Acta Polytechnica https://doi.org/10.14311/AP.2023.63.0171 Acta Polytechnica 63(3):171–178, 2023 © 2023 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague QUALITATIVE SIGN STABILITY OF LINEAR TIME INVARIANT DESCRIPTOR SYSTEMS Madhusmita Chanda, Mamoni Paitandia, Mahendra Kumar Guptaa, b, ∗ a National Institute of Technology Jamshedpur, Department of Mathematics, 831014 Jamshedpur, India b Indian Institute of Technology Bhubaneswar, School of Basic Sciences, Khordha, 752050 Odisha, India ∗ corresponding author: mkgupta@iitbbs.ac.in Abstract. This article discusses assessing the instability of a continuous linear homogeneous time- invariant descriptor system. Some necessary conditions and sufficient conditions are derived to establish the stability of a matrix pair by the fundamentals of qualitative ecological principles. The proposed conditions are derived using only the qualitative (sign) information of the matrix pair elements. Based on these conditions, the instability of a matrix pair can easily be determined, without any magnitude information of the matrix pair elements and without numerical eigenvalues calculations. With the proposed theory, Magnitude Dependent Stable, Magnitude Dependent Unstable, and Qualitative Sign Stable matrix pairs can be distinguished. The consequences of the proposed conditions and some illustrative examples are discussed. Keywords: Descriptor systems, stability of a matrix pair, qualitative sign instability, interactions and interconnections, characteristic polynomial. 1. Introduction The concept of stability of a matrix and a matrix pair is very fundamental to the control theory, and it is an important property to be analysed for all practical control systems. A continuous homogeneous linear time-invariant descriptor system, i.e. differential alge- braic equations (DAEs) can be written as: Eẋ(t) = Ax(t) , (1) where x(t) ∈ Rn is the state vector and E, A ∈ Rn×n are the constant matrices [1]. When E = I (identity matrix), system (1) is well known as a state space system. System (1) is called regular if det(λE − A) is not identically zero as a polynomial of λ [2, 3]. A regular system (1) is said to be stable if and only if the matrix pair (E, A) is a stable matrix pair, i.e. all of its eigenvalues have negative real parts. In or- der to find the eigenvalues of the matrix pair (E, A), we have to determine the roots of the characteristic equation det(λE − A) = 0. It is remarkable that when matrix E is singular, the number of eigenval- ues of the matrix pair (E, A) is less than n. This numerical eigenvalue calculation of a matrix pair of a higher order is a computationally intensive effort. To overcome this drawback, economists have introduced the concept of ‘qualitative stability’ and ecologists have derived some necessary and sufficient conditions for the stability of a matrix using only the sign in- formation of matrix elements. Nonetheless, in the literature, this problem is addressed only for state space systems, i.e. when E = I, where eigenvalues of only matrix A are checked. This paper extends these results for checking the eigenvalues of matrix pair (E, A). In this paper, the word ‘quantitative’ is used for both magnitude and sign information, and the word ‘qualitative’ strictly for the sign information with no magnitude information of matrix elements. Matrix pairs which are stable, independent of their magnitudes with only sign information are denoted as Qualitative Sign Stable (QLSS) matrix pairs and Qualitative Sign Unstable are denoted as ‘QLSU’ ma- trix pairs. Matrix pairs, whose stability/instability depend upon the magnitude information of the matrix pair elements, are denoted as Magnitude Dependent Stable/Unstable (MDS/U) matrix pairs. With the knowledge of qualitative sign structure, we can now discuss the stability of a matrix pair. The analysis of stability of matrices has evoked various research directions. The way non-engineers, such as ecologists and economists, have tackled this problem without even having any magnitude informa- tion is fascinating. In [4], the stability problem of a matrix is studied in a purely qualitative environment assuming that quantitative information is unavailable. Article [5] provides some sufficient conditions for the qualitative stability of an ecosystem by simply conclud- ing the mutual qualitative effects on member species via signed digraphs, whereas necessary conditions for the qualitative stability are presented in [6]. In [7], linear systems are studied based on the qualitative theory. Some conditions concerning the structural qualitative stability of a system are proposed in [8]. A graph-theoretic analysis based on sign patterns of a real square matrix is used to conclude the stability of a linear system in [9]. In [10], it has been shown that in a complex ecological system, when species in- teract as predator-prey, the system can still be stable. In [11, 12], ecological-sign stability pricnciples of a matrix are transformed into mathematical principles 171 https://doi.org/10.14311/AP.2023.63.0171 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en M. Chand, M. Paitandi, M. K. Gupta Acta Polytechnica to encounter stability problems in engineering control systems. The qualitative analysis of control systems is explained in [13]. The stability of the continuous-time linear state space system is explained in [14] and the stability of discrete-time system is explained in [15]. The series of papers [16], [17], and [18] were attempts to find the conditions for stability/instability of real matrices using qualitative reasoning. In [16], few con- ditions for qualitative sign instability of a matrix are derived in terms of the nature of interactions and in- terconnections, taken from ecological principles. The stability analysis of a matrix using these conditions requires only the qualitative (sign) information of the matrix elements (no need for any quantitative infor- mation). In [17], an alternative sufficient condition is proposed by combining the concepts of both quantita- tive (magnitude and sign) as well as the qualitative (only sign) information of the matrix elements. This condition possesses a convexity promotion property with respect to stability. A new necessary and suffi- cient condition is proposed in [18], for the stability of any real matrix that does not need the informa- tion of the characteristic polynomial, and it is based on matrix entries’ sign information only. Article [19] studies asymptotic stability criteria for time-delayed systems. Remarkable works have been done in [20, 21] on matrices with stable sign patterns. However, all the existing research is focused on the stability of a matrix confining its utility to only linear normal state space systems, and improving on these papers, this paper generalises some of these conditions for qualita- tive stability/instability of a matrix pair that have a relatively broader scope in the analysis of linear square descriptor systems in control engineering. To the best of our knowledge, this is the first work discussing the qualitative sign stability for a matrix pair. For a proper perspective, let us consider the follow- ing matrix pairs (Ei, Ai), i = 1, 2: E1 =   −3 −2 0.1 0.4 −1 0.9 −0.3 −8 −0.7 5 3 1 0.1 2 1.4 5   , A1 =   2 −1 −3 0.5 0.3 −1 0.4 −2 −0.1 3 −0.5 4 −2 −0.2 1 0.7   , E2 =   1 −0.2 0.8 −3.5 −4.1 1.9 −2.7 −3.3 4 −0.1 −1.2 −0.4 2 −0.5 −4.8 −0.3 −1.7 −4.3 −3.7 0.2 0.6 −2.5 0.9 −5.2 −2.3   , A2 =   −1 0.2 −3 −0.7 5 −0.9 1.3 2.4 −1.2 −3.1 4.8 −1.1 −2 −0.4 4.3 2.7 1 −1.8 1.7 −0.7 −5 −3.2 0.3 −2.9 −1.5   . It is very much difficult to decide the stabil- ity/instability of the above matrix pairs with the numerical eigenvalue calculation. But with the neces- sary and sufficient conditions presented in this paper, we can conclude that the matrix pair (E1, A1) is a Qualitative Sign Unstable (QLSU) matrix pair and the matrix pair (E2, A2) is an MDS/U matrix pair. This is concluded just by a simple visualisation of the nature of interactions and interconnections of the matrix pair. Note that, the matrix pairs of order 2 do not have any interconnection terms and thus are trivial for our studies. Hence, we focus on matrix pairs of order 3 and higher. In the next section, the matrix pair elemental sign structures are briefly reviewed and few basic ‘Qualita- tive Sign Matrix Pair Indices’ are developed for for- mulating the conditions for qualitative sign instability. In Section 3, the necessary and sufficient conditions for qualitative instability are proposed, which is the main result of this paper. Section 4 discusses the implications of these conditions and illustrates few examples for a clear visualisation of the importance of qualitative stability. In Section 5, we discuss the conclusions drawn from this paper. 2. Qualitative Sign Matrix Pair Indices For assessing the stability, first, we have to visualise an n×n matrix pair in the following structured way. Just for a simplified view, let us illustrate the structure using a 4 × 4 matrix pair: E =   e11 e12 e13 e14 e21 e22 e23 e24 e31 e32 e33 e34 e41 e42 e43 e44   and A =   a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34 a41 a42 a43 a44   . The entire Matrix Pair Sign Structure is completely specified by the diagonal elements and the off-diagonal link structures (Interactions and Interconnections). This matrix pair consists of: • Diagonal elements: eii and aii, • Interactions of the form eij eji and aij aji, • Interactions of the form eij aji, • Interconnections of the form bij bjk . . . bmi, where bij = eij or aij . Now, we look at the signs of the entries and use the following sign convention in the rest of this paper. We use the letter ‘P’ for the ‘+’ (positive) sign, the letter ‘N’ for the ‘−’ (negative) sign, and ‘0’ for the zero entry. We label the interactions of the matrix pair 172 vol. 63 no. 3/2023 Qualitative Sign Stability of Linear Time Invariant Descriptor systems Notations Np number of positive diagonal elements Nz number of zero diagonal elements Nng number of negative diagonal elements Npz number of non-negative diagonal ele- ments Table 1. Notations for number of diagonal elements. using this sign convention. The possible off-diagonal links or interactions of a matrix pair are: • Mutualism link: (PP) link, • Competition link: (NN) link, • Predation-Prey, Prey-Predation links: (PN) link and (NP) link, • Ammensalism link: (N0) link and (0N) link, • Commensalism link: (P0) link and (0P) link, • Null link: (00) link. We further categorise these links in the following way. All the Mutualism (PP) links and the Competition (NN) links are collectively labeled as ‘Same Sign (SS) links’. All the Predation-Prey (PN) links and the Prey-Predation (NP) links are collectively labeled as ‘Opposite Sign (OS) links’. Similarly, all the Ammen- salism (N0 and 0N) links and the Commensalism (P0 and 0P) links are collectively labeled as ‘Zero Sign (ZS) links’. Finally, the Null (00) links are labeled as ‘Zero Zero’ (ZZ) links. For a ‘structural zero link’, we label them as SZZ links and for Elemental Zero links, we label them as EZZ links. • Same Sign (SS) links: PP (++) links and NN (−−) links, • Opposite Sign (OS) links: PN (+−) links and NP (−+) links, • Zero Sign (ZS) links: N0 links, 0N links, P0 links, and 0P links, • Zero Zero (ZZ) links: 00 links. Based on this sign convention, the matrix pair ele- mental sign structure of an n × n matrix pair are elaborated in the following ways: 2.1. Diagonal Elements eii and aii The number of diagonal elements of different signs is mentioned in Table 1. The total number of diagonal elements of the matrix pair is 2n. The number of non-negative diagonal elements can be written as: Npz = Np + Nz . Let us assume that Npz is not zero and define: ηpz = Npz 2n , ηng = Nng 2n . Notations Ntl total number of links of E and A Nss total number of SS links of E and A Nzs total number of ZS links of E and A Nos total number of OS links of E and A Nszz total number of SZZ links of E and A Nezz total number of EZZ links of E and A Nlc total number of active links of E and A Ngood number of ‘Good’ links of E and A Nbad number of ‘Bad’ links of E and A Table 2. Notations for number of links of matrices E and A. ∴ ηpz + ηng = 1 . From an ecological perspective, the information about how a species affects itself is provided by the diagonal elements. The positive sign signifies that the species helps to increase its own population, zero signifies that the species has no effect on itself, and the negative sign signifies that it is self-regulatory. That is why Elemental + (positive) and 0 (zero) signs are considered as ‘Bad’ signs and Elemental − (negative) signs are considered as ‘Good’ signs in a row or column of a matrix pair [11]. 2.2. Interactions of the form eij eji and aij aji All the products of off-diagonal elements of the form eij eji connecting only two distinct nodes (indices) of matrix E are known as Interactions of matrix E. Similarly, all the products of off-diagonal elements of the form aij aji connecting only two distinct nodes (indices) of matrix A are known as Interactions of matrix A. Table 2 includes information on the number of dif- ferent links of matrices E and A. The total number of links (interactions) of this form is: Ntl = [1 + 2 + 3 + . . . + (n − 1)] × 2 = n(n − 1) and it can be expressed as: Ntl = Nss + Nzs + Nos + Nszz + Nezz . (2) We now take out the structural zero links from any further discussion [16] and denote the total number of ‘active links’ as Nlc. Thus: Nlc = Nss + Nzs + Nos + Nezz . (3) From an ecological viewpoint, it is noted that Same Sign (SS) links (i.e. PP and NN links) of this form are highly detrimental to stability whereas, Opposite Sign (OS) links (i.e. PN and NP links) of this form are conducive to stability [10, 22]. So, from the stability 173 M. Chand, M. Paitandi, M. K. Gupta Acta Polytechnica Notations N ′ tl total number of links of (E, A) N ′ ss total number of SS links of (E, A) N ′ zs total number of ZS links of (E, A) N ′ os total number of OS links of (E, A) N ′ szz total number of SZZ links of (E, A) N ′ ezz total number of EZZ links of (E, A) N ′ lc total number of active links of (E, A) N ′ good number of ‘Good’ links of (E, A) N ′ bad number of ‘Bad’ links of (E, A) Table 3. Notations for number of links of matrix pair (E, A). point of view, SS links, ZS links, and ZZ links of this form are considered as ‘Bad’ links, and OS links of this form are considered as ‘Good’ links. Hence: Ngood = Nos , (4) Nbad = Nss + Nzs + Nezz . (5) Let us define: ηbad = Nbad Nlc ηgood = Ngood Nlc ∴ ηbad + ηgood = 1 . Remark 1. The above indices are already defined for matrices. Now we are, for the first time, going to define another form of interaction and chain for matrix pairs. 2.3. Interactions of the form eij aji All the products of off-diagonal elements of the form eij aji connecting one node of matrix E and the corre- sponding node of matrix A are known as Interactions of the matrix pair (E, A). Table 3 lists the number of different links of matrix pair (E, A). The total number of links (interactions) of this form is: N ′ tl = n(n − 1) and it can be expressed as: N ′ tl = N ′ ss + N ′ zs + N ′ os + N ′ szz + N ′ ezz . (6) We now take out the structural zero links from any further discussion and denote the total number of ‘active links’ as N ′ lc. Thus: N ′ lc = N ′ ss + N ′ zs + N ′ os + N ′ ezz . (7) In ecological literature, it is realised that Same Sign (SS) links (i.e. PP and NN links) of this form are con- ducive to stability whereas Opposite Sign (OS) links (i.e. PN and NP links) of this form are detrimental to stability. So, SS links and ZS links of this form are considered as ‘Good’ links, and OS links of this form are considered as ‘Bad’ links. Hence: N ′ good = N ′ ss + N ′ zs + N ′ ezz , (8) N ′ bad = N ′ os . (9) Let us define: η ′ bad = N ′ bad N ′ lc η ′ good = N ′ good N ′ lc ∴ η ′ bad + η ′ good = 1 . Let us define ζbad as Potentially Destabilising Sign Matrix Pair Index and ζgood as the Potentially Stabil- ising Sign Matrix Pair Index. So: ζgood = ηng + ηgood + η ′ good , (10) ζbad = ηpz + ηbad + η ′ bad . (11) Also define ζnet as the Net Matrix Pair Stabilisation Index given by ζnet = ζgood − ζbad . (12) Let us define the index, known as ‘Chain’. The elemental structure of the form eij ejiaij aji is called a ‘Chain’. The chain containing at least three ‘+’ sign is known as ‘+ chain’ and the chain containing at least three ‘−’ sign is known as ‘− chain’. Using these ‘Qualitative Sign Matrix Pair Indices’, we discuss the qualitative stability of a matrix pair and derive few conditions for the qualitative sign in- stability. 3. Conditions for Qualitative Sign Stability and Instability In this section, the main results are presented. Here we focus on the case of matrix pairs with diagonal elements containing only a mixture of positive and neg- ative elements, i.e. with Npz = Np. Also, we assume that Nbad = Nss and N ′ good = N ′ ss. A series of neces- sary and sufficient conditions for stability/instability of a matrix pair are presented here. 174 vol. 63 no. 3/2023 Qualitative Sign Stability of Linear Time Invariant Descriptor systems 3.1. Necessary Conditions for Qualitative Stability of Matrix Pair For qualitative reasoning, we need to know the num- ber and nature of the above-defined interactions and interconnections. But it is not possible to find the number and nature of all the interconnection terms of a matrix pair. So, for the qualitative sign stability, all the interconnection terms of a matrix pair need to be zero. Furthermore, any matrix pair (E, A) with det(A) = 0 has always at least one zero eigenvalue, that makes the matrix pair unstable. Hence, the non- singularity of matrix A is also necessarily required for QLSS matrix pair. Let us consider a matrix pair (E, A) with entries eij and aij , respectively. Based on the above discus- sion, we are enlisting the two important ‘necessary’ conditions for ‘qualitative sign stability’: • C1: bij bjk . . . bqr bri = 0, where bij = eij or aij for any sequences of three or more distinct indices i, j, k, . . . , q, r. • C2: det(A) ̸= 0. Here, C1 is the necessarily required condition for Qualitative Sign Stability, i.e. the condition which makes the matrix pair stable, independent of magni- tudes, and C2 is the necessarily required condition for the stability of the matrix pair (E, A), independent of any qualitative or quantitative information of the matrix pair elements. More details about the concept of Qualitative Sign Stability is discussed in [11, 14]. The Net Matrix Pair Stabilisation Index ζnet serves as an index to indicate the likelihood of matrix pair being stable or unstable in a Qualitative way. Negative values indicate that the matrix pair is more likely to be unstable, and positive values indicate that the matrix pair is more likely to be stable. The higher the value, the higher the probability, see [16]. The qualitative sign stability of a matrix pair depends upon the stabilising strength of the matrix pair. When the matrix pair is more potentially stabilised, then ζnet is non-negative, and when it is more potentially destabilised, then ζnet is negative. So, the qualitative stability is connected with the value of ζnet. With this observation, we have the following condi- tions: • ζnet varies in an interval given by −3 ≤ ζnet ≤ 3. • ζnet < 0 specifies that the matrix pair is MDU, that means for these MDU matrix pairs, there al- ways exist magnitudes that make this matrix pair unstable. • ζnet ≥ 0 specifies that the matrix pair is MDS, that means for these MDS matrix pairs, there always exist magnitudes that make this matrix pair stable. • A given (non-QLSS) matrix pair is QLSU only if −3 ≤ ζnet < 0 (Necessary condition for QLSU). A matrix pair, which is not Qualitative Sign Stable (QLSS), is said to be a non-QLSS matrix pair. 3.2. A Necessary Condition for Instability of a Matrix In a matrix pair (E, A), if we substitute E by the Identity matrix I, then the above necessary condition for QLSU will be the same as that of the matrix A. With the fundamentals of ecological principles, the matrix pair (I, A) is visualised in the following structured way. Let us illustrate the structure by using a 4 × 4 matrix pair. I =   1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1   , A =   a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34 a41 a42 a43 a44   According to the qualitative stability concept of matrix pair: ηpz = n + Npz (A) 2n = 1 2 + Npz (A) 2n ηng = Nng (A) 2n ηbad = Nlc(I) + Nbad(A) Nlc = 1 2 + Nbad(A) Nlc ηgood = Ngood(A) Nlc η ′ bad = 0 η ′ good = 1 ∴ ζnet = ζgood − ζbad = 1 2 [ Nng (A) n + Ngood(A) Nlc(A) − Npz (A) n − Nbad(A) Nlc(A) ] = 1 2 [ζgood(A) − ζbad(A)] = 1 2 ζnet(A) . Since the matrix pair (I, A) is QLSU only if ζnet < 0, therefore, the matrix A is QLSU only if ζnet(A) < 0. This is the necessary condition for a matrix to be QLSU, which is a particular case of a matrix pair. The necessary and sufficient conditions of a matrix to be QLSU are discussed extensively in [16]. In this paper, we generalise the Identity matrix I to any matrix E and propose few necessary and sufficient conditions for the qualitative sign instability for a matrix pair. 3.3. A Necessary Condition for a Matrix Pair to be QLSU We know that ζnet is a real function and it takes on discrete values. So there may be a situation that not all the diagonal elements have to be positive and not all the links have to be ‘Bad’ links to make the matrix pair QLSU. That means, for a matrix pair to be QLSU, the number of positive diagonal elements can be less than 2n and the number of Bad links in the matrix pair can be less than Nlc, not necessarily equal to Nlc. Now, we calculate the minimum number of Bad links needed to make a matrix pair unstable and denote it as N ∗u . For a matrix pair with Np ̸= 0, let us define N ∗u as follows: 175 M. Chand, M. Paitandi, M. K. Gupta Acta Polytechnica N ∗u =the closest higher upper integer of (ηng + 1 2 ) times Nlc, including when that is itself an integer. Let us state a theorem. Theorem. A matrix pair is QLSU only if the total number of bad links in it is ≥ N ∗u i.e. Nbad + N ′ bad ≥ N ∗ u . Proof. For a QLSU matrix pair, ζnet < 0 =⇒ ηng + ηgood + η ′ good < ηpz + ηbad + η ′ bad =⇒ ηng + 1 − ηbad + 1 − η ′ bad < 1 − ηng + ηbad + η ′ bad =⇒ ηng + 1 2 < ηbad + η ′ bad =⇒ ηng + 1 2 < Nbad + N ′ bad Nlc =⇒ [ηng + 1 2 ] × Nlc < Nbad + N ′ bad . Thus, the total number of Bad links needed to make a matrix pair QLSU is greater than or equal to N ∗u . This is a necessary condition for a matrix pair to be QLSU. 3.4. Sufficiency Guidelines For QLSU Matrix Pair While observing the expression for the determinant of a matrix pair, we find that a diagonal element is always mul- tiplied by the link elements surrounded by it. That means the row and column elements associated with a diagonal element play a vital role to assess the stability/instability. These observations provide us with some ‘guidelines’ for sufficiency for QLSU [16]. • Guideline 1: A matrix pair with ζnet < 0, is likely QLSU if a positive diagonal element eii or aii is surrounded by + chains. (The above guideline is the result of the idea that pos- itive diagonal elements along with + chains promote instability.) • Guideline 2: A matrix pair with ζnet < 0, is likely QLSU if a negative diagonal element eii or aii is surrounded by − chains. (The above guideline is the result of the idea that neg- ative diagonal elements along with − chains promote instability.) 4. Illustrative examples for instability of a matrix pair By now, we have few necessary conditions for QLSU and few guidelines for sufficiency for QLSU. Once the necessary condition ζnet < 0 is satisfied, we can make a QLSU matrix pair, with the appropriate placement of the ‘Bad’ and ‘Good’ links in it. Let us consider some examples. Example 1. Consider a 4 × 4 matrix pair with diagonal elements as shown below: E =   − ∗ ∗ ∗ ∗ + ∗ ∗ ∗ ∗ + ∗ ∗ ∗ ∗ +   and A =   + ∗ ∗ ∗ ∗ − ∗ ∗ ∗ ∗ − ∗ ∗ ∗ ∗ +   Suppose there is no ZZ link in the matrix pair. For the given matrix pair (E, A), ηng = 3 8 , ηpz = 5 8 Nlc =12 N ∗ u =11 Thus, Nbad + N ′ bad ≥ 11. Any matrix pair with the given diagonal element struc- ture and satisfying the above condition have ζnet < 0, and thus the matrix pair is an MDU matrix pair. Example 2. Let us consider a matrix pair with the con- ditions given in Example 1: E =   − − + + − + − − − + + + + + + +   and A =   + − − + + − + − − + − + − − + +   Here Nbad = 7 , Ngood = 5 N ′ bad = 5 , N ′ good = 7 ∴ Nbad + N ′ bad = 7 + 5 = 12 > 11 = N ∗ u . Hence, the necessary condition is satisfied, and now we check the sufficiency guidelines required for QLSU. Here, all the negative diagonal elements are surrounded by − chains. Therefore, this is a QLSU matrix pair. It is noted that once there is at least one positive diago- nal element, we can assess the Qualitative Sign Instability by computing the relative distribution of the Bad links together with the Good links. Example 3. Let us discuss the stability of the matrix pair given below: E =   − − − + − − + + + + + + + + − − − + + + + + + − −   and A =   − + + − + + + + + + + − + + + − + − + − + + + − −   For the given matrix pair (E, A), ηpz = 6 10 , ηng = 4 10 ηbad = 3 5 , ηgood = 2 5 η ′ bad = 1 2 , η ′ good = 1 2 ∴ ζnet = − 2 5 < 0 . Here all the positive diagonal elements are surrounded by + chains. Hence, as per the sufficiency guidelines for QLSU, the given matrix pair is QLSU. Thus, this particular quantitative matrix pair is unstable without any need of eigenvalue calculations. 176 vol. 63 no. 3/2023 Qualitative Sign Stability of Linear Time Invariant Descriptor systems Example 4. Consider the sign pattern of the matrix pair given below: E = [ − + − − + + + − + ] and A = [ + + − − + + + − − ] This matrix pair (E, A) has ηpz = 4 6 , ηng = 2 6 ηbad = 0 , ηgood = 1 η ′ bad = 1 , η ′ good = 0 making ζnet = −1 3 < 0. Since, ζnet < 0, the necessary condition is satisfied. But the sufficient conditions are not satisfied. Hence, it is an MDU (not a QLSU) matrix pair. It should be noted that we are not stating that the matrix pair with this Elemental Sign Structure is always unstable. We are simply stating that the elemental sign structure of the above matrix pair guarantees that there exist magnitudes which would definitely make this matrix unstable. For example, the following matrix pair (E1, A1) with the above sign structure is unstable, E1 = [ −1.6132 2.0118 −1.6806 −0.0021 0.5791 0.2139 0.2017 −2.1852 1.7419 ] , A1 = [ 0.2462 1.8614 −0.7201 −1.8764 1.4531 1.9056 3.4318 −0.1567 −1.7543 ] , while the following matrix pair (E2, A2) having the same sign structure is stable. E2 = [ −1.0132 0.0118 −1.0006 −1.0021 0.4001 0.2110 0.0213 −0.0411 0.2015 ] , A2 = [ 1.8112 21.5624 −1.0207 −0.9016 1.0172 1.0061 9.6512 −0.0600 −5.0234 ] . Example 5. Consider the matrix pair (E, A) with a sign structure given by: E =   + − + − − + − − + − − − + − − − − − − + + − + − −   and A =   − + − − + − + + − − + − − − + + + − + − − − + − −   This matrix pair (E, A) has ηpz = 2 5 , ηng = 3 5 ηbad = 2 5 , ηgood = 3 5 η ′ bad = 1 5 , η ′ good = 4 5 ∴ ζnet = 1 > 0 . Here, ζnet > 0, but the necessary condition for Qual- itative Sign Stability C1 is not satisfied. Hence, it is an MDS/U matrix pair and is a non-QLSS matrix pair. 5. Conclusion This paper addresses the issue of determining the stabil- ity/instability of a matrix pair that arises in a continuous linear homogeneous time-invariant system. The conditions for a matrix to be QLSU have already been discussed in the earlier works. In this research, we generalise the iden- tity matrix I to any matrix E and propose few necessary conditions and sufficient conditions for qualitative sign instability of a matrix pair. The proposed conditions are very simple and are based on the number and nature of the diagonal elements and the number and nature of the off-diagonal element pairs (links). This reflects that the Elemental Sign Structure of a matrix pair is an important contributor to the stability/instability. These conditions are extremely helpful for engineers and ecologists, in solv- ing stability-related problems. Acknowledgements The research work is supported by SERB, DST under grant no. SRG/2019/000451. References [1] V. K. Mishra, N. K. Tomar, M. K. 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Nature 483(7388):205–208, 2012. https://doi.org/10.1038/nature10832 178 https://doi.org/10.1007/s12080-007-0007-8 https://doi.org/10.2514/1.46196 https://doi.org/10.1109/ACC.2010.5530716 https://doi.org/10.1016/j.matcom.2013.11.005 https://doi.org/10.1007/978-1-4612-1346-8_2 https://doi.org/10.1016/0024-3795(89)90547-8 https://doi.org/10.23919/ACC.2018.8431691 https://doi.org/10.1016/j.rico.2021.100014 https://doi.org/10.1016/j.laa.2011.08.011 https://doi.org/10.13001/ela.2020.4929 https://doi.org/10.1038/nature10832 Acta Polytechnica 63(3):171–178, 2023 1 Introduction 2 Qualitative Sign Matrix Pair Indices 2.1 Diagonal Elements eii and aii 2.2 Interactions of the form eijeji and aijaji 2.3 Interactions of the form eijaji 3 Conditions for Qualitative Sign Stability and Instability 3.1 Necessary Conditions for Qualitative Stability of Matrix Pair 3.2 A Necessary Condition for Instability of a Matrix 3.3 A Necessary Condition for a Matrix Pair to be QLSU 3.4 Sufficiency Guidelines For QLSU Matrix Pair 4 Illustrative examples for instability of a matrix pair 5 Conclusion Acknowledgements References