EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 4, 2019, 1360-1370
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Optimal Solution Properties of an Overdetermined
System of Linear Equations

Vedran Novoselac1,∗, Zlatko Pavić 1

1 Mechanical Engineering Faculty in Slavonski Brod, University of Osijek,
Trg Ivane Brlić Mažuranić 2, 35000 Slavonski Brod, Croatia

Abstract. The paper considers the solution properties of an overdetermined system of linear
equations in a given norm. The problem is observed as a minimization of the corresponding
functional of the errors. Presenting the main results of p norm it is shown that the functional
is convex. Following the convex properties we examine minimization properties showing that the
problem possesses regression, scale, and affine equivariant properties. As an example we illustrated
the problem of finding the weighted mean and weighted median of the data.

2010 Mathematics Subject Classifications: 15A06, 65F20, 58K70

Key Words and Phrases: Linear equations, overdetermined systems, equivariance

1. Introduction

A system of linear equations denoted as Ax = b where

A =

 a11 . . . a1n
...

...
...

am1 . . . amn

 ∈ Rm×n, x =

 x1
...
xn

 ∈ Rn, b =

 b1
...
bm

 ∈ Rm. (1)

If A is m × n matrix, with m > n, then it is said that the linear system of equations is
overdetermined. In general, such a system will have no solution, i.e. it is inconsistent, i.e.
b /∈ R(A). Instead the solution with the smallest error ‖b− Ax‖p is observed using some
p ∈ [1,∞〉 norme, where the problem is to find a vector x ∈ Rn such that

min
x∈Rn

‖b−Ax‖p, p ∈ [1,∞〉. (2)

Figure 1 presents the problem of an overdetermined system of linear equations.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3517

Email addresses: Vedran.Novoselac@sfsb.hr (V. Novoselac), Zlatko.Pavic@sfsb.hr (Z. Pavić)

http://www.ejpam.com 1360 c© 2019 EJPAM All rights reserved.



V. Novoselac, Z. Pavić / Eur. J. Pure Appl. Math, 12 (4) (2019), 1360-1370 1361

Figure 1: Graphic illustration of an overdetermined system Ax = b.

Problem (2) can be observed as a minimization problem of a functional

fp(x) = ‖b−Ax‖p =

 m∑
i=1

|bi −
n∑

j=1

aijxj |p
 1

p

, p ∈ [1,∞〉. (3)

Many problems can be presented as an overdetermined system of linear equations such as
the problem of determinating linear models for fitting experimental data. It is intended to
help researchers fit appropriate curves to their data. Curve fitting, also known as regression
analysis, is a common technique for modelling data [13]. The problem of determining n-
dimensional hyperplane, in order to have its graph pass as close as possible to given points
in sense of p norme, can be presented as an overdetermined system of linear equations [2].

In linear regression analysis it is most frequently assumed that errors, i.e. so called
’outliers’, can occur in measured values of the independent variable. In this case, if Eu-
clidian norm (p = 2) is used, vector x ∈ Rn is obtained in the sense of Least Squares (LS)
problem by minimizing (3). In many technical and other applications using the p = 1
norm is much more interesting. Because of robust properties of p = 1 norm the outliers
in data should not affect the obtained results. In literature this problem is known as the
Least Absolute Deviation (LAD) problem, and is an efficient method for outlier detection
[11, 14].

For that purpose some properties of functional fp : Rn → R are shown in Section 2,
especially taking into account the equivariant properties of a solution of an overdetermined
system of linear equations. In Section 3, we analyzed equivariant properties of the weighted
mean and weighted median of the data, which are a reduced case of an overdetermined
system of linear equations, where p = 1, p = 2 is observed, and n = 1 respectively.

2. Properties of the Functional fp

In this section we present some properties of functional fp : Rn → R in order to prove
equivariant properties. Following the Minkowski inequality presented in Theorem 9, which
is proven in Section 5, presented in appendix, we give directly the next theorem.

Theorem 1. Functional fp : Rn → R is convex on Rn.

The importance of the extremum problems in applied mathematics leads us to the
general study of extremum of functional. It is not easy to know the extremum points



V. Novoselac, Z. Pavić / Eur. J. Pure Appl. Math, 12 (4) (2019), 1360-1370 1362

for differentiable functions, because it is not always possible to solve ∇fp(x) = 0 to
calculate critical points. In the situation when p = 2, the problem is known as the LS
problem and it always has a solution. In this case x∗ minimizes f2 if and only if x∗ solves
normal equation system ATAx = AT b. It can be shown that if A ∈ Rm×n has full rank
n, then the LS problem has a unique minimizer. In that case global extremum can be
presented as x∗ = A+b, where matrix A+ = (ATA)−1AT is usually called the Moore-
Penrose inverse [3]. In most general cases, convex functional fp has a particularly simple
extremal structure, and there are algorithms to calculate extremum points, supposing its
existance [15]. Knowing whether or not a local minimum is also global, is one of the most
important questions in optimization [7]. The assumption of convexity gives a positive
answer to this question as it is stated in the following theorem.

Theorem 2. A local minimum x∗ of a functional fp : Rn → R is always a global extremum.

Proof. Suppose that x∗ is a local minimum of fp, that is, there is an open neighborhood
U of x∗ where fp(x

∗) ≤ fp(x), ∀x ∈ U.We prove that fp(x
∗) ≤ fp(y∗) for arbitrary y∗ ∈ Rn.

Consider the convex combination (1 − λ)x∗ + λy∗, for λ ∈ [0, 1], the convex combination
approaches to x∗ as λ → 0. Therefore for small enough λ, (1 − λ)x∗ + λy∗ is in the
neighborhood U . Then

fp(x
∗) ≤ fp((1− λ)x∗ + λy∗)
≤ (1− λ)fp(x

∗) + λfp(y
∗).

(4)

Rearranging terms, we have fp(x
∗) ≤ fp(y∗).

From the Theorem 2, it directly follows that the local minimum of a convex functional
is necessarily the global minimum. However, this minimum is not necessarily unique, a
sufficient condition is strict convexity. Let us now discuss some equivariant properties
of functional fp. We considered three types of equivariance: regression, scale, and affine
equivariance [11]. The next theorem shows that functional fp possesses these three types
of equvivarance.

Theorem 3. Let A ∈ Rm×n, m > n, b ∈ Rm, and x∗ ∈ Rn such that

fp(x
∗) = min

x∈Rn
‖b−Ax‖p, p ∈ [1,∞〉. (5)

Then

(a) For an arbitrary v ∈ Rn, vector x∗ + v is a solution of

min
x∈Rn

‖b+Av −Ax‖p. (6)

(b) For any c ∈ R, vector c x∗ is a solution of

min
x∈Rn

‖c b−Ax‖p. (7)



V. Novoselac, Z. Pavić / Eur. J. Pure Appl. Math, 12 (4) (2019), 1360-1370 1363

(c) For a nonsingular matrix B ∈ Rn×n, vector B−1x∗ is a solution of

min
x∈Rn

‖b−ABx‖p. (8)

Proof. Let us discuss:

(a) Let y∗ be a solution of (6). Then

‖b+Av −Ay∗‖p = ‖b−A(y∗ − v)‖p ≥ ‖b−Ax∗‖p, (9)

whereby the equation is only if y∗ − v = x∗.

(b) For c = 0 the assertion is obvious. Let c 6= 0, and y∗ such that is a solution of (7).
Then

‖cb−Ay∗‖p = |c|
∥∥∥∥b−A(y∗c

)∥∥∥∥
p

≥ |c|‖b−Ax∗‖p, (10)

whereby the equation is only if y∗

c = x∗.

(c) Let y∗ be a solution of (8). Then

‖b−ABy∗‖p ≥ ‖b−Ax∗‖p, (11)

whereby the equation is only if By∗ = x∗.

Corollary 1. Let A ∈ Rm×n, m > n, b ∈ Rm, and x∗ ∈ Rn such that

fp(x
∗) = min

x∈Rn
‖b−Ax‖p, p ∈ [1,∞〉. (12)

Then for an arbitrary v ∈ Rn, c ∈ R, and nonsingular matrix B ∈ Rn×n, vector
cB−1(x∗ + v) is a solution of

min
x∈Rn

‖c(b+Av)−ABx‖p. (13)

3. Weighted Mean and Weighted Median of the Data

In this section we will illustrate the equivariant properties of the weighted mean and
weighted median of the data. As we mentioned, the problem of the weighted mean and
weighted median are essentially reduced to solving an overdetermined system of linear
equations. Numerous applications of this problem can be found in various branches of
applied research, like image processing [10], or methods for outlier detection [11].

Let a ∈ Rm be the vector data with corresponding positive vector data weights
w ∈ Rm

+ . If we denote by A = [ p
√
w1, . . . , p

√
wm]T and b = [ p

√
w1a1, . . . , p

√
wmam]T in

function fp : R→ R, where n = 1, then follows that

f2(x) = ‖b−Ax‖2 =

√√√√ m∑
i=1

wi(ai − x)2, (14)



V. Novoselac, Z. Pavić / Eur. J. Pure Appl. Math, 12 (4) (2019), 1360-1370 1364

is convex and attains its global minimum on the set R, which is denoted by
x∗ = mean(w, a) and called the weighted mean of the data. If
w1 = · · · = wm = 1, the global minimum of the corresponding functional (14) is de-
noted by x∗ = mean(a) and called the mean of the data. Analogously, a real number
which minimizes function

f1(x) = ‖b−Ax‖1 =

m∑
i=1

wi|ai − x|, (15)

is called the weighted median of the data and is denoted by x∗ = med(w, a). If
w1 = · · · = wm = 1, the global minimum of the corresponding function (15) is denoted by
x∗ = med(a) and called the median of the data. In the Figure 2 we present the example
of function fp, for p = 2 and p = 1 norm. Functions are generated with data vector
a = [1, 2, 3, 4, 5]T and corresponding weights vector w = [1, 1, 2, 1, 1]T presented in Figure
2(a) for p = 2, and for p=1 in Figure 2(b). In Theorem 4 it is shown that the minimum
of the function f2, i.e. weighted mean, always achieved unique minimum. Considering
the different data weights w = [1, 3, 2, 1, 1]T , presented in Figure 2(c), it can be seen that
the construction of the minimum of f1, i.e. weighted median, directly depends on it, and
achieved its minimum on interval [a(2), a(3)], in contrast to the situation in Figure 2(b)
where the minimum is unique. Also, it can be seen that function f1 is a piecewise linear
function. This property of functional f1 is considered for finding a minimum, where details
are presented in Theorem 5. In the sequel we give solutions for minimizing problems (14)
and (15), i.e. for the weighted mean and weighted median of the data.

2

4

6

8

y

x

f2

x∗ = mean(w, a)

5

10

15

20

y

x

f1

a(1) a(2) a(3) a(4) a(5)

5

10

15

20

25

y

x

f1

a(1) a(2) a(3) a(4) a(5)

(a) (b) (c)

Figure 2: Function fp : R→ R.

Theorem 4. Let a ∈ Rm, m ≥ 2, be the data vector with corresponding data weights
w ∈ Rm

+ . Then

mean(w, a) =
1

W

m∑
i=1

wiai, W =

m∑
i=1

wi. (16)

Proof. Because the functional defined by (14) is derivable, the minimum is attained by

finding a solution of ∂f2(x)
∂x = 0.



V. Novoselac, Z. Pavić / Eur. J. Pure Appl. Math, 12 (4) (2019), 1360-1370 1365

Corollary 2. Let a ∈ Rm, m ≥ 2, be the data vector, then

mean(a) =
1

m

m∑
i=1

ai. (17)

Theorem 5. Let a ∈ Rm, m ≥ 2, be the data vector with corresponding data weights
w ∈ Rm

+ . Let a(1) ≤ a(2) ≤ . . . ≤ a(m) denote ordered observation and
0 < w(1) ≤ w(2) ≤ . . . ≤ w(m) corresponding weights. Thereby with the denotation

L :=
{
l :

l∑
i=1

w(i) ≤
W

2

}
, l ∈ {1, . . . ,m}, W =

m∑
i=1

wi, (18)

(a) if L = ∅, then med(w, a) = a(1);

(b) if L 6= ∅, then with the denotation l0 = maxL there holds:

(i) if
l0∑
i=1

w(i) <
W
2 , then med(w, a) = a(l0+1);

(ii) if
l0∑
i=1

w(i) = W
2 , then med(w, a) ∈ [a(l0), a(l0+1)].

Proof. Notice that on each interval

〈−∞, a(1)〉, 〈a(1), a(2)〉, . . . , 〈a(m−1), a(m)〉, 〈a(m),∞〉,

function f1 : R → R defined by (15) is linear, and thereby derivable. The slopes of those
linear functions are consecutively kl, l = 0, . . . ,m, where

k0 = −
m∑
i=1

wi, km =
m∑
i=1

wi (19)

and for l = 1, . . . ,m− 1

kl = 2

l∑
i=1

w(i) −
m∑
i=1

wi = kl−1 + 2w(l). (20)

If L = ∅, then for every l = 1, . . . ,m− 1, is k0 < 0 < kl. It follows that f1 is strongly
decreasing on 〈−∞, a(1)〉 and strongly increasing on 〈a(1),∞〉, therefore the minimum of
f1 is attained for x∗ = a(1).

If L 6= ∅, then kl+1 − kl = 2w(l+1) > 0, and the sequence (kl) is increasing and

k0 < k1 < · · · < kl0 ≤ 0 < kl0+1 < · · · < km. (21)

If k(l0) < 0, it follows from (21) that f1 is decreasing on 〈−∞, a(l0+1)〉 and increasing
on 〈a(l0+1),∞〉, therefore the minimum of f1 is attained for x∗ = a(l0+1)



V. Novoselac, Z. Pavić / Eur. J. Pure Appl. Math, 12 (4) (2019), 1360-1370 1366

If k(l0) = 0, it follows from (21) that f1 is decreasing on 〈−∞, a(l0)〉, is constant on
[a(l0), a(l0)+1] and increasing on 〈a(l0+1),∞〉, therefore the minimum of f1 is attained at
every point x∗ ∈ [a(l0), a(l0+1)].

Figure 3(a) present the value of data vector a = [1, 2, 3, 4, 5]T . In this situation it is
obvious that the weighted mean and weighted median with corresponding data weights
w = [1, 1, 2, 1, 1]T of the observed data are equal. Suppose that in some situation two
outliers are added, e.g. because of a copying or transmission error. Figure 3(b) displays
such a situation, where the two last data have moved up and away from its original
position. These are so called outliers, and they have a large influence on the weighted
mean, i.e. LS problem, which is quite different from the weighted mean in Figure 3(a).

1 2 3 4 5

1

2

3

4

5

1

2

3

4

1 2 3 4 5

1

2

3

4

5

1

2

3

8

ai ai

i i

mean(w, a)

med(w, a)

mean(w, a)

med(w, a)

OUTLIERS

(a) (b)

Figure 3: Weighted mean and weighted median: (a) original data; (b) same data as in part (a), but with two
outliers.

In the sequel, the corollary is mentioned, which specializes the situation for the case
if the weights of the data are not assigned, or if all weights are mutually equal. Also, a
description of the pseudo-halving property is mentioned [1], which follows directly from
Theorem 5.

Corollary 3. Let a ∈ Rm, m ≥ 2, be the data vector and
y(1) ≤ y(2) ≤ . . . ≤ y(m) denoted ordered observation. Then follows

(a) if m is odd (m = 2l0 + 1), med(a) = a(l0+1);

(b) if m is even (m = 2l0), med(a) is every number from the segment [a(l0), a(l0+1)].

Corollary 4. Let a ∈ Rm, m ≥ 2, be the data vector with corresponding data weights
w ∈ Rm

+ . Let a(1) ≤ a(2) ≤ . . . ≤ a(m) denote ordered observation and 0 < w(1) ≤ w(2) ≤
. . . ≤ w(m) corresponding weights. Then there holds that the pseudo-halving property

∑
a(i)<x∗

w(i) ≤
W

2
and

∑
a(i)>x∗

w(i) ≤
W

2
, W =

m∑
i=1

wi. (22)



V. Novoselac, Z. Pavić / Eur. J. Pure Appl. Math, 12 (4) (2019), 1360-1370 1367

Theorem 6. Let a ∈ Rm, m ≥ 2, be the data vector with corresponding data weights
w ∈ Rm

+ . Then for α, β, γ ∈ R, α > 0, there holds that

(a) mean(αw, βa+ γe) = βmean(w, a) + γ;

(b) med(αw, βa+ γe) = βmed(w, a) + γ,

where e = [1, . . . , 1]T ∈ Rm.

Proof. First we prove (a), while the proof of (b) is going analogue of (a). Notice that
equality (a) holds if and only if

mean(αw, a) = mean(w, a), and (23)

mean(αw, βa+ γe) = βmean(w, a) + γ. (24)

Property (23) is trivial to prove. Property (24) follows immediately from Theorem 3, i.e.
from regression and scale equivariant property.

The next figure presents the weighted mean and weighted median equivariance prop-
erties presented in Theorem 6. For example we observed data vector a and weight vector
w from Figure 3(a). Parameter α > 0 do not have influence on results of the weighted
mean and median. For other parameters we observed case β = −2, and γ = 25.

1 2 3 4 5

1
2
3
4
5

23

21

19

17

15

1 2 3 4 5

1
2
3
4
5

23

21

19

17

15

βai + γ

ai

i i

mean(αw, βa+ γe)

mean(αw, a)

med(αw, βa+ γe)

med(αw, a)

(a) (b)

Figure 4: Equivariance properties: (a) weighted mean; (b) weighted median.

4. Conclusion

Considering the properties of an overdetermined system of linear equations it is estab-
lished that the solution of the system possesses regression, scale, and affine equivariant
properties in observed norm p ∈ [1,∞〉. As an example we observed the problem of finding
the weighted mean and weighted median of the data.



V. Novoselac, Z. Pavić / Eur. J. Pure Appl. Math, 12 (4) (2019), 1360-1370 1368

5. Appendix - Discrete Forms of Inequalities

A set S ⊆ Rn is said to be convex if, for all x, y ∈ S and all λ ∈ [0, 1], the point
(1 − λ)x + λy also belongs to S, i.e. (1 − λ)x + λy ∈ S. The sum (1 − λ)x + λy is
called binomial convex combination. It can be easily seen that if we have m observations
x1, . . . , xm ∈ S in convex set S, and λ1, . . . , λm nonnegative number such that

∑m
i=1 λi = 1,

then
∑m

i=1 λixi ∈ S. A point of this type is known as a m-member convex combination of
x1, . . . , xm.

Let S ⊆ Rn be convex, a functional f : S → R is said to be convex if the inequality

f((1− λ)x+ λy) ≤ (1− λ)f(x) + λf(y) (25)

holds for all points x, y ∈ S and coefficient λ ∈ [0, 1]. If the inequality (25) is strict for all
x, y ∈ S, then f(x) is called strictly convex.

Using mathematical induction, the inequality in formula (25) can be extended to m-
membered convex combinations.

Theorem 7. (Discrete form of Jensen’s inequality) Let S ⊆ Rn be a convex set, let

xi ∈ S be points, and let λi ∈ [0, 1] be coefficients such that
m∑
i=1

λi = 1. Then each convex

function f : S → R satisfies the inequality

f

(
m∑
i=1

λixi

)
≤

m∑
i=1

λif(xi). (26)

Theorem 8. (Discrete form of Hölder’s inequality) Let x, y ∈ Rn be points, and let
p, q ∈ 〈0,∞〉 be numbers such that 1/p+ 1/q = 1. Then we have the inequality

m∑
i=1

|xiyi| ≤

(
m∑
i=1

|xi|p
) 1

p
(

m∑
i=1

|yi|q
) 1

q

. (27)

Proof. Assuming that all points yi are different from zero, formula (27) can be obtained
from formula (26) as follows. Using the points

|xi||yi|−
q
p

as xi, the coefficients

λi =
|yi|q

m∑
i=1
|yi|q

,

and the convex function xp, we get 1
m∑
i=1
|yi|q

m∑
i=1

|yi|q|xi||yi|−
q
p


p

≤ 1
m∑
i=1
|yi|q

m∑
i=1

|yi|q
(
|xi||yi|−

q
p

)p
.



V. Novoselac, Z. Pavić / Eur. J. Pure Appl. Math, 12 (4) (2019), 1360-1370 1369

Since q − q/p = 1, it follows that 1
m∑
i=1
|yi|q

m∑
i=1

|xiyi|


p

≤ 1
m∑
i=1
|yi|q

m∑
i=1

|xi|p.

Taking the p-th root, multiplying by
∑m

i=1 |yi|q, and using the exponent 1/q instead of
1− 1/p, we achieve the inequality in formula (27).

If some yj is equal to zero, then xjyj = 0 does not increase the left side of formula
(27), but xj 6= 0 increases the right side.

Utilizing the vectors x = [x1, . . . , xn]T , y = [y1, . . . , yn]T and z = [x1y1, . . . , xnyn]T ,
formula (27) can be expressed by the norms,

‖z‖1 ≤ ‖x‖p‖y‖q. (28)

Theorem 9. (Discrete form of Minkowski’s inequality) Let x, y ∈ Rn be points,
and let p ∈ [1,∞) be a number. Then we have the inequality(

m∑
i=1

|xi + yi|p
) 1

p

≤

(
m∑
i=1

|xi|p
) 1

p

+

(
m∑
i=1

|yi|p
) 1

p

. (29)

Proof. If all xi and yi are equal to zero, then formula (29) trivially holds. If p = 1, then
the inequality in formula (29) follows from the simple triangle inequality
|xi + yi| ≤ |xi|+ |yi|. If some of the points are different from zero, and if p > 1, the
inequality in formula (29) can be derived by using the simple triangle inequality, and the
inequality in formula (27). In this intention, we have

m∑
i=1

|xi + yi|p ≤
m∑
i=1

|xi||xi + yi|p−1 +
m∑
i=1

|yi||xi + yi|p−1

≤

(
m∑
i=1

|xi|p
)1

p
(

m∑
i=1

|xi + yi|(p−1)q
)1

q

+

(
m∑
i=1

|yi|p
)1

p
(

m∑
i=1

|xi + yi|(p−1)q
)1

q

=

( m∑
i=1

|xi|p
) 1

p

+

(
m∑
i=1

|yi|p
) 1

p

( m∑
i=1

|xi + yi|p
) 1

q

because (p−1)q = p. Dividing by
(∑n

i=1 |xi +yi|p
)1/q

, and putting 1/p instead of 1−1/q,
we obtain the inequality in formula (29).

Using p norms of the vectors x, y and x + y, the inequality in formula (29) takes the
form

‖x+ y‖p ≤ ‖x‖p + ‖y‖p. (30)



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