three components: Membership degree (µ): The degree to which an element belongs to the set. Non-membership degree (ν): The degree to which an element does not belong to the set. Hesitation degree (π): The degree of uncertainty or hesitation, calculated as π=1−μ−ν 1.A 2×2 Intuitionistic fuzzy matrix: A=[ (0.7, 0.2) (0.5 0.3) (0.6 0.1) (0.4 0.4) ] Membership=0.7, non –membership=0.2 and hesitation degree=1-(0.7+0.2) =0.1 2. A 2×3 Intuitionistic fuzzy matrix: B=[ 〈0.8 0.1〉 〈0.6 0.2〉 〈0.5 0.3〉 〈0.7 0.2〉 〈0.4 0.4〉 〈0.3 0.5〉 ] Membership=0.3, non –membership=0.5 and hesitation degree=1-(0.3+0.5) =0.2 3.A 3×3 Intuitionistic fuzzy matrix: C=[ 〈0.6 0.3〉 〈0.7 0.2〉 〈0.5 0.4〉 〈0.8 0.1〉 〈0.4 0.5〉 〈0.6 0.3〉 〈0.3 0.6〉 〈0.5 0.3〉 〈0.7 0.2〉 ] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2929 https://internationalpubls.com Membership=0.3, non–membership=0.5 and hesitation degree=1-(0.3+0.5) =0.2 3.3 Application of intuitionistic fuzzy matrix: 3.3.1 Decision-Making Problems (Multi-Criteria Decision Analysis): Intuitionistic fuzzy matrices are often used in decision-making problems where decisions depend on multiple criteria, and there is uncertainty in the assessment of each criterion. Example: In a decision-making process for selecting the best supplier from a set of alternatives, each criterion (e.g., cost, quality, delivery time) may have uncertainty. Intuitionistic fuzzy matrices help incorporate both membership (degree of suitability) and non-membership (degree of unsuitability) values into the decision matrix. 3.3.2 Pattern Recognition: In pattern recognition, intuitionistic fuzzy matrices can be used to represent the degree of similarity and dissimilarity between patterns, making it easier to classify patterns with uncertainties. Example: In image processing or speech recognition, patterns may not always be perfectly clear. Intuitionistic fuzzy matrices provide a way to handle imprecise or incomplete data, allowing more flexible and accurate classification. 3.3.3 Data Mining and Clustering: Intuitionistic fuzzy clustering algorithms, where each data point belongs to multiple clusters with varying degrees of membership and non-membership, can benefit from intuitionistic fuzzy matrices. Example: In customer segmentation, instead of classifying customers strictly into one segment, intuitionistic fuzzy clustering allows a customer to belong to multiple segments with different degrees of membership and non-membership, representing their preferences or behaviours more flexibly. 3.3.4 Control Systems: In fuzzy control systems, where control rules are designed based on uncertain or imprecise data, intuitionistic fuzzy matrices help incorporate the uncertainty of the system's parameters. Example: In temperature control of a system, intuitionistic fuzzy matrices can help model the uncertainty in sensor readings and adjust the control system’s response accordingly. 3.3.5 Optimization Problem: In optimization problems with imprecise constraints or objective functions, intuitionistic fuzzy matrices can be used to represent the uncertainty in the data, allowing for more robust optimization results. Example: In supply chain optimization, the cost or demand may have uncertainty. Using intuitionistic fuzzy matrices to model these uncertainties helps in obtaining optimal solutions under varying levels of confidence. 3.3.6 Fault Diagnosis and Reliability Analysis: Intuitionistic fuzzy matrices are useful in fault diagnosis systems where the exact cause of a fault is not immediately clear, and multiple possible faults need to be considered with varying levels of confidence. Example: In industrial systems, intuitionistic fuzzy matrices can help in analysing machine faults by considering various possible failure modes, each with a degree of membership (how likely the fault is) and non-membership (how unlikely it is). 3.3.7 Expert Systems and Knowledge Representation: Intuitionistic fuzzy matrices can enhance expert systems by handling uncertainty in expert knowledge and decision rules. Example: In medical diagnosis systems, intuitionistic fuzzy matrices can model the uncertainty in diagnosing a disease based on symptoms and test results, where the certainty of a diagnosis may not be absolute Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2930 https://internationalpubls.com 3.3.8 Recommendation Systems: In recommendation systems, intuitionistic fuzzy matrices can be used to represent the preferences of users and the uncertainty in their preferences for different items or services. Example: In an e-commerce platform, users may have different levels of preference for products, but these preferences may also be uncertain or ambiguous. Intuitionistic fuzzy matrices can help provide better personalized recommendations by taking into account both positive preferences and the lack of preference. 3.3. 9 Social Network Analysis: In analysing social networks, intuitionistic fuzzy matrices can model relationships between individuals, where relationships may have different levels of strength and uncertainty. Example: In social media platforms, the relationship strength between two users may be uncertain (not just strong or weak) and could vary over time. Intuitionistic fuzzy matrices help represent this uncertainty in social network analysis. 3.3.10 Medical Diagnostics and Health Monitoring: Intuitionistic fuzzy matrices can be applied in the representation of medical data, where diagnostic results may have both degrees of certainty and uncertainty, especially when dealing with incomplete or ambiguous information. Example: In health monitoring, an intuitionistic fuzzy matrix can represent various symptoms and their relationships to potential diseases, with both membership and non-membership values indicating the degree of certainty about the presence of a condition. 4 Overview on Normalization and Cartesian product of Intuitionistic Fuzzy Matrices: 4.1Definition:The normalization of an intuitionistic fuzzy matrix (IFM) refers to the process of transforming an intuitionistic fuzzy matrix into a standardized form while preserving its essential characteristics .An intuitionistic fuzzy matrix is a matrix in which each element is represented as an intuitionistic fuzzy number (IFM) ,typically denoted as a pair (µ𝑖𝑗 , 𝜗𝑖𝑗), where, µ𝑖𝑗is the membership degree of the element (i, j),representing the degree of belonging to a set. 𝜗𝑖𝑗 is the non - membership degree, representing the degree of not belonging to a set. The hesitation degree is given by 𝜋𝑖𝑗 = 1 − (µ𝑖𝑗 + 𝜗𝑖𝑗) Normalization in this context ensures that the fuzzy matrix values are on the same scale, which is important for: Reducing the impact of extreme values. Making the matrix more uniform for better comparison and analysis. Ensuring consistency when aggregating or combining intuitionistic fuzzy information across different decision criteria or alternatives. 4.2Normalization Methods for Intuitionistic Fuzzy Matrices: Normalization of IFMs helps to standardize the values of the matrix to a common range, typically between 0 and 1, to ensure that they are comparable across different criteria or decision-making factors. Here are some common normalization methods: 4.2.1 Linear Normalization: In this method, the values of the membership, non-membership, and uncertainty degrees are normalized based on their maximum and minimum values across the entire matrix. The formula for normalization of an element aij = (µ(x), ν(x), π(x)) is given as 𝑎𝑖𝑗=( µ(x)−µ𝑚𝑖𝑛 µ𝑚𝑎𝑥−µ𝑚𝑖𝑛 , ν(x)−𝜗𝑚𝑖𝑛 𝜗𝑚𝑎𝑥−𝜗𝑚𝑖𝑛 , π(x)−𝜋𝑚𝑖𝑛 𝜋𝑚𝑎𝑥−𝜋𝑚𝑖𝑛 )Where µ𝑚𝑖𝑛, µ𝑚𝑎𝑥 are the minimum and maximum membership values across all elements. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2931 https://internationalpubls.com 𝜗𝑚𝑖𝑛, 𝜗𝑚𝑎𝑥 are the minimum and maximum non-membership values across all elements. 𝜋𝑚𝑖𝑛, 𝜋𝑚𝑎𝑥 are the minimum and maximum uncertainty values across all elements. 4.2.2Column-Wise Normalization: In this approach, normalization is done separately for each column of the Intuitionistic Fuzzy Matrix. Each element in a column is normalized by dividing by the sum of the membership, non-membership, and uncertainty values in that column. For an element aij = (µ(x), ν(x), π(x)) the normalization formula is 𝑎𝑖𝑗 = ( µ(x) ∑ µ𝑘𝑗 𝑚 𝑘=1 , ϑ(x) ∑ 𝜗𝑘𝑗 𝑚 𝑘=1 , μ(x) ∑ 𝜋𝑘𝑗 𝑚 𝑘=1 ) Where m is the number of rows in the matrix. 4.3.3 Max-Min Normalization: This method normalizes the elements of the Intuitionistic Fuzzy Matrix using the maximum and minimum values across each row. For an element aij=(µ(x), ν(x), π(x), the normalization is: 𝑎𝑖𝑗 = ( µ(x)−min (µ) max(µ)−min (µ) , ϑ(x)−min (𝜗) max(𝜗)−min (𝜗) , π(x)−min (π) max(π)−min (π) ) this method uses the maximum and minimum values for each degree across the matrix to scale the values. 4.3.4. Vector –Based Normalization: This method normalizes each element in the matrix by considering the vector magnitude of its components. Given that each element is a triplet, the normalization is based on the euclined norm or magnitude of the triplet vector (µ(x), ν(x), π(x)). The formula for normalization is: 𝑎𝑖𝑗 = ( µ(x) √µ(x)2+ϑ(x)2+π(x)2 , ϑ(x) √µ(x)2+ϑ(x)2+π(x)2 , π(x) √µ(x)2+ϑ(x)2+π(x)2 ) this method ensures that the sum of the squares of the components of each element in the matrix equals1 effectively standardizing the values. 4.4 Examples: Steps for Normalization of Intuitionistic Fuzzy Matrices: 1.Identify the maximum and minimum values for each component (membership, non-membership, and indeterminacy) across the entire matrix. 2.Normalize each component by applying a normalization function. The most common normalization method is min-max normalization, where the values are mapped to a desired range, typically [0, 1]. For each component 𝑥𝑖𝑗 of the matrix (whether it's the membership, non-membership, or indeterminacy), the normalized value 𝑥′𝑖𝑗 can be computed as = 𝑥𝑖𝑗−𝑚𝑖𝑛(𝑥) max(𝑥)−min(𝑥) , where min(x) and max(x) are the minimum and maximum values of the respective component across the entire matrix. 3.Adjust the indeterminacy (if required). After normalizing the membership and non-membership components, you may need to adjust the indeterminacy to ensure that the sum of membership, non- membership, and indeterminacy still equals 1. you normalize the membership and non-membership components first; you can calculate the indeterminacy for each element as: 𝜋′𝑖𝑗 = 1 − µ′ 𝑖𝑗 − 𝜗′ 𝑖𝑗 This ensures that the sum of the normalized membership, non-membership, and indeterminacy remains 1 for each element. Example of Normalization: Consider an intuitionistic fuzzy matrix with the following entries A = ( (0.6,0.2 ,0.2) (0.8,0.1 ,0.1) (0.4,0.3 ,0.3) (0.7,0.2 ,0.1) ) Step 1: Identify the max and min values for each component. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2932 https://internationalpubls.com For membership (µ) : max(µ)=0.8 ,min(µ)=0.4 For non-membership (𝜗) : max(𝜗)=0.3 ,min(𝜗)=0.1 For indeterminacy (𝜋) : max(𝜋)=0.3 ,min(𝜋)=0.1 Step 2: Normalize the membership and non-membership components using min-max normalization. For each 𝑎𝑖𝑗 = (µ𝑖𝑗, 𝜗𝑖𝑗 , 𝜋𝑖𝑗) ,we apply the normalization for µ11 = 0.6: µ′11 = 0.6−0.4 0.8−0.4 = 0.2 0.4 =0.5 For 𝜗11 = 0.2 𝜗′11 = 0.2−0.1 0.3−0.1 = 0.1 0.2 =0.5 For 𝜋11 = 0.2 𝜋′11 = 1 − 0.5 = 0.5 = 0 Now, apply the same normalization for the other elements of the matrix. Step 3: Adjust indeterminacy. After normalizing the membership and non-membership values, the indeterminacy values are recalculated as: 𝜋′ 𝑖𝑗 = 1 − µ′ 𝑖𝑗 − 𝜗′ 𝑖𝑗 The final normalized matrix will have the form 𝐴′ = ( (0.5,0.5 ,0) (1 ,0 ,0) (0, 0) (0.75 ,0.25) ) 2.Consider a 2.2 intuitionistic fuzzy matrix 𝐴 = [ (0.5 ,0.3) (0.7 ,0.2) (0.4 ,0.4) (0.6 ,0.1) ] using the Normalization: µ′11 = 0.5 0.5+0.3 = 0.625 𝜗′11 = 0.3 0.5+0.3 = 0.37 µ′12 = 0.7 0.7+0.2 = 0.777 𝜗′12 = 0.2 0.7+0.2 = 0.22 µ′21 = 0.4 0.4+0.4 = 0.5 𝜗′21 = 0.4 0.4+0.4 = 0 µ′22 = 0.6 0.6+0.1 = 0.857 𝜗′22 = 0.1 0.6+0.1 = 0.143 The normalised matrix becomes 𝐴′ = ( (0.625 , 0.375) (0.777 , 0.222) (0.5 , 0.5) (0.857 , 0.143) ) 4.5. Applications of Normalization of Intuitionistic Fuzzy Matrices 1. Multi-Criteria Decision Making (MCDM) 2. Pattern Recognition 3.Machine learning &AI 1. Multi-Criteria Decision Analysis (MCDA): When dealing with multi-criteria decision-making problems, intuitionistic fuzzy matrices allow capturing the hesitancy or uncertainty associated with expert judgments. Normalizing the IFM helps standardize the fuzzy decision criteria to a common scale, making it easier to compare alternatives across different criteria and achieve more reliable ranking. Fuzzy Topsis Method: In methods like TOPSIS (Technique for Order Preference by Similarity to Ideal Solution), normalization of intuitionistic fuzzy matrices allows a consistent comparison of alternatives by transforming the fuzzy values into a comparable scale, reducing inconsistencies and ambiguities in decision-making. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2933 https://internationalpubls.com 2.Image Processing: Fuzzy Image Segmentation: In image processing, intuitionistic fuzzy matrices can be used to represent uncertain or vague image boundaries. Normalization helps in adjusting pixel intensities and related fuzzy values (membership and non-membership) to ensure proper image segmentation, enhancing the performance of algorithms that rely on fuzzy logic. 3.Pattern Recognition and Classification: Feature Extraction and Classification: In pattern recognition, normalized intuitionistic fuzzy matrices can be used to extract features from data sets where uncertainty is present. The normalization ensures that the features, expressed in fuzzy terms, are on the same scale, allowing more effective classification and pattern recognition by algorithms such as fuzzy clustering or fuzzy decision trees. 4.Control Systems: Fuzzy Logic Controllers: In control systems, intuitionistic fuzzy matrices can represent uncertain inputs and outputs. Normalizing the IFM ensures that the fuzzy rules and control outputs are consistent, leading to better control decisions in real-time applications such as robotic navigation, industrial automation, and HVAC systems. 5.Risk Assessment and Reliability Analysis: Risk Modelling: In assessing the risk in uncertain environments, intuitionistic fuzzy matrices allow the modelling of both membership (degree of risk) and non-membership (degree of safety). Normalization of such matrices ensures that risk factors are represented proportionately, aiding in better decision- making for risk mitigation strategies. 6.Data Fusion: Sensor Data Fusion: In applications like sensor networks or multi-source information integration, normalization of intuitionistic fuzzy matrices can combine data from multiple sources while adjusting for different degrees of uncertainty across sensors. This leads to a more accurate and reliable fused data set. 7.Healthcare and Medical Diagnosis: Medical Decision Support Systems: In healthcare, intuitionistic fuzzy matrices are used to represent patient data, symptoms, and diagnostic criteria under uncertainty. Normalization helps to align various medical parameters and judgment values, ensuring that the final diagnosis or treatment plan is consistent with all available information. 4.6 Importance: Consistent Comparison of Data: Normalization ensures that the elements in the intuitionistic fuzzy matrix are scaled to a standard range (typically [0, 1]), making it easier to compare and combine information from different sources. Without normalization, the data might be inconsistent and difficult to interpret, especially when the membership and non-membership values vary across different scales or ranges. Improving Accuracy in Decision Making: In decision-making problems (like multi-criteria decision analysis or decision support systems), normalization helps to make sure that all the criteria or attributes are comparable on the same scale. This improves the accuracy and reliability of the decision-making process. Eliminating Bias Due to Scale Differences: Different elements of an intuitionistic fuzzy matrix can have different scales, leading to biases in the analysis. Normalization removes such biases by ensuring that all values contribute equally to the outcome. This is important when the matrix represents the relationship between various factors or alternatives in a decision-making problem. Handling Uncertainty Effectively: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2934 https://internationalpubls.com Intuitionistic fuzzy sets deal with both membership and non-membership, leaving room for uncertainty (indeterminacy). Normalization allows better handling and representation of this uncertainty, ensuring that the resulting matrix values still remain meaningful even when there is incomplete or imprecise data. Facilitating Aggregation of Data: When working with multiple intuitionistic fuzzy matrices, normalization allows for the proper aggregation of data from various sources. This is particularly important in fields such as multi-criteria decision analysis, where data from different criteria need to be aggregated into a final decision. Simplifying Further Mathematical Operations: Normalized values in intuitionistic fuzzy matrices make it easier to apply further operations like addition, multiplication, or averaging, which are common in matrix-based decision-making methods (e.g., AHP, TOPSIS, etc.). Without normalization, the results of these operations may be skewed due to differences in scale between different elements of the matrix. Ensuring Feasibility and Validity of Results: For certain algorithms or applications, it's necessary that the values in the intuitionistic fuzzy matrix sum to a specific value (such as 1). Normalization ensures that the sum of membership and non- membership functions, along with the degree of indeterminacy, remains valid for further processing. Enhancing Robustness of Algorithms Many machines learning or optimization algorithms require data in a specific form for stable convergence. Normalized intuitionistic fuzzy matrices help improve the robustness and convergence speed of these algorithms, preventing issues caused by extreme or unbalanced values. We define over the set of IFS, two modal operators which transform every IFS into fuzzy set. These operators are similar to the operator’s ‘necessity’ and ‘possibility’ defined in some modal logics. This idea is drawn from the modal operators on IFS proposed by [3]. 5. Overview on Normalization and Cartesian product of Intuitionistic Fuzzy Matrices 5.1 Definition: [Modal operators’ necessity]: Let X be nonempty. If A is an IFS drawn from X, then A = {x, μA(x): xX} = {x, μA(x), 1 − 𝜇𝐴(x): xX} Definition: [Modal operators’ possibility]: Let X be nonempty. If A is an IFS drawn from X, then A = {x, 1 − 𝜗𝐴(x): xX} = {x, 1 − 𝜗𝐴(x), 𝜗𝐴(x): xX} Definition: If 𝐴𝐸 and 𝐵𝐹 are two IFSs over different universes E and F gives as 𝐴𝐸 = {x, μA(x), 𝜗𝐴(x): xE} and 𝐵𝐹= {y, 𝜇𝐵(y), 𝜗𝐵(y): yF} Definition: If 𝐴𝐸 ̅̅̅̅ and𝐵𝐹 ̅̅̅̅ are two IFSs over different universes E and F gives as 𝐴𝐸 ̅̅̅̅ = {x, 𝜗𝐴(x), μA(x): xE} and 𝐵𝐹 ̅̅̅̅ = {y, 𝜗𝐵(y), 𝜇𝐵(y): yF} Definition: If 𝐴𝐸 and 𝐵𝐹 are two IFSs over different universes E and F gives as 𝐴𝐸 = {x, μA(x), 𝜗𝐴(x): xE} and 𝐵𝐹= {y, 𝜇𝐵(y), 𝜗𝐵(y): yF} Definition: If 𝐴𝐸 ̅̅̅̅ and𝐵𝐹 ̅̅̅̅ are two IFSs over different universes E and F gives as 𝐴𝐸 ̅̅̅̅ = {x, 𝜗𝐴(x), μA(x): xE} and 𝐵𝐹 ̅̅̅̅ = {y, 𝜗𝐵(y), 𝜇𝐵(y): yF} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2935 https://internationalpubls.com 5.2 Definition: Let E be non-empty universe set. The normalization of intuitionistic fuzzy sets A denoted by Norm (A) is defined as Norm (A) = {x, 𝜇𝑁𝑜𝑟𝑚(𝐴)(x), 𝜗𝑁𝑜𝑟𝑚(𝐴)(x): xE}, where μNorm(A)(x) = μA(x) sup (μA(x)) and ϑNorm(A)(x) = ϑA(x)− inf (ϑA(x)) 1−inf (ϑA(x)) . Definition5.2.1: If AE and BF are two IFSs over different universes sets E and F gives as AE = {x, μA(x), ϑA(x): xE} and BF= {y, μB(y), ϑB(y): yF}. 1.AEBF = {(x, y), min (μA(x), μB(y)), max (ϑA(x), ϑB(x)): xE, yF} 2. AEBF = {(x, y), max (μA(x), μB(y)), min (ϑA(x), ϑB(x)): xE, yF} 3. AEBF = {(x, y), (μA(x) + μB(y)) - μA(x). μB(y), ϑA(x).ϑB(y): xE, yF} 4. AEBF = {(x, y), μA(x). μB(y), ϑA(x) +ϑB(y) - ϑA(x).ϑB(y): xE, yF} 5. AE@BF = {(x, y), μA(x)+ μB(y) 2 , ϑA(x)+ϑB(y) 2 : xE, yF} 6. A#BF = {(x, y), 2μA(x).μB(y) μA(x)+ϑB(y) , 2ϑA(x).ϑB(y) ϑA(x)+ϑB(y) : xE, yF} 7. AE$BF = {(x, y),√μA(x). μB(y) , √ϑA(x). ϑB(y)xE, yF} 8. AE*BF = {(x, y), μA(x)+ μB(y) 2(μA(x). μB(y)+1) , ϑA(x)+ϑB(y) 2(ϑA(x).ϑB(x)+1) : xE, yF } 9. AE∆BF = {(x, y), μA(x)+ μB(y) μA(x)+ μB(y)+ϑA(x)+ϑB(y) , ϑA(x)+ϑB(y) μA(x)+ μB(y)+ϑA(x)+ϑB(y) ∶ xE, yF} Theorem 5.2.1: If E and F be two universal sets. For every normalization of complex intuitionistic fuzzy matrices 𝐴𝐸 𝑎𝑛𝑑 𝐵𝐹 are in E and F then Norm 𝐴𝐸 ∩ 𝑁𝑜𝑟𝑚 𝐵𝐹 is also normalization of complex intuitionistic fuzzy matrices. Proof: If Norm 𝐴𝐸 and 𝑁𝑜𝑟𝑚 𝐵𝐹 are two normalization intuitionistic fuzzy matrices over different unive𝑟𝑠𝑒𝑠 𝑠𝑒𝑡𝑠 𝐸 𝑎𝑛𝑑 𝐹. If we consider two 2 × 2 normalization intuitionistic fuzzy matrices, Norm 𝐴𝐸𝑎𝑛𝑑 𝑁𝑜𝑟𝑚 𝐵𝐹. Let Norm 𝐴𝐸 = ( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11) 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12) 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21) 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22) 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22)) ) and 𝑁𝑜𝑟𝑚 𝐵𝐹 = ( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11) 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12) 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑖𝑦21) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑖𝑦22) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)) ) are two normalization intuitionistic fuzzy matrices. Now applying both side on intersection, we have Norm 𝐴𝐸  𝑁𝑜𝑟𝑚 𝐵𝐹 = ( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11) 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12) 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21) 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22) 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22)) ) ∩ ( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11) 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12) 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑖𝑦21) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑖𝑦22) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)) ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2936 https://internationalpubls.com Calculate, 𝑋11, 𝑋12, 𝑋21and 𝑋22, we have X11 =(𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11) 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11)) (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11) 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)) X12 =(𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12) 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12))  (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12) 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12)) X21 =(𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21) 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21))  (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑖𝑦21) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21)) X22 =(𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22) 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22))  (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑖𝑦22) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)) Applying Formula AEBF = {(x, y), min (μA(x), μB(y)), max (ϑA(x), ϑB(x)): xE, yF} X11 ={min ( 𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11), 𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11), max (NormµE11 (𝛾11 + 𝑖𝛿11), NormγE11 (𝜇11 + 𝑖𝜃11)} where ∣μ11∣= (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11))2 + (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛽11))2 < 1, ∣ν11∣= (NormµE11 (𝛾11))2 + (NormγE11 (𝛿11)) 2 < 1, ∣μ11∣+∣ν11∣ ≤ 1. X12 ={min ( NormϑE12 (𝛼12 + 𝑖𝛽12), NormδF12 (𝑥12 + 𝑖𝑦12), min (NormµE12 (𝛾12 + 𝑖𝛿12), NormγE12 (𝜇12 + 𝑖𝜃12)} where ∣μ12∣= (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12))2 + (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛽12))2 < 1, ∣ν11∣= (NormµE12 (𝛾12))2 + (NormγE12 (𝛿12)) 2 < 1, ∣μ12∣+∣ν12∣ ≤ 1. X21 ={max( NormϑE21 (𝛼21 + 𝑖𝛽21), NormδF21 (𝑥21 + 𝑖𝑦21), min (NormµE21 (𝛾21 + 𝑖𝛿21), NormγE21 (𝜇21 + 𝑖𝜃21)} where ∣μ21∣= (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21))2 + (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛽21))2 < 1, ∣ν21∣= (NormµE21 (𝛾21))2 + (NormγE21 (𝛿21)) 2 < 1, ∣μ21∣+∣ν21∣ ≤ 1. X22 ={max( NormϑE22 (𝛼22, 𝑥22) + 𝑖𝛽22), iNormδF22 (+𝑖𝑦22), min (NormµE22 (𝛾22 + 𝑖𝛿22), NormγE22 (𝜇22 + 𝑖𝜃22)} where ∣μ22∣= (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22))2 + (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛽22))2 < 1, ∣ν11∣= (NormµE22 (𝛾22))2 + (NormγE22 (𝛿22)) 2 < 1, ∣μ22∣+∣ν22∣ ≤ 1. X11 =(𝑁𝑜𝑟𝑚𝜗𝐸11 (min (𝛼11, 𝑥11) + 𝑖 min (𝛽11, 𝑦11))), NormµE11 (max (𝛾11, 𝜇11) + 𝑖max (𝛿11, 𝜃11)) Where ∣μ11∣= (𝑁𝑜𝑟𝑚𝜗𝐸11 (min(𝛼11, 𝑥11))2 + (𝑁𝑜𝑟𝑚𝜗𝐸11 min(𝛽11, 𝑦11))2 +(𝑁𝑜𝑟𝑚𝛿𝐹11 𝑚𝑖𝑛 (𝛾11, 𝜇11))2 +(𝑁𝑜𝑟𝑚𝛿𝐹11 𝑚𝑖𝑛 (𝛾11, 𝜇11))2 < 1, ∣ν11∣= (NormµE11 𝑚𝑎𝑥(𝛾11, 𝜇11))2 +(𝑁𝑜𝑟𝑚𝛾𝐸11 𝑚𝑎𝑥(𝛿11, 𝜃11))2 < 1, ∣μ11∣+∣ν11∣ ≤ 1. X12 =(𝑁𝑜𝑟𝑚𝜗𝐸12 (min (𝛼12, 𝑥12) + 𝑖 min (𝛽12, 𝑦12)), NormµE12 (max (𝛾12, 𝜇12) + 𝑖max (𝛿12, 𝜃12)) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2937 https://internationalpubls.com where ∣μ12∣=(𝑁𝑜𝑟𝑚𝜗𝐸12 𝑚𝑖𝑛 (𝛼12, 𝑥12))2 +(𝑁𝑜𝑟𝑚𝛿𝐹12 𝑚𝑖𝑛 (𝛽12, 𝑦12))2 < 1, ∣ν12∣= (NormµE12 𝑚𝑎𝑥(𝛾12, 𝜇12))2 +(𝑁𝑜𝑟𝑚𝛾𝐸12 𝑚𝑎𝑥( (𝛿12, 𝜃12))2 < 1, ∣μ12∣+∣ν12∣ ≤ 1. X21 ={( NormϑE21 𝑚𝑖𝑛 (𝛼21, 𝑥21) + iNormδF21 (𝛽21, 𝑦21), max (NormµE21 (𝛾21, 𝜇21 ) + iNormγE21 (𝛿21, 𝜃21)} where ∣μ21∣=(𝑁𝑜𝑟𝑚𝜗𝐸21 𝑚𝑖𝑛 (𝛼21, 𝑥21))2 +(𝑁𝑜𝑟𝑚𝛿𝐹21 𝑚𝑖𝑛 (𝛽21, 𝑦21))2 < 1, ∣ν21∣= (NormµE21 𝑚𝑎𝑥(𝛾21, 𝜇21))2 +(𝑁𝑜𝑟𝑚𝛾𝐸21 𝑚𝑎𝑥( (𝛿21, 𝜃21))2 < 1, ∣μ21∣+∣ν21∣ ≤ 1. X22 ={max( NormϑE22 (𝛼22, 𝑥22)), iNormδF22 𝑚𝑖𝑛 (𝛽22, 𝑦22), max (NormµE22 (𝛾22, 𝜇21), NormγE22 (𝛿22, 𝜃22)} where ∣μ22∣=(𝑁𝑜𝑟𝑚𝜗𝐸22 𝑚𝑖𝑛 (𝛼22, 𝑥22))2 +(𝑁𝑜𝑟𝑚𝛿𝐹22 𝑚𝑖𝑛 (𝛽22, 𝑦22))2 < 1, ∣ν21∣= (NormµE22 𝑚𝑎𝑥(𝛾22, 𝜇22))2 +(𝑁𝑜𝑟𝑚𝛾𝐸22 𝑚𝑎𝑥( (𝛿22, 𝜃22))2 < 1, ∣μ22∣+∣ν22∣ ≤ 1. We have Norm 𝐴𝐸  𝑁𝑜𝑟𝑚 𝐵𝐹 = ( 𝑋11 𝑋12 𝑋21 𝑋22 ) is also normalization of complex intuitionistic fuzzy matrix. Example5.2.2: If 𝐴𝐸=( (0.6 + 0.2i, 0.3 + 0.1i)(0.4 + 0.1i, 0.5 + 0.2i) (0.5 + 0.3i, 0.2 + 0.1i)(0.3 + 0.2i, 0.6 + 0.1i) ) and BF = ( (0.4 + 0.3i, 0.5 + 0.1i)(0.3 + 0.4i, 0.4 + 0.2i) (0.6 + 0.2i, 0.2 + 0.2i)(0.5 + 0.1i, 0.4 + 0.3i) ) are normalization of complex intuitionistic fuzzy matrix two intuitionistic fuzzy matrices over different universe E and F then Norm 𝐴𝐸  𝑁𝑜𝑟𝑚 𝐵𝐹 is also normalization of complex intuitionistic fuzzy matrix. Proof: Given 𝑨𝑬=( (0.6 + 0.2i, 0.3 + 0.1i)(0.4 + 0.1i, 0.5 + 0.2i) (0.5 + 0.3i, 0.2 + 0.1i)(0.3 + 0.2i, 0.6 + 0.1i) ) and 𝑩𝑭 = ( (0.4 + 0.3i, 0.5 + 0.1i)(0.3 + 0.4i, 0.4 + 0.2i) (0.6 + 0.2i, 0.2 + 0.2i)(0.5 + 0.1i, 0.4 + 0.3i) ) Norm 𝐴𝐸 = ( (1 + 0.67𝑖 0.125 + 0𝑖) (0.67 + 0.33𝑖 0.30 + 0.10𝑖) (0.83 + 1𝑖 0 + 0𝑖) (0.5 + 0.67𝑖 0.5 + 0𝑖) ) 𝑁𝑜𝑟𝑚 𝐵𝐹= ( (0.67 + 0.75𝑖 0.375 + 0𝑖) (0.5 + 1𝑖 0.25 + 0.11𝑖) (1 + 0.5𝑖 0 + 0.11𝑖) (0.83 + 0.25𝑖 0.25 + 0.22𝑖) ) Norm 𝐴𝐸𝑁𝑜𝑟𝑚 𝐵𝐹 = ( (1 + 0.67𝑖 0.125 + 0𝑖) (0.67 + 0.33𝑖 0.30 + 0.10𝑖) (0.83 + 1𝑖 0 + 0𝑖) (0.5 + 0.67𝑖 0.5 + 0𝑖) )  ( (0.67 + 0.75𝑖 0.375 + 0𝑖) (0.5 + 1𝑖 0.25 + 0.11𝑖) (1 + 0.5𝑖 0 + 0.11𝑖) (0.83 + 0.25𝑖 0.25 + 0.22𝑖) ) 𝑋11 = (1 + 0.67𝑖 0.125 + 0𝑖) ∩ (0.67 + 0.75𝑖 0.375 + 0𝑖) 𝑋12 = (0.67 + 0.33𝑖 0.30 + 0.10𝑖) ∩ (0.5 + 1𝑖 0.25 + 0.11𝑖) 𝑋21 = (0.83 + 1𝑖 0 + 0𝑖) ∩ (1 + 0.5𝑖 0 + 0.11𝑖) 𝑋22 = (0.5 + 0.67𝑖 0.5 + 0𝑖) ∩ (0.83 + 0.25𝑖 0.25 + 0.22𝑖) Applying Formula AEBF = {(x, y), min (μA(x), μB(y)), max (ϑA(x), ϑB(x)): xE, yF} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2938 https://internationalpubls.com 𝑋11 ={𝑚𝑖𝑛(1 + 0.67𝑖, 0.67 + 0.75𝑖) , 𝑚𝑎𝑥(0.125 + 0𝑖, 0.375 + 0𝑖)} 𝑋12 ={𝑚𝑖𝑛(0.67 + 0.33𝑖, 0.5 + 1𝑖) , 𝑚𝑎𝑥(0.30 + 0.10𝑖, 0.25 + 0.11𝑖)} 𝑋21 ={𝑚𝑖𝑛(0.83 + 1𝑖, 1 + 0.5𝑖) , 𝑚𝑎𝑥(0 + 0𝑖, 0 + 0.11𝑖)} 𝑋22 ={𝑚𝑖𝑛(0.5 + 0.67𝑖, 0.83 + 0.25𝑖) , 𝑚𝑎𝑥(0.5 + 0𝑖, 0.25 + 0.22𝑖)} 𝑋11 ={0.67+0.67i, 0.375+0i} 𝑋12 ={0.5+0.33i, 0.3+0.11i} 𝑋21 ={0.83+0.5i, 0+0.11i} 𝑋22 ={0.5+0.25i, 0.5+0.22i} We have Norm 𝐴𝐸  𝑁𝑜𝑟𝑚 𝐵𝐹 = ( (0.67 + 0.67i, 0.375 + 0i ) (0.5 + 0.33i, 0.3 + 0.11i) (0.83 + 0.5i, 0 + 0.11i) (0.5 + 0.25i, 0.5 + 0.22i) ) is also normalization of complex intuitionistic fuzzy matrix. Theorem 5.2.3: If E and F be two universal sets. For every normalization of complex intuitionistic fuzzy matrices 𝐴𝐸 𝑎𝑛𝑑 𝐵𝐹 are in E and F then Norm 𝐴𝐸 ∪ 𝑁𝑜𝑟𝑚 𝐵𝐹 is also normalization of complex intuitionistic fuzzy matrices. Proof: If Norm 𝐴𝐸 and 𝑁𝑜𝑟𝑚 𝐵𝐹 are two normalization intuitionistic fuzzy matrices over different unive𝑟𝑠𝑒𝑠 𝑠𝑒𝑡𝑠 𝐸 𝑎𝑛𝑑 𝐹. If we consider two 2 × 2 normalization intuitionistic fuzzy matrices, Norm 𝐴𝐸𝑎𝑛𝑑 𝑁𝑜𝑟𝑚 𝐵𝐹. Let Norm 𝐴𝐸 = ( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11) 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12) 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21) 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22) 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22)) ) and 𝑁𝑜𝑟𝑚 𝐵𝐹 = ( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11) 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12) 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑖𝑦21) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑖𝑦22) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)) ) are two normalization intuitionistic fuzzy matrices. Norm𝐴𝐸 ∪ 𝑁𝑜𝑟𝑚𝐵𝐹 = ( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11) 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12) 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21) 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22) 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22)) ) ∪ ( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11) 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12) 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑖𝑦21) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑖𝑦22) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)) ) Calculate, 𝑋11, 𝑋12, 𝑋21and 𝑋22, we have. Now applying both side on intersection, we have Norm 𝐴𝐸 ∪ 𝑁𝑜𝑟𝑚 𝐵𝐹 = ( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11) 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12) 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21) 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22) 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22)) ) ∪ ( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11) 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12) 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑖𝑦21) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑖𝑦22) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)) ) Calculate, 𝑋11, 𝑋12, 𝑋21and 𝑋22, we have Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2939 https://internationalpubls.com X11 =(𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11) 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11)) ∪ (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11) 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)) X12 =(𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12) 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12)) ∪ (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12) 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12)) X21 =(𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21) 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21)) ∪ (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑖𝑦21) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21)) X22 =(𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22) 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22)) ∪ (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑖𝑦22) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)) Applying formula AEBF = {(x, y), max (μA(x), μB(y)), min (ϑA(x), ϑB(x)): xE, yF} X11 ={max ( 𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11), 𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11), min (NormµE11 (𝛾11 + 𝑖𝛿11), NormγE11 (𝜇11 + 𝑖𝜃11)} where ∣μ11∣= (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11))2 + (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛽11))2 < 1, ∣ν11∣= (NormµE11 (𝛾11))2 + (NormγE11 (𝛿11)) 2 < 1, ∣μ11∣+∣ν11∣ ≤ 1. X12 ={max ( NormϑE12 (𝛼12 + 𝑖𝛽12), NormδF12 (𝑥12 + 𝑖𝑦12), min (NormµE12 (𝛾12 + 𝑖𝛿12), NormγE12 (𝜇12 + 𝑖𝜃12)} where ∣μ12∣= (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12))2 + (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛽12))2 < 1, ∣ν11∣= (NormµE12 (𝛾12))2 + (NormγE12 (𝛿12)) 2 < 1, ∣μ12∣+∣ν12∣ ≤ 1. X21 ={max( NormϑE21 (𝛼21 + 𝑖𝛽21), NormδF21 (𝑥21 + 𝑖𝑦21), min (NormµE21 (𝛾21 + 𝑖𝛿21), NormγE21 (𝜇21 + 𝑖𝜃21)} where ∣μ21∣= (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21))2 + (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛽21))2 < 1, ∣ν21∣= (NormµE21 (𝛾21))2 + (NormγE21 (𝛿21)) 2 < 1, ∣μ21∣+∣ν21∣ ≤ 1. X22 ={max ( NormϑE22 (𝛼22, 𝑥22) + 𝑖𝛽22), iNormδF22 (+𝑖𝑦22), min (NormµE22 (𝛾22 + 𝑖𝛿22), NormγE22 (𝜇22 + 𝑖𝜃22)} where ∣μ22∣= (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22))2 + (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛽22))2 < 1, ∣ν11∣= (NormµE22 (𝛾22))2 + (NormγE22 (𝛿22)) 2 < 1, ∣μ22∣+∣ν22∣ ≤ 1. X11 =(𝑁𝑜𝑟𝑚𝜗𝐸11 (max (𝛼11, 𝑥11) + 𝑖 max (𝛽11, 𝑦11))), NormµE11 (min (𝛾11, 𝜇11) + 𝑖min (𝛿11, 𝜃11)) Where ∣μ11∣= (𝑁𝑜𝑟𝑚𝜗𝐸11 (max(𝛼11, 𝑥11))2 + (𝑁𝑜𝑟𝑚𝜗𝐸11 max(𝛽11, 𝑦11))2 +(𝑁𝑜𝑟𝑚𝛿𝐹11 𝑚𝑖𝑛 (𝛾11, 𝜇11))2 +(𝑁𝑜𝑟𝑚𝛿𝐹11 𝑚𝑖𝑛 (𝛾11, 𝜇11))2 < 1, ∣ν11∣= (NormµE11 𝑚𝑖𝑛(𝛾11, 𝜇11))2 +(𝑁𝑜𝑟𝑚𝛾𝐸11 𝑚𝑖𝑛(𝛿11, 𝜃11))2 < 1, ∣μ11∣+∣ν11∣ ≤ 1. X12 =(𝑁𝑜𝑟𝑚𝜗𝐸12 (max (𝛼12, 𝑥12) + 𝑖 max (𝛽12, 𝑦12)), NormµE12 (min (𝛾12, 𝜇12) + 𝑖min (𝛿12, 𝜃12)) where ∣μ12∣=(𝑁𝑜𝑟𝑚𝜗𝐸12 𝑚𝑎𝑥 (𝛼12, 𝑥12))2 +(𝑁𝑜𝑟𝑚𝛿𝐹12 𝑚𝑎𝑥 (𝛽12, 𝑦12))2 < 1, ∣ν12∣= (NormµE12 𝑚𝑖𝑛(𝛾12, 𝜇12))2 +(𝑁𝑜𝑟𝑚𝛾𝐸12 𝑚𝑖𝑛( (𝛿12, 𝜃12))2 < 1, ∣μ12∣+∣ν12∣ ≤ 1. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2940 https://internationalpubls.com X21 ={( NormϑE21 𝑚𝑎𝑥 (𝛼21, 𝑥21) + iNormδF21 𝑚𝑎𝑥(𝛽21, 𝑦21), (NormµE21 𝑚𝑖𝑛(𝛾21, 𝜇21 ) + iNormγE21 𝑚𝑖𝑛(𝛿21, 𝜃21)} where ∣μ21∣=(𝑁𝑜𝑟𝑚𝜗𝐸21 𝑚𝑎𝑥 (𝛼21, 𝑥21))2 +(𝑁𝑜𝑟𝑚𝛿𝐹21 𝑚𝑎𝑥 (𝛽21, 𝑦21))2 < 1, ∣ν21∣= (NormµE21 𝑚𝑖𝑛(𝛾21, 𝜇21))2 +(𝑁𝑜𝑟𝑚𝛾𝐸21 𝑚𝑖𝑛( (𝛿21, 𝜃21))2 < 1, ∣μ21∣+∣ν21∣ ≤ 1. X22 ={max( NormϑE22 (𝛼22, 𝑥22)), iNormδF22 𝑚𝑎𝑥 (𝛽22, 𝑦22), min (NormµE22 (𝛾22, 𝜇21), NormγE22 (𝛿22, 𝜃22)} where ∣μ22∣=(𝑁𝑜𝑟𝑚𝜗𝐸22 𝑚𝑎𝑥 (𝛼22, 𝑥22))2 +(𝑁𝑜𝑟𝑚𝛿𝐹22 𝑚𝑎𝑥 (𝛽22, 𝑦22))2 < 1, ∣ν21∣= (NormµE22 𝑚𝑖𝑛(𝛾22, 𝜇22))2 +(𝑁𝑜𝑟𝑚𝛾𝐸22 𝑚𝑖𝑛( (𝛿22, 𝜃22))2 < 1, ∣μ22∣+∣ν22∣ ≤ 1. We have Norm𝐴𝐸 ∪ 𝑁𝑜𝑟𝑚𝐵𝐹= ( 𝑋11 𝑋12 𝑋21 𝑋22 ) is normalization of complex intuitionistic fuzzy matrix. Example5.2.4: If 𝐴𝐸=( (0.6 + 0.2i, 0.3 + 0.1i)(0.4 + 0.1i, 0.5 + 0.2i) (0.5 + 0.3i, 0.2 + 0.1i)(0.3 + 0.2i, 0.6 + 0.1i) ) and BF = ( (0.4 + 0.3i, 0.5 + 0.1i)(0.3 + 0.4i, 0.4 + 0.2i) (0.6 + 0.2i, 0.2 + 0.2i)(0.5 + 0.1i, 0.4 + 0.3i) ) are normalization of complex intuitionistic fuzzy matrix two intuitionistic fuzzy matrices over different universe E and F then Norm 𝐴𝐸  𝑁𝑜𝑟𝑚 𝐵𝐹 is also normalization of complex intuitionistic fuzzy matrix. Proof: Given 𝑨𝑬=( (0.6 + 0.2i, 0.3 + 0.1i)(0.4 + 0.1i, 0.5 + 0.2i) (0.5 + 0.3i, 0.2 + 0.1i)(0.3 + 0.2i, 0.6 + 0.1i) ) and 𝑩𝑭 = ( (0.4 + 0.3i, 0.5 + 0.1i)(0.3 + 0.4i, 0.4 + 0.2i) (0.6 + 0.2i, 0.2 + 0.2i)(0.5 + 0.1i, 0.4 + 0.3i) ) Norm 𝐴𝐸 = ( (1 + 0.67𝑖 0.125 + 0𝑖) (0.67 + 0.33𝑖 0.30 + 0.10𝑖) (0.83 + 1𝑖 0 + 0𝑖) (0.5 + 0.67𝑖 0.5 + 0𝑖) ) 𝑁𝑜𝑟𝑚 𝐵𝐹= ( (0.67 + 0.75𝑖 0.375 + 0𝑖) (0.5 + 1𝑖 0.25 + 0.11𝑖) (1 + 0.5𝑖 0 + 0.11𝑖) (0.83 + 0.25𝑖 0.25 + 0.22𝑖) ) Norm𝐴𝐸 ∪ 𝑁𝑜𝑟𝑚𝐵𝐹 = ( (1 + 0.67𝑖 0.125 + 0𝑖) (0.67 + 0.33𝑖 0.30 + 0.10𝑖) (0.83 + 1𝑖 0 + 0𝑖) (0.5 + 0.67𝑖 0.5 + 0𝑖) ) ∪ ( (0.67 + 0.75𝑖 0.375 + 0𝑖) (0.5 + 1𝑖 0.25 + 0.11𝑖) (1 + 0.5𝑖 0 + 0.11𝑖) (0.83 + 0.25𝑖 0.25 + 0.22𝑖) ) 𝑋11 = (1 + 0.67𝑖 0.125 + 0𝑖) ∪ (0.67 + 0.75𝑖 0.375 + 0𝑖) 𝑋12 = (0.67 + 0.33𝑖 0.30 + 0.10𝑖) ∪ (0.5 + 1𝑖 0.25 + 0.11𝑖) 𝑋21 = (0.83 + 1𝑖 0 + 0𝑖) ∪ (1 + 0.5𝑖 0 + 0.11𝑖) 𝑋22 = (0.5 + 0.67𝑖 0.5 + 0𝑖) ∪ (0.83 + 0.25𝑖 0.25 + 0.22𝑖) 𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 AEBF = {(x, y), max (μA(x), μB(y)), min (ϑA(x), ϑB(x)): xE, yF} 𝑋11 ={max(1 + 0.67𝑖, 0.67 + 0.75𝑖) , min(0.125 + 0𝑖, 0.375 + 0𝑖)} 𝑋12 ={max(0.67 + 0.33𝑖, 0.5 + 1𝑖) , min(0.30 + 0.10𝑖, 0.25 + 0.11𝑖)} 𝑋21 ={max(0.83 + 1𝑖, 1 + 0.5𝑖) , min(0 + 0𝑖, 0 + 0.11𝑖)} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2941 https://internationalpubls.com 𝑋22 ={max(0.5 + 0.67𝑖, 0.83 + 0.25𝑖) , min(0.5 + 0𝑖, 0.25 + 0.22𝑖)} 𝑋11 ={1+0.75i, 0.125+0i} 𝑋12 ={0.67+1i, 0.25+0.10i} 𝑋21 ={1+1i, 0+0i} 𝑋22 ={0.83+0.67i, 0.25+0i} We have Norm𝐴𝐸 ∪ 𝑁𝑜𝑟𝑚𝐵𝐹 = ( (1 + 0.75i, 0.125 + 0i) (0.67 + 1i, 0.25 + 0.10i) (1 + 1i, 0 + 0i) (0.83 + 0.67i, 0.25 + 0i) ) is also normalization of complex intuitionistic fuzzy matrix. Theorem 5.2.5: If E and F be two universal sets. For every normalization of complex intuitionistic fuzzy matrices 𝐴𝐸 𝑎𝑛𝑑 𝐵𝐹 are in E and F then Norm𝐴𝐸𝑁𝑜𝑟𝑚𝐵𝐹 is also normalization of complex intuitionistic fuzzy matrix. Proof: If Norm 𝐴𝐸 and 𝑁𝑜𝑟𝑚 𝐵𝐹 are two normalization intuitionistic fuzzy matrices over different unive𝑟𝑠𝑒𝑠 𝑠𝑒𝑡𝑠 𝐸 𝑎𝑛𝑑 𝐹. If we consider two 2 × 2 normalization intuitionistic fuzzy matrices, Norm 𝐴𝐸𝑎𝑛𝑑 𝑁𝑜𝑟𝑚 𝐵𝐹. Let Norm 𝐴𝐸 = ( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11) 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12) 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21) 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22) 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22)) ) and 𝑁𝑜𝑟𝑚 𝐵𝐹 = ( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11) 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12) 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑖𝑦21) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑖𝑦22) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)) ) are two normalization intuitionistic fuzzy matrices. Norm𝐴𝐸𝑁𝑜𝑟𝑚𝐵𝐹 = ( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11) 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12) 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21) 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22) 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22)) ) ( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11) 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12) 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑖𝑦21) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑖𝑦22) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)) ) Calculate, 𝑋11, 𝑋12, 𝑋21and 𝑋22, we have. Now applying both side on AEBF = {(x, y), (μA(x) + μB(y)) - μA(x). μB(y), ϑA(x).ϑB(y): xE, yF}, we have Norm𝐴𝐸𝑁𝑜𝑟𝑚𝐵𝐹 = ( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11) 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12) 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21) 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22) 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22)) ) ( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11) 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12) 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑖𝑦21) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑖𝑦22) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)) ) Calculate, 𝑋11, 𝑋12, 𝑋21and 𝑋22, we have X11 =(𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11) 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11)) +(𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11) 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2942 https://internationalpubls.com Where ∣μ11∣= (𝑁𝑜𝑟𝑚𝜗𝐸11 (max(𝛼11, 𝑥11))2 + (𝑁𝑜𝑟𝑚𝜗𝐸11 max(𝛽11, 𝑦11))2 +(𝑁𝑜𝑟𝑚𝛿𝐹11 𝑚𝑖𝑛 (𝛾11, 𝜇11))2 +(𝑁𝑜𝑟𝑚𝛿𝐹11 𝑚𝑖𝑛 (𝛾11, 𝜇11))2 < 1, ∣ν11∣= (NormµE11 𝑚𝑖𝑛(𝛾11, 𝜇11))2 +(𝑁𝑜𝑟𝑚𝛾𝐸11 𝑚𝑖𝑛(𝛿11, 𝜃11))2 < 1, ∣μ11∣+∣ν11∣ ≤ 1. X12 =(𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12) 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12)) + (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12) 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12)) X21 =(𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21) 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21)) + (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑖𝑦21) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21)) X22 =(𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22) 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22)) + (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑖𝑦22) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)) 𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 AEBF = {(x, y), (μA(x) + μB(y)) - μA(x). μB(y), ϑA(x).ϑB(y): xE, yF} X11 ={𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11) + 𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11) − 𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11). 𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11), 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11). 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)} X12 ={𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12) + 𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12) − 𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12). 𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12), 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12). 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12) } X21 ={𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21) + 𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑖𝑦21) − 𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21). 𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑦21), 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21). 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21) } X22 ={𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22) + 𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑖𝑦22) − 𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22). 𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑦22), 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22). 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)} X11 = {𝑁𝑜𝑟𝑚𝜗𝐸11+ 𝛿𝐹11 ((𝛼11 + 𝑥11 − 𝛼11. 𝑥11) + 𝑖(𝛽11+𝑦11 − 𝛽11.𝑦11)), 𝑁𝑜𝑟𝑚µ𝐸11+𝛾𝐹11 (𝛾11. 𝜇11 + 𝑖𝛿11. 𝜃11)} Where ∣μ11∣= (𝑁𝑜𝑟𝑚𝜗𝐸11+ 𝛿𝐹11 ((𝛼11 + 𝑥11 − 𝛼11. 𝑥11))2 + (𝑁𝑜𝑟𝑚𝜗𝐸11+ 𝛿𝐹11 (𝛽11 + 𝑦11 − 𝛽11. 𝑦11))2 < 1 ∣ ν11 ∣= (𝑁𝑜𝑟𝑚µ𝐸11+𝛾𝐹11 (𝛾11. 𝜇11)2 + 𝑁𝑜𝑟𝑚µ𝐸11+𝛾𝐹11 ( 𝛿11. 𝜃11)2 < 1, ∣μ11∣+∣ν11∣ ≤ 1. X12 = {𝑁𝑜𝑟𝑚𝜗𝐸12+ 𝛿𝐹12 ((𝛼12 + 𝑥12 − 𝛼12. 𝑥12) + 𝑖(𝛽12+𝑦12 − 𝛽12.𝑦12)), 𝑁𝑜𝑟𝑚µ𝐸12+𝛾𝐹12 (𝛾12. 𝜇12 + 𝑖𝛿12. 𝜃12)} Where ∣μ12∣= (𝑁𝑜𝑟𝑚𝜗𝐸12+ 𝛿𝐹12 ((𝛼12 + 𝑥12 − 𝛼12. 𝑥12))2 + (𝑁𝑜𝑟𝑚𝜗𝐸12+ 𝛿𝐹12 (𝛽12 + 𝑦12 − 𝛽12. 𝑦12))2 < 1 ∣ ν12 ∣= (𝑁𝑜𝑟𝑚µ𝐸12+𝛾𝐹12 (𝛾12. 𝜇12)2 + 𝑁𝑜𝑟𝑚µ𝐸12+𝛾𝐹12 ( 𝛿12. 𝜃12)2 < 1, ∣μ12∣+∣ν12∣ ≤ 1. X21 = {𝑁𝑜𝑟𝑚𝜗𝐸21+ 𝛿𝐹21 ((𝛼21 + 𝑥21 − 𝛼21. 𝑥21) + 𝑖(𝛽21+𝑦21 − 𝛽21.𝑦21)), 𝑁𝑜𝑟𝑚µ𝐸21+𝛾𝐹21 (𝛾21. 𝜇21 + 𝑖𝛿21. 𝜃21)} Where ∣μ21∣= (𝑁𝑜𝑟𝑚𝜗𝐸21+ 𝛿𝐹21 ((𝛼21 + 𝑥21 − 𝛼21. 𝑥21))2 + (𝑁𝑜𝑟𝑚𝜗𝐸21+ 𝛿𝐹21 (𝛽21 + 𝑦21 − 𝛽21. 𝑦21))2 < 1 ∣ ν21 ∣= (𝑁𝑜𝑟𝑚µ𝐸21+𝛾𝐹21 (𝛾21. 𝜇21)2 + 𝑁𝑜𝑟𝑚µ𝐸21+𝛾𝐹21 ( 𝛿21. 𝜃21)2 < 1, ∣μ21∣+∣ν21∣ ≤ 1. X22 = {𝑁𝑜𝑟𝑚𝜗𝐸22+ 𝛿𝐹22 ((𝛼22 + 𝑥22 − 𝛼22. 𝑥22) + 𝑖(𝛽22+𝑦22 − 𝛽22.𝑦22)), 𝑁𝑜𝑟𝑚µ𝐸22+𝛾𝐹22 (𝛾22. 𝜇22 + 𝑖𝛿22. 𝜃22)} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2943 https://internationalpubls.com Where ∣μ22∣= (𝑁𝑜𝑟𝑚𝜗𝐸22+ 𝛿𝐹22 ((𝛼22 + 𝑥22 − 𝛼22. 𝑥22))2 + (𝑁𝑜𝑟𝑚𝜗𝐸22+ 𝛿𝐹22 (𝛽22 + 𝑦22 − 𝛽22. 𝑦22))2 < 1 ∣ ν22 ∣= (𝑁𝑜𝑟𝑚µ𝐸22+𝛾𝐹22 (𝛾22. 𝜇22)2 + 𝑁𝑜𝑟𝑚µ𝐸22+𝛾𝐹22 ( 𝛿22. 𝜃22)2 < 1, ∣μ22∣+∣ν22∣ ≤ 1. We have Norm𝐴𝐸𝑁𝑜𝑟𝑚𝐵𝐹 = ( 𝑋11 𝑋12 𝑋21 𝑋22 ) is also intuitionistic fuzzy matrix. Example5.2.6: : If 𝐴𝐸=( (0.6 + 0.2i, 0.3 + 0.1i)(0.4 + 0.1i, 0.5 + 0.2i) (0.5 + 0.3i, 0.2 + 0.1i)(0.3 + 0.2i, 0.6 + 0.1i) ) and BF = ( (0.4 + 0.3i, 0.5 + 0.1i)(0.3 + 0.4i, 0.4 + 0.2i) (0.6 + 0.2i, 0.2 + 0.2i)(0.5 + 0.1i, 0.4 + 0.3i) ) are normalization of complex intuitionistic fuzzy matrix two intuitionistic fuzzy matrices over different universe E and F then Norm𝐴𝐸𝑁𝑜𝑟𝑚𝐵𝐹 is also normalization of complex intuitionistic fuzzy matrix. Proof: Given 𝑨𝑬=( (0.6 + 0.2i, 0.3 + 0.1i)(0.4 + 0.1i, 0.5 + 0.2i) (0.5 + 0.3i, 0.2 + 0.1i)(0.3 + 0.2i, 0.6 + 0.1i) ) and 𝑩𝑭 = ( (0.4 + 0.3i, 0.5 + 0.1i)(0.3 + 0.4i, 0.4 + 0.2i) (0.6 + 0.2i, 0.2 + 0.2i)(0.5 + 0.1i, 0.4 + 0.3i) ) Norm 𝐴𝐸 = ( (1 + 0.67𝑖 0.125 + 0𝑖) (0.67 + 0.33𝑖 0.30 + 0.10𝑖) (0.83 + 1𝑖 0 + 0𝑖) (0.5 + 0.67𝑖 0.5 + 0𝑖) ) 𝑁𝑜𝑟𝑚 𝐵𝐹= ( (0.67 + 0.75𝑖 0.375 + 0𝑖) (0.5 + 1𝑖 0.25 + 0.11𝑖) (1 + 0.5𝑖 0 + 0.11𝑖) (0.83 + 0.25𝑖 0.25 + 0.22𝑖) ) Norm𝐴𝐸𝑁𝑜𝑟𝑚𝐵𝐹 = ( (1 + 0.67𝑖 0.125 + 0𝑖) (0.67 + 0.33𝑖 0.30 + 0.10𝑖) (0.83 + 1𝑖 0 + 0𝑖) (0.5 + 0.67𝑖 0.5 + 0𝑖) )  ( (0.67 + 0.75𝑖 0.375 + 0𝑖) (0.5 + 1𝑖 0.25 + 0.11𝑖) (1 + 0.5𝑖 0 + 0.11𝑖) (0.83 + 0.25𝑖 0.25 + 0.22𝑖) ) 𝑋11 = (1 + 0.67𝑖 0.125 + 0𝑖)(0.67 + 0.75𝑖 0.375 + 0𝑖) 𝑋12 = (0.67 + 0.33𝑖 0.30 + 0.10𝑖) (0.5 + 1𝑖 0.25 + 0.11𝑖) 𝑋21 = (0.83 + 1𝑖 0 + 0𝑖)  (1 + 0.5𝑖 0 + 0.11𝑖) 𝑋22 = (0.5 + 0.67𝑖 0.5 + 0𝑖)(0.83 + 0.25𝑖 0.25 + 0.22𝑖) 𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 AEBF = {(x, y), (μA(x) + μB(y)) - μA(x). μB(y), ϑA(x).ϑB(y): xE, yF} AEBF = ( 1 + 0.9175𝑖, 0.046 + 0𝑖 0.835 + 1𝑖, 0.075 + 0.011𝑖 1 + 1𝑖, 0 + 0𝑖 0.915 + 0.752𝑖, 0.125 + 0𝑖 ) 𝑋11 ={1+0.9175i, 0.046+0i} 𝑋12 ={0.835+1i, 0.075+0.011i} 𝑋21 ={1+1i, 0+0i} 𝑋22 ={0.915+0.752i, 0.125+0i} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2944 https://internationalpubls.com We have Norm𝐴𝐸𝑁𝑜𝑟𝑚𝐵𝐹 = ( (1 + 0.9175i, 0.046 + 0i) (0.835 + 1i, 0.075 + 0.011i) (1 + 1i, 0 + 0i) (0.915 + 0.752i, 0.125 + 0i) ) is also normalization of complex intuitionistic fuzzy matrix. Theorem5.2.7: If E and F be two universal sets. For every normalization of complex intuitionistic fuzzy matrices 𝐴𝐸 𝑎𝑛𝑑 𝐵𝐹 are in E and F then Norm𝐴𝐸𝑁𝑜𝑟𝑚𝐵𝐹 is also normalization of complex intuitionistic fuzzy matrix. Proof: If Norm 𝐴𝐸 and 𝑁𝑜𝑟𝑚 𝐵𝐹 are two normalization intuitionistic fuzzy matrices over different unive𝑟𝑠𝑒𝑠 𝑠𝑒𝑡𝑠 𝐸 𝑎𝑛𝑑 𝐹. If we consider two 2 × 2 normalization intuitionistic fuzzy matrices, Norm 𝐴𝐸𝑎𝑛𝑑 𝑁𝑜𝑟𝑚 𝐵𝐹. Let Norm 𝐴𝐸 = ( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11) 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12) 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21) 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22) 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22)) ) and 𝑁𝑜𝑟𝑚 𝐵𝐹 = ( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11) 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12) 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑖𝑦21) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑖𝑦22) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)) ) are two normalization intuitionistic fuzzy matrices. Norm𝐴𝐸𝑁𝑜𝑟𝑚𝐵𝐹 = ( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11) 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12) 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21) 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22) 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22)) ) ( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11) 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12) 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑖𝑦21) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑖𝑦22) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)) ) Calculate, 𝑋11, 𝑋12, 𝑋21and 𝑋22, we have X11 =(𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11) 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11)) (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11) 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)) Where ∣μ11∣= (𝑁𝑜𝑟𝑚𝜗𝐸11 (max(𝛼11, 𝑥11))2 + (𝑁𝑜𝑟𝑚𝜗𝐸11 max(𝛽11, 𝑦11))2 +(𝑁𝑜𝑟𝑚𝛿𝐹11 𝑚𝑖𝑛 (𝛾11, 𝜇11))2 +(𝑁𝑜𝑟𝑚𝛿𝐹11 𝑚𝑖𝑛 (𝛾11, 𝜇11))2 < 1, ∣ν11∣= (NormµE11 𝑚𝑖𝑛(𝛾11, 𝜇11))2 +(𝑁𝑜𝑟𝑚𝛾𝐸11 𝑚𝑖𝑛(𝛿11, 𝜃11))2 < 1, ∣μ11∣+∣ν11∣ ≤ 1. X12 =(𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12) 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12))  (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12) 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12)) X21 =(𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21) 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21))  (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑖𝑦21) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21)) X22 =(𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22) 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22))  (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑖𝑦22) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)) 𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 AEBF = {(x, y), μA(x). μB(y), ϑA(x) +ϑB(y) - ϑA(x).ϑB(y): xE, yF} X11 ={𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11). 𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11), 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11) + 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11) − 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11). 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2945 https://internationalpubls.com X12 ={𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12). 𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12), 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12) + 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12) − 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12). 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12)} X21 ={𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21). 𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑦21), 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21) + 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21) − 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21). 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21)} X22 ={𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22). 𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑦22), 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22) + 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22) − 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22). 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)} X11 = {𝑁𝑜𝑟𝑚𝜗𝐸11+ 𝛿𝐹11 (𝛼11. 𝑥11) + 𝑖(𝛽11.𝑦11), 𝑁𝑜𝑟𝑚µ𝐸11+𝛾𝐹11 (𝛾11 + 𝜇11 − 𝛾11. 𝜇11 + 𝑖(𝛿11 + 𝜃11 − 𝛿11. 𝜃11)} Where ∣μ11∣= (𝑁𝑜𝑟𝑚𝜗𝐸11+ 𝛿𝐹11 (𝛼11. 𝑥11))2 + (𝑁𝑜𝑟𝑚𝜗𝐸11+ 𝛿𝐹11 (𝛽11. 𝑦11)2 < 1 ∣ ν11 ∣= (𝑁𝑜𝑟𝑚µ𝐸11+𝛾𝐹11 (𝛾11 + 𝜇11 − 𝛾11. 𝜇11)2 + 𝑁𝑜𝑟𝑚µ𝐸11+𝛾𝐹11 ( 𝛿11 + 𝜃11 − 𝛿11. 𝜃11)2 < 1, ∣μ11∣+∣ν11∣ ≤ 1. X12 = {𝑁𝑜𝑟𝑚𝜗𝐸12+ 𝛿𝐹12 (𝛼12. 𝑥12) + 𝑖(𝛽12.𝑦12), 𝑁𝑜𝑟𝑚µ𝐸12+𝛾𝐹12 (𝛾12 + 𝜇12 − 𝛾12. 𝜇12) + 𝑖(𝛿12 + 𝜃12 − 𝛿12. 𝜃12)} Where ∣μ12∣= (𝑁𝑜𝑟𝑚𝜗𝐸12+ 𝛿𝐹12 (𝛼12. 𝑥12)2 + (𝑁𝑜𝑟𝑚𝜗𝐸12+ 𝛿𝐹12 (𝛽12. 𝑦12)2 < 1 ∣ ν12 ∣= (𝑁𝑜𝑟𝑚µ𝐸12+𝛾𝐹12 (𝛾12 + 𝜇12 − 𝛾12. 𝜇12)2 + 𝑁𝑜𝑟𝑚µ𝐸12+𝛾𝐹12 ( 𝛿12 + 𝜃12 − 𝛿12. 𝜃12)2 < 1, ∣μ12∣+∣ν12∣ ≤ 1. X21 = {𝑁𝑜𝑟𝑚𝜗𝐸21+ 𝛿𝐹21 (𝛼21. 𝑥21) + 𝑖(𝛽21.𝑦21), 𝑁𝑜𝑟𝑚µ𝐸21+𝛾𝐹21 (𝛾21 + 𝜇21 − 𝛾21. 𝜇21 + 𝑖(𝛿21 + 𝜃21 − 𝛿21. 𝜃21} Where ∣μ21∣= (𝑁𝑜𝑟𝑚𝜗𝐸21+ 𝛿𝐹21 (𝛼21. 𝑥21)2 + (𝑁𝑜𝑟𝑚𝜗𝐸21+ 𝛿𝐹21 (𝛽21. 𝑦21)2 < 1 ∣ ν21 ∣= (𝑁𝑜𝑟𝑚µ𝐸21+𝛾𝐹21 (𝛾21 + 𝜇21 − 𝛾21. 𝜇21)2 + 𝑁𝑜𝑟𝑚µ𝐸21+𝛾𝐹21 ( 𝛿21 + 𝜃21 − 𝛿21. 𝜃21)2 < 1, ∣μ21∣+∣ν21∣ ≤ 1. X22 = {𝑁𝑜𝑟𝑚𝜗𝐸22+ 𝛿𝐹22 (𝛼22. 𝑥22) + 𝑖(𝛽22.𝑦22), 𝑁𝑜𝑟𝑚µ𝐸22+𝛾𝐹22 (𝛾22 + 𝜇22 − 𝛾22. 𝜇22) + 𝑖(𝛿22 + 𝜃22 − 𝛿22. 𝜃22)} Where ∣μ22∣= (𝑁𝑜𝑟𝑚𝜗𝐸22+ 𝛿𝐹22 (𝛼22. 𝑥22)2 + (𝑁𝑜𝑟𝑚𝜗𝐸22+ 𝛿𝐹22 (𝛽22. 𝑦22)2 < 1 ∣ ν22 ∣= (𝑁𝑜𝑟𝑚µ𝐸22+𝛾𝐹22 (𝛾22 + 𝜇22 − 𝛾22. 𝜇22)2 + 𝑁𝑜𝑟𝑚µ𝐸22+𝛾𝐹22 ( 𝛿22 + 𝜃22 − 𝛿22. 𝜃22)2 < 1, ∣μ22∣+∣ν22∣ ≤ 1. We have Norm𝐴𝐸𝑁𝑜𝑟𝑚𝐵𝐹 = ( 𝑋11 𝑋12 𝑋21 𝑋22 ) is also normalization of complex intuitionistic fuzzy matrix Example5.2.8: If 𝐴𝐸=( (0.6 + 0.2i, 0.3 + 0.1i)(0.4 + 0.1i, 0.5 + 0.2i) (0.5 + 0.3i, 0.2 + 0.1i)(0.3 + 0.2i, 0.6 + 0.1i) ) and BF = ( (0.4 + 0.3i, 0.5 + 0.1i)(0.3 + 0.4i, 0.4 + 0.2i) (0.6 + 0.2i, 0.2 + 0.2i)(0.5 + 0.1i, 0.4 + 0.3i) ) are normalization of complex intuitionistic fuzzy matrix two intuitionistic fuzzy matrices over different universe E and F then Norm𝐴𝐸𝑁𝑜𝑟𝑚𝐵𝐹 is also normalization of complex intuitionistic fuzzy matrix. Proof: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2946 https://internationalpubls.com Given 𝑨𝑬=( (0.6 + 0.2i, 0.3 + 0.1i)(0.4 + 0.1i, 0.5 + 0.2i) (0.5 + 0.3i, 0.2 + 0.1i)(0.3 + 0.2i, 0.6 + 0.1i) ) and 𝑩𝑭 = ( (0.4 + 0.3i, 0.5 + 0.1i)(0.3 + 0.4i, 0.4 + 0.2i) (0.6 + 0.2i, 0.2 + 0.2i)(0.5 + 0.1i, 0.4 + 0.3i) ) Norm 𝐴𝐸 = ( (1 + 0.67𝑖 0.125 + 0𝑖) (0.67 + 0.33𝑖 0.30 + 0.10𝑖) (0.83 + 1𝑖 0 + 0𝑖) (0.5 + 0.67𝑖 0.5 + 0𝑖) ) 𝑁𝑜𝑟𝑚 𝐵𝐹= ( (0.67 + 0.75𝑖 0.375 + 0𝑖) (0.5 + 1𝑖 0.25 + 0.11𝑖) (1 + 0.5𝑖 0 + 0.11𝑖) (0.83 + 0.25𝑖 0.25 + 0.22𝑖) ) Apply the formula AEBF = {(x, y), μA(x). μB(y), ϑA(x) +ϑB(y) - ϑA(x).ϑB(y): xE, yF AEBF = ( (1 + 0.67𝑖 0.125 + 0𝑖) (0.67 + 0.33𝑖 0.30 + 0.10𝑖) (0.83 + 1𝑖 0 + 0𝑖) (0.5 + 0.67𝑖 0.5 + 0𝑖) )  ( (0.67 + 0.75𝑖 0.375 + 0𝑖) (0.5 + 1𝑖 0.25 + 0.11𝑖) (1 + 0.5𝑖 0 + 0.11𝑖) (0.83 + 0.25𝑖 0.25 + 0.22𝑖) ) 𝑋11 = (1 + 0.67𝑖 0.125 + 0𝑖)(0.67 + 0.75𝑖 0.375 + 0𝑖) 𝑋12 = (0.67 + 0.33𝑖 0.30 + 0.10𝑖) (0.5 + 1𝑖 0.25 + 0.11𝑖) 𝑋21 = (0.83 + 1𝑖 0 + 0𝑖)  (1 + 0.5𝑖 0 + 0.11𝑖) 𝑋22 = (0.5 + 0.67𝑖 0.5 + 0𝑖)(0.83 + 0.25𝑖 0.25 + 0.22𝑖) 𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 AEBF = {(x, y), μA(x). μB(y), ϑA(x) +ϑB(y) - ϑA(x).ϑB(y): xE, yF AEBF = ( 0.67 + 0.502𝑖, 0.453 + 0𝑖 0.335 + 0.33𝑖, 0.65 + 0.199𝑖 0.83 + 0.5𝑖, 0 + 0.11𝑖 0.415 + 0.1678𝑖, 0.625 + 0.22𝑖 ) 𝑋11 ={0.807+1i, 0.0.453+0i} 𝑋12 ={0.403+0.667i, 0.65+0.199i} 𝑋21 ={1+0.99i, 0+0.11i} 𝑋22 ={0.5+0.332i, 0.625+022i} We have Norm𝐴𝐸𝑁𝑜𝑟𝑚𝐵𝐹 = ( (0.807 + 1i, 0.0.453 + 0i) (0.403 + 0.667i, 0.65 + 0.199i) (1 + 0.99i, 0 + 0.11i) (0.5 + 0.332i, 0.625 + 022i) ) is also normalization of complex intuitionistic fuzzy matrix. Theorem 5.2.9: If E and F be two universal sets. For every normalization of complex intuitionistic fuzzy matrices 𝐴𝐸 𝑎𝑛𝑑 𝐵𝐹 are in E and F then Norm𝐴𝐸@𝑁𝑜𝑟𝑚𝐵𝐹 is also normalization of complex intuitionistic fuzzy matrix. Proof: If Norm 𝐴𝐸 and 𝑁𝑜𝑟𝑚 𝐵𝐹 are two normalization intuitionistic fuzzy matrices over different unive𝑟𝑠𝑒𝑠 𝑠𝑒𝑡𝑠 𝐸 𝑎𝑛𝑑 𝐹. If we consider two 2 × 2 normalization intuitionistic fuzzy matrices, Norm 𝐴𝐸𝑎𝑛𝑑 𝑁𝑜𝑟𝑚 𝐵𝐹. Let Norm 𝐴𝐸 = ( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11) 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12) 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21) 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22) 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22)) ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2947 https://internationalpubls.com and 𝑁𝑜𝑟𝑚 𝐵𝐹 = ( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11) 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12) 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑖𝑦21) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑖𝑦22) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)) ) are two normalization intuitionistic fuzzy matrices. Norm𝐴𝐸@𝑁𝑜𝑟𝑚𝐵𝐹 = ( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11) 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12) 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21) 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22) 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22)) ) @ ( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11) 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12) 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑖𝑦21) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑖𝑦22) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)) ) Calculate, 𝑋11, 𝑋12, 𝑋21and 𝑋22, we have X11 =(𝑁𝑜𝑟𝑚𝜗𝐸11 (𝛼11 + 𝑖𝛽11) 𝑁𝑜𝑟𝑚µ𝐸11 (𝛾11 + 𝑖𝛿11)) @(𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑥11 + 𝑖𝑦11) 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝜇11 + 𝑖𝜃11)) Where ∣μ11∣= (𝑁𝑜𝑟𝑚𝜗𝐸11 (max(𝛼11, 𝑥11))2 + (𝑁𝑜𝑟𝑚𝜗𝐸11 max(𝛽11, 𝑦11))2 +(𝑁𝑜𝑟𝑚𝛿𝐹11 𝑚𝑖𝑛 (𝛾11, 𝜇11))2 +(𝑁𝑜𝑟𝑚𝛿𝐹11 𝑚𝑖𝑛 (𝛾11, 𝜇11))2 < 1, ∣ν11∣= (NormµE11 𝑚𝑖𝑛(𝛾11, 𝜇11))2 +(𝑁𝑜𝑟𝑚𝛾𝐸11 𝑚𝑖𝑛(𝛿11, 𝜃11))2 < 1, ∣μ11∣+∣ν11∣ ≤ 1. X12 =(𝑁𝑜𝑟𝑚𝜗𝐸12 (𝛼12 + 𝑖𝛽12) 𝑁𝑜𝑟𝑚µ𝐸12 (𝛾12 + 𝑖𝛿12)) @ (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑥12 + 𝑖𝑦12) 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝜇12 + 𝑖𝜃12)) X21 =(𝑁𝑜𝑟𝑚𝜗𝐸21 (𝛼21 + 𝑖𝛽21) 𝑁𝑜𝑟𝑚µ𝐸21 (𝛾21 + 𝑖𝛿21)) @ (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑥21 + 𝑖𝑦21) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝜇21 + 𝑖𝜃21)) X22 =(𝑁𝑜𝑟𝑚𝜗𝐸22 (𝛼22 + 𝑖𝛽22) 𝑁𝑜𝑟𝑚µ𝐸22 (𝛾22 + 𝑖𝛿22)) @ (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑥22 + 𝑖𝑦22) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝜇22 + 𝑖𝜃22)) 𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 AE@BF = {(x, y), μA(x)+ μB(y) 2 , ϑA(x)+ϑB(y) 2 : xE, yF} X11 ={𝑁𝑜𝑟𝑚𝜗𝐸11@𝛿𝐹11 ( 𝛼11+𝑥11+𝑖(𝛽11+𝑦11) 2 ) , 𝑁𝑜𝑟𝑚µ𝐸11@𝛾𝐹11 ( 𝛾11+𝜇11+𝑖(𝛿11+𝜃11) 2 )} Where ((𝑁𝑜𝑟𝑚𝜗𝐸11@𝛿𝐹11 ( 𝛼11+𝑥11 2 ) X12 ={𝑁𝑜𝑟𝑚𝜗𝐸11@𝛿𝐹11 ( 𝛼11+𝑥11+𝑖(𝛽11+𝑦11) 2 ) , 𝑁𝑜𝑟𝑚µ𝐸11@𝛾𝐹11 ( 𝛾11+𝜇11+𝑖(𝛿11+𝜃11) 2 )} X21 ={𝑁𝑜𝑟𝑚𝜗𝐸11@𝛿𝐹11 ( 𝛼11+𝑥11+𝑖(𝛽11+𝑦11) 2 ) , 𝑁𝑜𝑟𝑚µ𝐸11@𝛾𝐹11 ( 𝛾11+𝜇11+𝑖(𝛿11+𝜃11) 2 )} X22 ={𝑁𝑜𝑟𝑚𝜗𝐸11@𝛿𝐹11 ( 𝛼11+𝑥11+𝑖(𝛽11+𝑦11) 2 ) , 𝑁𝑜𝑟𝑚µ𝐸11@𝛾𝐹11 ( 𝛾11+𝜇11+𝑖(𝛿11+𝜃11) 2 )} X11 = {𝑁𝑜𝑟𝑚𝜗𝐸11+ 𝛿𝐹11 (𝛼11. 𝑥11) + 𝑖(𝛽11.𝑦11), 𝑁𝑜𝑟𝑚µ𝐸11+𝛾𝐹11 (𝛾11 + 𝜇11 − 𝛾11. 𝜇11 + 𝑖(𝛿11 + 𝜃11 − 𝛿11. 𝜃11)} Where ∣μ11∣= (𝑁𝑜𝑟𝑚𝜗𝐸11+ 𝛿𝐹11 (𝛼11. 𝑥11))2 + (𝑁𝑜𝑟𝑚𝜗𝐸11+ 𝛿𝐹11 (𝛽11. 𝑦11)2 < 1 ∣ ν11 ∣= (𝑁𝑜𝑟𝑚µ𝐸11+𝛾𝐹11 (𝛾11 + 𝜇11 − 𝛾11. 𝜇11)2 + 𝑁𝑜𝑟𝑚µ𝐸11+𝛾𝐹11 ( 𝛿11 + 𝜃11 − 𝛿11. 𝜃11)2 < 1, ∣μ11∣+∣ν11∣ ≤ 1. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2948 https://internationalpubls.com X12 = {𝑁𝑜𝑟𝑚𝜗𝐸12+ 𝛿𝐹12 (𝛼12. 𝑥12) + 𝑖(𝛽12.𝑦12), 𝑁𝑜𝑟𝑚µ𝐸12+𝛾𝐹12 (𝛾12 + 𝜇12 − 𝛾12. 𝜇12) + 𝑖(𝛿12 + 𝜃12 − 𝛿12. 𝜃12)} Where ∣μ12∣= (𝑁𝑜𝑟𝑚𝜗𝐸12+ 𝛿𝐹12 (𝛼12. 𝑥12)2 + (𝑁𝑜𝑟𝑚𝜗𝐸12+ 𝛿𝐹12 (𝛽12. 𝑦12)2 < 1 ∣ ν12 ∣= (𝑁𝑜𝑟𝑚µ𝐸12+𝛾𝐹12 (𝛾12 + 𝜇12 − 𝛾12. 𝜇12)2 + 𝑁𝑜𝑟𝑚µ𝐸12+𝛾𝐹12 ( 𝛿12 + 𝜃12 − 𝛿12. 𝜃12)2 < 1, ∣μ12∣+∣ν12∣ ≤ 1. X21 = {𝑁𝑜𝑟𝑚𝜗𝐸21+ 𝛿𝐹21 (𝛼21. 𝑥21) + 𝑖(𝛽21.𝑦21), 𝑁𝑜𝑟𝑚µ𝐸21+𝛾𝐹21 (𝛾21 + 𝜇21 − 𝛾21. 𝜇21 + 𝑖(𝛿21 + 𝜃21 − 𝛿21. 𝜃21} Where ∣μ21∣= (𝑁𝑜𝑟𝑚𝜗𝐸21+ 𝛿𝐹21 (𝛼21. 𝑥21)2 + (𝑁𝑜𝑟𝑚𝜗𝐸21+ 𝛿𝐹21 (𝛽21. 𝑦21)2 < 1 ∣ ν21 ∣= (𝑁𝑜𝑟𝑚µ𝐸21+𝛾𝐹21 (𝛾21 + 𝜇21 − 𝛾21. 𝜇21)2 + 𝑁𝑜𝑟𝑚µ𝐸21+𝛾𝐹21 ( 𝛿21 + 𝜃21 − 𝛿21. 𝜃21)2 < 1, ∣μ21∣+∣ν21∣ ≤ 1. X22 = {𝑁𝑜𝑟𝑚𝜗𝐸22+ 𝛿𝐹22 (𝛼22. 𝑥22) + 𝑖(𝛽22.𝑦22), 𝑁𝑜𝑟𝑚µ𝐸22+𝛾𝐹22 (𝛾22 + 𝜇22 − 𝛾22. 𝜇22) + 𝑖(𝛿22 + 𝜃22 − 𝛿22. 𝜃22)} Where ∣μ22∣= (𝑁𝑜𝑟𝑚𝜗𝐸22+ 𝛿𝐹22 (𝛼22. 𝑥22)2 + (𝑁𝑜𝑟𝑚𝜗𝐸22+ 𝛿𝐹22 (𝛽22. 𝑦22)2 < 1 ∣ ν22 ∣= (𝑁𝑜𝑟𝑚µ𝐸22+𝛾𝐹22 (𝛾22 + 𝜇22 − 𝛾22. 𝜇22)2 + 𝑁𝑜𝑟𝑚µ𝐸22+𝛾𝐹22 ( 𝛿22 + 𝜃22 − 𝛿22. 𝜃22)2 < 1, ∣μ22∣+∣ν22∣ ≤ 1. We have Norm𝐴𝐸𝑁𝑜𝑟𝑚𝐵𝐹 = ( 𝑋11 𝑋12 𝑋21 𝑋22 ) is also intuitionistic fuzzy matrix Example5.2.10: If 𝑨𝑬=( (.9, .4) (.4, .1) (.5, .3) (.8, .3) ) and𝑩𝑭 = ( (.8, .6) (.6, .3) (.7, .5) (.9, .4) ) are two intuitionistic fuzzy matrices over different universe E and F then Norm𝐴̅ 𝐸@𝑁𝑜𝑟𝑚�̅�𝐹is also fuzzy matrix set. Proof: Given 𝑨𝑬 =( (.9, .4) (.4, .1) (.5, .3) (.8, .3) ) and 𝑩𝑭 = ( (.8, .6) (.6, .3) (.7, .5) (.9, .4) ) 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐴 =( ( .8,1) (0, .3) (.2, .8) (.7, .8) ) and 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐵=( (.5, 1) (0, .5) (.3, .8) (.8, .7) ) 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐴@ 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ ̅ 𝐵 = ( ( .8,1) (0, .3) (.2, .8) (.7, .8) ) @ ( (.5, 1) (0, .5) (.3, .8) (.8, .7) ) 𝑋11 = ( .8, 1)@(. 5, 1) 𝑋12 = (0, .3)@(0, .5) 𝑋21 = (.2, .8) @ (.3, .8) 𝑋22 = (. 7, .8)@(.8, .7) 𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝐴𝐸 ̅̅̅̅ @𝐵𝐹 ̅̅̅̅ = {〈x, y〉 (𝜗𝐴(x) +𝜗𝐵(y)) 2 , ( µ𝐴(x) . µ𝐵(y)) 2 : xE, y𝐹 } 𝑋11 = ( . 8 + .5 2 ) , ( 1 . 1 2 ) 𝑋12 = ( 0 + 0 2 ) , ( . 3 . 5 2 ) 𝑋21 = ( . 2 + .3 2 ) , ( . 8 . 8 2 ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2949 https://internationalpubls.com 𝑋22 = ( . 7 + .8 2 ) , ( . 8 . 7 2 ) 𝑋11 ={.7, .5} 𝑋12 ={0, .07} 𝑋21 ={.3, .3} 𝑋22 ={.8, .3} We have Norm𝐴̅ 𝐸@ 𝑁𝑜𝑟𝑚�̅�𝐹 = ( (.7, .5) (0, .07) (.3, .3) (.8 , .3) ) is also intuitionistic fuzzy matrix Theorem 5.2.6: If E and F be two universal sets. For every normalization of an intuitionistic fuzzy matrices 𝐴𝐸 ̅̅̅̅ 𝑎𝑛𝑑 𝐵𝐹 ̅̅̅̅ are in E and F then Norm𝐴̅ 𝐸 $ 𝑁𝑜𝑟𝑚�̅�𝐹is also normalization of an intuitionistic fuzzy matrices. Proof: If 𝑁𝑜𝑟𝑚𝐴 and 𝑁𝑜𝑟𝑚𝐵 are two normalization intuitionistic fuzzy matrices over different unive𝑟𝑠𝑒𝑠 𝑠𝑒𝑡𝑠 𝐸 𝑎𝑛𝑑 𝐹. 𝐼𝑓 𝑤𝑒 𝑐𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑡𝑤𝑜 2 × 2 𝑚𝑎𝑡𝑟𝑖𝑥 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐴 =( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸11(𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸12(𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸21 (𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸22 (𝑥)) ) and 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐵=( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹11(𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹12(𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝑦)) ) 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐴 $ 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐵 = ( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸11 (𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸12 (𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸21 (𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸22 (𝑥)) ) $ ( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹11(𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹12(𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝑦)) ) 𝑋11 =(𝑁𝑜𝑟𝑚𝜗𝐸11 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸11(𝑥)) $ (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹11(𝑦)) 𝑋12 =(𝑁𝑜𝑟𝑚𝜗𝐸12 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸12(𝑥)) $ (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹12(𝑦)) 𝑋21 =(𝑁𝑜𝑟𝑚𝜗𝐸21 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸21 (𝑥)) $ (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝑦)) 𝑋22 =(𝑁𝑜𝑟𝑚𝜗𝐸22 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸22 (𝑥)) $ (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝑦)) 𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝐴𝐸 ̅̅̅̅ $ 𝐵𝐹 ̅̅̅̅ = {〈x, y〉, √𝜗𝐴(x) . 𝜗𝐵(y), √ µ𝐴 (x) . µ𝐵(y) , : xE, y𝐹 } 𝑋11 = (√𝑁𝑜𝑟𝑚𝜗𝐸11 (𝑥). 𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑦) , √𝑁𝑜𝑟𝑚µ𝐸11 (𝑥). 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝑦)) 𝑋12 = (√𝑁𝑜𝑟𝑚𝜗𝐸12 (𝑥). 𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑦) , √𝑁𝑜𝑟𝑚µ𝐸12 (𝑥). 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝑦)) 𝑋21 = (√𝑁𝑜𝑟𝑚𝜗21 (𝑥). 𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑦) , √𝑁𝑜𝑟𝑚µ𝐸21 (𝑥). 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝑦)) 𝑋22 = (√𝑁𝑜𝑟𝑚𝜗𝐸22 (𝑥). 𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑦) , √𝑁𝑜𝑟𝑚µ𝐸22 (𝑥). 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝑦)) We have Norm𝐴̅ 𝐸$ 𝑁𝑜𝑟𝑚�̅�𝐹 = ( 𝑋11 𝑋12 𝑋21 𝑋22 ) is also intuitionistic fuzzy matrix Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2950 https://internationalpubls.com Example: If 𝑨𝑬=( (.9, .4) (.4, .1) (.5, .3) (.8, .3) ) and𝑩𝑭 = ( (.8, .6) (.6, .3) (.7, .5) (.9, .4) ) are two intuitionistic fuzzy matrices over different universe E and F then Norm𝐴̅ 𝐸$𝑁𝑜𝑟𝑚�̅�𝐹is also fuzzy matrix set. Proof: Given 𝑨𝑬 =( (.9, .4) (.4, .1) (.5, .3) (.8, .3) ) and 𝑩𝑭 = ( (.8, .6) (.6, .3) (.7, .5) (.9, .4) ) 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐴 =( ( .8,1) (0, .3) (.2, .8) (.7, .8) ) and 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐵=( (.5, 1) (0, .5) (.3, .8) (.8, .7) ) 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐴$ 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ ̅ 𝐵 = ( ( .8,1) (0, .3) (.2, .8) (.7, .8) ) $ ( (.5, 1) (0, .5) (.3, .8) (.8, .7) ) 𝑋11 = ( .8, 1)$(. 5, 1) 𝑋12 = (0, .3) $(0, .5) 𝑋21 = (.2, .8) $(.3, .8) 𝑋22 = (. 7, .8)$(.8, .7) 𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝐴𝐸 ̅̅̅̅ $ 𝐵𝐹 ̅̅̅̅ = {〈x, y〉, √𝜗𝐴(x) . 𝜗𝐵(y), √ µ𝐴 (x) . µ𝐵(y) , : xE, y𝐹 } 𝑋11 = (√. 8 .5 , √1 . 1 ) 𝑋12 = (√0 . 0 , √. 3 .5 ) 𝑋21 = (√. 2 .3 , √. 8 .8 ) 𝑋22 = (√. 7 .8 , √. 8 .7 ) 𝑋11 ={.6, .1} 𝑋12 ={0 , .4} 𝑋21 ={.2, .8} 𝑋22 ={.7 , .7} We have Norm𝐴̅ 𝐸$ 𝑁𝑜𝑟𝑚�̅�𝐹 = ( (.6, .1) (0 , .4) (.2, .8) (.7 , .7) ) is also intuitionistic fuzzy matrix. Theorem 5.2.7: If E and F be two universal sets. For every normalization of an intuitionistic fuzzy matrices 𝐴𝐸 ̅̅̅̅ 𝑎𝑛𝑑 𝐵𝐹 ̅̅̅̅ are in E and F then Norm𝐴̅ 𝐸# 𝑁𝑜𝑟𝑚�̅�𝐹 is also normalization of an intuitionistic fuzzy matrices. Proof: If 𝑁𝑜𝑟𝑚𝐴 and 𝑁𝑜𝑟𝑚𝐵 are two normalization intuitionistic fuzzy matrices over different unive𝑟𝑠𝑒𝑠 𝑠𝑒𝑡𝑠 𝐸 𝑎𝑛𝑑 𝐹. 𝐼𝑓 𝑤𝑒 𝑐𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑡𝑤𝑜 2 × 2 𝑚𝑎𝑡𝑟𝑖𝑥 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐴 =( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸11(𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸12(𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸21 (𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸22 (𝑥)) ) and 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐵=( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹11(𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹12(𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝑦)) ) 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐴 # 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐵 = ( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸11 (𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸12 (𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸21 (𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸22 (𝑥)) ) # Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2951 https://internationalpubls.com ( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹11(𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹12(𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝑦)) ) 𝑋11 =(𝑁𝑜𝑟𝑚𝜗𝐸11 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸11(𝑥)) # (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹11(𝑦)) 𝑋12 =(𝑁𝑜𝑟𝑚𝜗𝐸12 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸12(𝑥)) # (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹12(𝑦)) 𝑋21 =(𝑁𝑜𝑟𝑚𝜗𝐸21 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸21 (𝑥)) # (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝑦)) 𝑋22 =(𝑁𝑜𝑟𝑚𝜗𝐸22 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸22 (𝑥)) # (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝑦)) 𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝐴𝐸 ̅̅̅̅ # 𝐵𝐹 ̅̅̅̅ = {〈x, y〉, 2𝜗𝐴(x) .𝜗𝐵(y) 𝜗𝐴(x)+𝜗𝐵(y) , 2µ𝐴 (x) .µ𝐵(y) µ𝐴 (x)+ µ𝐵(y) ∶ xE, y𝐹 } 𝑋11 = ( 2𝑁𝑜𝑟𝑚𝜗𝐸11 (𝑥). 𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑦) 𝑁𝑜𝑟𝑚𝜗𝐸11 (𝑥) + 𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑦) , 2𝑁𝑜𝑟𝑚µ𝐸11 (𝑥). 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝑦) 𝑁𝑜𝑟𝑚µ𝐸11 (𝑥) + 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝑦) ) 𝑋12 = ( 2𝑁𝑜𝑟𝑚𝜗𝐸12 (𝑥). 𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑦) 𝑁𝑜𝑟𝑚𝜗𝐸12 (𝑥) + 𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑦) , 2𝑁𝑜𝑟𝑚µ𝐸12 (𝑥). 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝑦) 𝑁𝑜𝑟𝑚µ𝐸12 (𝑥) + 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝑦) ) 𝑋21 = ( 2𝑁𝑜𝑟𝑚𝜗𝐸21 (𝑥). 𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑦) 𝑁𝑜𝑟𝑚𝜗𝐸21 (𝑥) + 𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑦) , 2𝑁𝑜𝑟𝑚µ𝐸21 (𝑥). 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝑦) 𝑁𝑜𝑟𝑚µ𝐸21 (𝑥) + 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝑦) ) 𝑋22 = ( 2𝑁𝑜𝑟𝑚𝜗𝐸22 (𝑥). 𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑦) 𝑁𝑜𝑟𝑚𝜗𝐸22 (𝑥) + 𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑦) , 2𝑁𝑜𝑟𝑚µ𝐸22 (𝑥). 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝑦) 𝑁𝑜𝑟𝑚µ𝐸22 (𝑥) + 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝑦) ) We have Norm𝐴̅ 𝐸# 𝑁𝑜𝑟𝑚�̅�𝐹 = ( 𝑋11 𝑋12 𝑋21 𝑋22 ) is also intuitionistic fuzzy matrix Example: If 𝑨𝑬=( (.9, .4) (.4, .1) (.5, .3) (.8, .3) ) and𝑩𝑭 = ( (.8, .6) (.6, .3) (.7, .5) (.9, .4) ) are two intuitionistic fuzzy matrices over different universe E and F then Norm𝐴̅ 𝐸$𝑁𝑜𝑟𝑚�̅�𝐹is also fuzzy matrix set. Proof: Given 𝑨𝑬 =( (.9, .4) (.4, .1) (.5, .3) (.8, .3) ) and 𝑩𝑭 = ( (.8, .6) (.6, .3) (.7, .5) (.9, .4) ) 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐴 =( ( .8,1) (0, .3) (.2, .8) (.7, .8) ) and 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐵=( (.5, 1) (0, .5) (.3, .8) (.8, .7) ) 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐴# 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ ̅ 𝐵 = ( ( .8,1) (0, .3) (.2, .8) (.7, .8) ) # ( (.5, 1) (0, .5) (.3, .8) (.8, .7) ) 𝑋11 = ( .8, 1)#(. 5, 1) 𝑋12 = (0, .3) #(0, .5) 𝑋21 = (. 2, .8)# (.3, .8) 𝑋22 = (. 7, .8)#(.8, .7) 𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑓𝑜𝑟𝑚𝑢𝑙𝑎𝐴𝐸 ̅̅̅̅ # 𝐵𝐹 ̅̅̅̅ = {〈x, y〉, 2𝜗𝐴(x) .𝜗𝐵(y) 𝜗𝐴(x)+𝜗𝐵(y) , 2µ𝐴 (x) .µ𝐵(y) µ𝐴 (x)+ µ𝐵(y) ∶ xE, y𝐹 } 𝑋11 = ( 2(. 8). (.5) (. 8) + (.5) , 2(1) . (1) 1 + 1 ) 𝑋12 = ( 2(0). (0) (0) + (0) , 2(.3) . (5) (.3) + (.5) ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2952 https://internationalpubls.com 𝑋21 = ( 2(. 2). (.3) (. 2) + (.3) , 2(.8) . (.8) (.8) + (.8) ) 𝑋22 = ( 2(. 7). (.8) (. 7) + (.8) , 2(.8) . (7) (.8) + (.7) ) 𝑋11 ={.6 , 1} 𝑋12 ={0 , .4} 𝑋21 ={.2, .8} 𝑋22 ={.7 ,.7} We have Norm𝐴̅ 𝐸$ 𝑁𝑜𝑟𝑚�̅�𝐹 = ( (.6, .1) (0 , .4) (.2, .8) (.7 , .7) ) is also intuitionistic fuzzy matrix. Theorem5.2.8: If E and F be two universal sets. For every normalization of an intuitionistic fuzzy matrices 𝐴𝐸 ̅̅̅̅ 𝑎𝑛𝑑 𝐵𝐹 ̅̅̅̅ are in E and F then Norm𝐴̅ 𝐸 ∗ 𝑁𝑜𝑟𝑚�̅�𝐹is also normalization of an intuitionistic fuzzy matrices. Proof: If 𝑁𝑜𝑟𝑚𝐴 and 𝑁𝑜𝑟𝑚𝐵 are two normalization intuitionistic fuzzy matrices over different unive𝑟𝑠𝑒𝑠 𝑠𝑒𝑡𝑠 𝐸 𝑎𝑛𝑑 𝐹. 𝐼𝑓 𝑤𝑒 𝑐𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑡𝑤𝑜 2 × 2 𝑚𝑎𝑡𝑟𝑖𝑥 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐴 =( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸11(𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸12(𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸21 (𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸22 (𝑥)) ) and 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐵=( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹11(𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹12(𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝑦)) ) 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐴 ∗ 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐵 = ( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸11 (𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸12 (𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸21 (𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸22 (𝑥)) ) ∗ ( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹11(𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹12(𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝑦)) ) 𝑋11 =(𝑁𝑜𝑟𝑚𝜗𝐸11 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸11(𝑥)) ∗ (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹11(𝑦)) 𝑋12 =(𝑁𝑜𝑟𝑚𝜗𝐸12 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸12(𝑥)) ∗ (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹12(𝑦)) 𝑋21 =(𝑁𝑜𝑟𝑚𝜗𝐸21 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸21 (𝑥)) ∗ (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝑦)) 𝑋22 =(𝑁𝑜𝑟𝑚𝜗𝐸22 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸22 (𝑥)) ∗ (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝑦)) 𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝐴𝐸 ̅̅̅̅ ∗ 𝐵𝐹 ̅̅̅̅ = {〈x, y〉, 𝜗𝐴(x)+𝜗𝐵(y) 2(𝜗𝐴(x).𝜗𝐵(y)+1) , µ𝐴 (x)+ µ𝐵(y) 2(µ𝐴(x).µ𝐵(y)+1) : xE, y𝐹 } 𝑋11 = ( 𝑁𝑜𝑟𝑚𝜗𝐸11 (𝑥) + 𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑦) 2(𝑁𝑜𝑟𝑚𝜗𝐸11 (𝑥). 𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑦) + 1) , 𝑁𝑜𝑟𝑚µ𝐸11 (𝑥) + 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝑦) 2(𝑁𝑜𝑟𝑚µ𝐸11 (𝑥). 𝑁𝑜𝑟𝑚𝛾𝐹11 (𝑦) + 1) ) 𝑋12 = ( 𝑁𝑜𝑟𝑚𝜗𝐸12 (𝑥) + 𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑦) 2(𝑁𝑜𝑟𝑚𝜗𝐸12 (𝑥). 𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑦) + 1) , 𝑁𝑜𝑟𝑚µ𝐸12 (𝑥) + 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝑦) 2(𝑁𝑜𝑟𝑚µ𝐸12 (𝑥). 𝑁𝑜𝑟𝑚𝛾𝐹12 (𝑦) + 1) ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2953 https://internationalpubls.com 𝑋21 = ( 𝑁𝑜𝑟𝑚𝜗𝐸21 (𝑥) + 𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑦) 2(𝑁𝑜𝑟𝑚𝜗𝐸21 (𝑥). 𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑦) + 1) , 𝑁𝑜𝑟𝑚µ𝐸21 (𝑥) + 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝑦) 2(𝑁𝑜𝑟𝑚µ𝐸21 (𝑥). 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝑦) + 1) ) 𝑋22 = ( 𝑁𝑜𝑟𝑚𝜗𝐸22 (𝑥) + 𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑦) 2(𝑁𝑜𝑟𝑚𝜗𝐸11 (𝑥). 𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑦) + 1) , 𝑁𝑜𝑟𝑚µ𝐸22 (𝑥) + 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝑦) 2(𝑁𝑜𝑟𝑚µ𝐸22 (𝑥). 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝑦) + 1) ) We have Norm𝐴̅ 𝐸 ∗ 𝑁𝑜𝑟𝑚�̅�𝐹 = ( 𝑋11 𝑋12 𝑋21 𝑋22 ) is also intuitionistic fuzzy matrix Example: If 𝑨𝑬=( (.9, .4) (.4, .1) (.5, .3) (.8, .3) ) and𝑩𝑭 = ( (.8, .6) (.6, .3) (.7, .5) (.9, .4) ) are two intuitionistic fuzzy matrices over different universe E and F then Norm𝐴̅ 𝐸 ∗ 𝑁𝑜𝑟𝑚�̅�𝐹is also fuzzy matrix set. Proof: Given 𝑨𝑬 =( (.9, .4) (.4, .1) (.5, .3) (.8, .3) ) and 𝑩𝑭 = ( (.8, .6) (.6, .3) (.7, .5) (.9, .4) ) 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐴 =( ( .8,1) (0, .3) (.2, .8) (.7, .8) ) and 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐵=( (.5, 1) (0, .5) (.3, .8) (.8, .7) ) 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐴 ∗ 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ ̅ 𝐵 = ( ( .8,1) (0, .3) (.2, .8) (.7, .8) ) ∗ ( (.5, 1) (0, .5) (.3, .8) (.8, .7) ) 𝑋11 = ( .8, 1) ∗ (. 5, 1) 𝑋12 = (0, .3) ∗ (0, .5) 𝑋21 = (.2, .8) ∗ (.3, .8) 𝑋22 = (. 7, .8) ∗ (.8, .7) 𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝐴𝐸 ̅̅̅̅ ∗ 𝐵𝐹 ̅̅̅̅ = {〈x, y〉, 𝜗𝐴(x)+𝜗𝐵(y) 2(𝜗𝐴(x).𝜗𝐵(y)+1) , µ𝐴 (x)+ µ𝐵(y) 2(µ𝐴(x).µ𝐵(y)+1) : xE, y𝐹 } 𝑋11 = ( (. 8) + (.5) 2((. 8). (. 5) + 1) , (1) + (1) 2(1.1 + 1) ) 𝑋12 = ( (0) + (0) 2((0). (0) + 1) , (. 3) + (.5) 2(.3 .5 + 1) ) 𝑋21 = ( (. 2) + (.3) 2((. 2). (. 3) + 1) , (. 8) + (.8) 2(.8 .8 + 1) ) 𝑋22 = ( (. 7) + (.8) 2((. 7). (. 8) + 1) , (. 8) + (.7) 2(.8 .7 + 1) ) 𝑋11 ={.5 , .5} 𝑋12 ={0 , .3} 𝑋21 ={.2, .5} 𝑋22 ={.5 ,.5} We have Norm𝐴̅ 𝐸 ∗ 𝑁𝑜𝑟𝑚�̅�𝐹 = ( (.5, .5) (0 , .3) (.2, .5) (.5 , .5) ) is also intuitionistic fuzzy matrix. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2954 https://internationalpubls.com Theorem 5.2.9: If E and F be two universal sets. For every normalization of an intuitionistic fuzzy matrices 𝐴𝐸 ̅̅̅̅ 𝑎𝑛𝑑 𝐵𝐹 ̅̅̅̅ are in E and F then Norm𝐴̅ 𝐸∆ 𝑁𝑜𝑟𝑚�̅�𝐹is also normalization of an intuitionistic fuzzy matrices. Proof: If 𝑁𝑜𝑟𝑚𝐴 and 𝑁𝑜𝑟𝑚𝐵 are two normalization intuitionistic fuzzy matrices over different unive𝑟𝑠𝑒𝑠 𝑠𝑒𝑡𝑠 𝐸 𝑎𝑛𝑑 𝐹. 𝐼𝑓 𝑤𝑒 𝑐𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑡𝑤𝑜 2 × 2 𝑚𝑎𝑡𝑟𝑖𝑥 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐴 =( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸11(𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸12(𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸21 (𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸22 (𝑥)) ) and 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐵=( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹11(𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹12(𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝑦)) ) 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐴 ∆ 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐵 = ( (𝑁𝑜𝑟𝑚𝜗𝐸11 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸11 (𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸12 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸12 (𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸21 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸21 (𝑥)) (𝑁𝑜𝑟𝑚𝜗𝐸22 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸22 (𝑥)) ) ∆ ( (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹11(𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹12(𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝑦)) (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝑦)) ) 𝑋11 =(𝑁𝑜𝑟𝑚𝜗𝐸11 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸11(𝑥)) ∆ (𝑁𝑜𝑟𝑚𝛿𝐹11 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹11(𝑦)) 𝑋12 =(𝑁𝑜𝑟𝑚𝜗𝐸12 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸12(𝑥)) ∆ (𝑁𝑜𝑟𝑚𝛿𝐹12 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹12(𝑦)) 𝑋21 =(𝑁𝑜𝑟𝑚𝜗𝐸21 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸21 (𝑥)) ∆ (𝑁𝑜𝑟𝑚𝛿𝐹21 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹21 (𝑦)) 𝑋22 =(𝑁𝑜𝑟𝑚𝜗𝐸22 (𝑥) 𝑁𝑜𝑟𝑚µ𝐸22 (𝑥)) ∆ (𝑁𝑜𝑟𝑚𝛿𝐹22 (𝑦) 𝑁𝑜𝑟𝑚𝛾𝐹22 (𝑦)) 𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝐴𝐸 ̅̅̅̅  𝐵𝐹 ̅̅̅̅ = {〈x, y〉, 𝜗𝐴(x)+𝜗𝐵(y) µ𝐴 (x)+ µ𝐵(y)+𝜗𝐴(x)+𝜗𝐵(y) , µ𝐴 (x)+ µ𝐵(y) µ𝐴 (x)+ µ𝐵(y)+𝜗𝐴(x)+𝜗𝐵(y) : xE, y𝐹 } X11 = NormϑE11 (x)+NormδF11 (y) NormµE11 (x)+NormγF11 (y)+NormϑE11 (x)+NormδF11 (y) , NormµE11 (x)+NormγF11 (y) NormµE11 (x)+NormγF11 (y)+NormϑE11 (x)+NormδF11 (y) X12 = NormϑE12 (x)+NormδF12 (y) NormµE12 (x)+NormγF12 (y)+NormϑE12 (x)+NormδF12 (y) , NormµE12 (x)+NormγF12 (y) NormµE12 (x)+NormγF12 (y)+NormϑE12 (x)+NormδF12 (y) X21 = Normϑ21 (x)+NormδF21 (y) NormµE21 (x)+NormγF21 (y)+NormϑE21 (x)+NormδF21 (y) , NormµE21 (x)+NormγF21 (y) NormµE21 (x)+NormγF21 (y)+NormϑE21 (x)+NormδF21 (y) X22 = NormϑE22 (x)+NormδF22 (y) NormµE22 (x)+NormγF22 (y)+NormϑE22 (x)+NormδF22 (y) , NormµE22 (x)+NormγF22 (y) NormµE22 (x)+NormγF22 (y)+NormϑE22 (x)+NormδF22 (y) We have Norm𝐴̅ 𝐸 𝑁𝑜𝑟𝑚�̅�𝐹 = ( 𝑋11 𝑋12 𝑋21 𝑋22 ) is also intuitionistic fuzzy matrix. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2955 https://internationalpubls.com Example: If 𝑨𝑬=( (.9, .4) (.4, .1) (.5, .3) (.8, .3) ) and𝑩𝑭 = ( (.8, .6) (.6, .3) (.7, .5) (.9, .4) ) are two intuitionistic fuzzy matrices over different universe E and F then Norm𝐴̅ 𝐸𝑁𝑜𝑟𝑚�̅�𝐹is also fuzzy matrix set. Proof: Given 𝑨𝑬 =( (.9, .4) (.4, .1) (.5, .3) (.8, .3) ) and 𝑩𝑭 = ( (.8, .6) (.6, .3) (.7, .5) (.9, .4) ) 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐴 =( ( .8,1) (0, .3) (.2, .8) (.7, .8) ) and 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐵=( (.5, 1) (0, .5) (.3, .8) (.8, .7) ) 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ 𝐴 𝑁𝑜𝑟𝑚̅̅ ̅̅ ̅̅ ̅̅ ̅ 𝐵 = ( ( .8,1) (0, .3) (.2, .8) (.7, .8) ) ( (.5, 1) (0, .5) (.3, .8) (.8, .7) ) 𝑋11 = ( .8, 1)(. 5, 1) 𝑋12 = (0, .3)(0, .5) 𝑋21 = (.2, .8) (.3, .8) 𝑋22 = (. 7, .8)(.8, .7) 𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝐴𝐸 ̅̅̅̅  𝐵𝐹 ̅̅̅̅ = {〈x, y〉, 𝜗𝐴(x)+𝜗𝐵(y) µ𝐴 (x)+ µ𝐵(y)+𝜗𝐴(x)+𝜗𝐵(y) , µ𝐴 (x)+ µ𝐵(y) µ𝐴 (x)+ µ𝐵(y)+𝜗𝐴(x)+𝜗𝐵(y) : xE, y𝐹 } 𝑋11 = (.8)+(.5) (1)+(1)+(.8)+(.5) , (1)+(1) (1)+(1)+(.8)+(.5) 𝑋12 = (0)+(0) (.3)+(.5)+(0)+(0) , (.3)+(.5) (.3)+(.5)+(0)+(0) 𝑋21 = (.2)+(.3) (.8)+(.8)+(.2)+(.3) , (.8)+(.8) (.8)+(.8)+(.2)+(.3) 𝑋22 = (.7)+(.8) (.8)+(.7)+(.7)+(.8) , (.8)+(.7) (.8)+(.7)+(.7)+(.8) 𝑋11 ={.4 , .6} 𝑋12 ={0 , 1} 𝑋21 ={.2, .8} 𝑋22 ={.5 ,.5} We have Norm𝐴̅ 𝐸 𝑁𝑜𝑟𝑚�̅�𝐹 = ( (.4, .6) (0 , .1) (.2, .8) (.5 , .5) ) is also intuitionistic fuzzy matrix. 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