the matrix of matrices exponential and application...2019/01/04 · the matrix of matrices...

International Journal of Mathematical AnalysisVol. 13, 2019, no. 2, 81 - 97

HIKARI Ltd, www.m-hikari.comhttps://doi.org/10.12988/ijma.2019.9210

The Matrix of Matrices Exponential

and Application

Z. Kishkaa, M. Saleema and A. Elrawyb,1

a Dept. of Math., Faculty of ScienceSohag University, Sohag, Egypt

b Dept. of Math., Faculty of ScienceSouth Valley University, Qena, Egypt

This article is distributed under the Creative Commons by-nc-nd Attribution License.

Copyright c© 2019 Hikari Ltd.

Abstract

In this paper, we introduce a definition of an exponential functionin the sense of the matrix of matrices (for short, MMs). After defin-ing the eigenvalue and the eigenvector of MMs, we solve the system ofa homogeneous linear matrix of first-order differential equations whosecoefficients are constant matrices. Existence and uniqueness of a so-lution have been proven for that system. Finally, some examples areestablished.

Mathematics Subject Classification: 15A16, 34B02, 15B02

Keywords: Matrix of matrices, Exponential function, System of a homo-geneous linear matrix of first-order differential equations

1 Introduction

Computation of matrix exponential is required in many applications such ascontrol theory [4, 6], Markov chain process [25] and nuclear magnetic reso-nance spectroscopy [7]. For several years, great efforts have been devoted tothe study of matrix [28, 27, 5] and matrix exponential (see [18, 11, 8]). A

1Corresponding author

82 Z. Kishka, M. Saleem and A. Elrawy

decently wide variety of methods for computing matrix exponential were in-troduced ( see for instance [16]). The interest in it stems from its key rolein solving a corresponding system of ordinary differential equations (ODEs).This subject dates back more than 100 years with the starting point of theLyapunov stability theorem. The asymptotic properties of matrix exponentialfunctions extend information on the long-time conduct of solutions of ODEs.(see [18, 9, 21, 1, 26]).The concept of the MMs was introduced by Kishka et al. (cf. [12, 13, 14, 15]).That proposed concept provides the possibility of extending many topics ofmatrices in addition to the benefits of saving time and effort done in manymathematical calculations that one can gain using them. In details, we be-gin this subject with introducing the concept of MMs, proving its validity andshowing some methods that is used to find the inverse of MMs [12]. Hadamardand Kronecker products over MMs have been introduced and their basic prop-erties in [13]. While in [14], we present MMs over semiring and we study someresults on a regular and invertible MMs over semirings. In [15], the system ofthe linear matrix equations are solved by using the concept of MMs.

Our first contribution in the current work is to define the MMs exponential.The second one is concerned with an application of MMs exponential to solvethe system of a homogeneous linear matrix of first-order differential equations.Moreover, the existence and uniqueness of a solution for the system are proven.The organization of the paper is as follows. In the next section, we givesome definitions that are going to be used later on. In section 3, we presentdefinitions of the eigenvalue, the eigenvector of MMs and MMs exponential.Section 4 is devoted to show some theories, the main results, together withtheir mathematical proofs. In section 5, we present a set of examples as anapplication to the previous theories. Finally, we draw some concluding remarksin section 6.

2 Preliminaries

To obviate lengthy scripts, the following notations are adopted throughout thisarticle;Let K be a field, K[t] be the ring of polynomial forms over K and Ml (K) be theset of all l× l matrices defined on K, I and O stand for the identity matrix andthe zero matrix in Ml (K), respectively. Also, we consider M

′

l (K) ⊂Ml (K) isa commutative subring with unity. We denote by Mm×n (Ml (K)) the set ofall m× n MMs over Ml (K) . The elements of Mm×n (Ml (K)) are denoted byA, B, C, ..., etc. Two important special MMs are the identity MMs I and thezero MMs O.Let A ∈Ml (K), then the l × l matrix exponential exp(A) by the power seriesexpansion

The matrix of matrices exponential and application 83

exp(A) =∑∞

i=0

Ai

i!.

Definition 2.1 [13] Let A ∈ Mm×n (Ml (K)) , then it can be written as arectangular table of elements Aij; i = 1, 2, ...,m and j = 1, 2, ..., n, as follows:

A =

A11 ... A1n...

. . ....

Am1 ... Amn

,

where Aij ∈Ml (K).

Definition 2.2 The elements Aii (i = 1, 2, ..., n) in square MMs A are calledprincipal diagonal, and the sum of the elements of diagonal square MMs iscalled trace of MMs

Tr (A) =n∑i=1

Aii.

Definition 2.3 [10] The determinant of n−square A = (Aij), where Aij ∈M

′

l (K), denoted by det (A) or |A| , is

det (A) =∑σ

(sgn σ)A1j1A2j2 ...Anjn ,

or

det (A) =∑σ

(sgn σ)A1σ(1)A2σ(2)...Anσ(n).

Observe that each term where A1j1A2j2 ...Anjn contains exactly one entry fromeach row and each column of A, σ = j1j2...jn ∈ Sn, where Sn is the set of allpermutations on {1, 2, ..., n} (see [19, 2]).

Before presenting this system of homogeneous linear matrix differentialequations, it is useful to introduce the eigenvalues, the eigenvectors of MMsand the MMs exponential.

3 Basic points of current work

We now define a vector of matrices, eigenvalue, eigenvectors of MMs and MMsexponential.

Definition 3.1 Let Ml (K) be a ring with unity, then the set

Vn = {(A1, A2, ..., An) : A1, A2, ..., An ∈Ml (K)} ,


is called vector of matrices; for short VMs.

Let A = (Aij)∈Mn

(M

′

l (K)), N ∈ M

′

l (K) and V ∈ Mn×1(M

′

l (K)). Con-

sider the eigenvalue problem for A

AV = NV , V 6= On×1∈Mn×1

(M

′

l (K)). (1)

The solution to this problem consists of identifying all possible values ofN (called the eigenvalues of MMs) and the corresponding non-zero vector ofmatrices (VMs) V (called the eigenvectors of MMs) that satisfy (1). We canrewrite (1) as

(A−NI)V = On×1. (2)

If (A−NI) is an invertible MMs, then V = On×1 is the unique solution.By definition, the zero VMs is not an eigenvector. Thus, in order to find non-trivial solutions to (2), one must demand that (A−NI) is not invertible orequivalently,

φ (N) := det (A−NI) = O, O ∈M ′

l (K) . (3)

Evaluating the determinant yields an nth order matrix polynomial in N , calledthe characteristic matrix polynomial, which we have denoted above by φ (N) .

The determinant in (3)

φ (N) = (−1)n [INn + E1Nn−1 + E2N

n−2 + ....+ En−1N + En],

where I, Ei ∈M′

l (K) , i = 1, 2, ..., n.

Theorem 3.2 Every MMs A is a root of its characteristic matrix polynomial.

Here we are going to present an example to show exactly how the eigenvaluesand the eigenvectors can be calculated in the sense of MMs.

Definition 3.3 Let A be n-MMs, then

exp(At) =∞∑k=0

1

k!(At)k .

It is easily to show that

exp(O) = exp(AO) = I,

andd

dtexp(At) = Aexp(At) = exp(At)A.


4 Main results

In this section, we define the system of a homogeneous matrix of first-orderdifferential equations. Also, we show some theories that are considered as anextension of Leonard [18], in case of the system of a homogeneous linear matrixof first-order differential equations, which are the main core of our study.Let us consider now the system of homogeneous linear matrix of first-orderdifferential equations defined as;

X′

1 = A11X1 + A12X2 + ...+ A1lXl,

...

X′

l = Al1X1 + Al2X2 + ...+ AllXl,

(4)

where Aij ∈ Ml (K) is a constant l × l matrix, Xi = (xij) is an unknownmatrix function, X

′i =

(x

′ij

)for all i, j = 1, 2, ..., l.

Now, we write the above system (4) in the form

X ′= AX , (5)

where X ′=(X

′i

), A = (Aij) and X = (Xi) ; i, j = 1, 2, ..., l.

The next theorem gives a general solution of the system (5) in the sense of theeigenvalues and the eigenvectors of MMs.

Theorem 4.1 Let A be n × n constant MMs with Ni eigenvalue and letV1,V2, ...,Vn be the eigenvectors corresponding to the Ni, i = 1, 2, ..., n. Then

{V1exp(N1t), V2(expN2t), ..., Vnexp(Nnt)}, (6)

is a fundamental solution set for the homogeneous system X ′= AX . Hence

the general solution of X ′= AX is

X =C1V1exp(N1t) + C2 V2exp(N2t)...+ Cn Vnexp(Nnt),

where C1, ...., Cn are arbitrary constants matrices.

Proof. SinceAV i = NiVi,

we haved

dt(Viexp(Nit)) = NiViexp(Nit) = A (Viexp(Nit)) ,

so each element of the set (6) is a solution of the system (5) . Now, we need the

following two results. The first theorem guarantees the existence of a uniquesolution to an initial value problem for an MMs differential equation, while thesecond theorem gives a method for constructing the MMs exponential from thesolutions to certain scalar initial value problems.


Theorem 4.2 Let A ∈Mn

(M

′

l (K))

be MMs with characteristic matrix poly-nomial

φ (N) = (−1)n [INn + E1Nn−1 + E2N

n−2 + ....+ En−1N + En],

then Ψ (t) = exp(At) is the unique solution to the nth order MMs differentialequation

Ψ(n) + E1Ψ(n−1) + ....+ En−1Ψ

′ + EnΨ = O, (7)

satisfying the initial conditions

Ψ (0) = I, Ψ′(0) = A,..., Ψ(n−1) (0) = An−1. (8)

Proof. Suppose that A ∈Mn

(M

′

l (K))

is MMs with characteristic matrixpolynomial

φ (N) = (−1)n [INn + E1Nn−1 + E2N

n−2 + ....+ En−1N + En].

If Ψ (t) = exp(At), then

Ψ′(t) = Aexp(At), Ψ

′′

(t) = A2exp(At),..., Ψ(n) (t) = Anexp(At),

so that

Ψ(n) + E1Ψ(n−1) + ....+ En−1Ψ

′ + EnΨ

= (−1)n(IAn + E1An−1 + E2An−2 + ....+ En−1A+ EnI

)exp(At)

= φ (A) exp(At) = O.

Now, we prove the uniqueness let Ψ1 and Ψ2 be solutions to the nth orderMMs differential equation (7) and the initial conditions (8) , and assume Ψ =Ψ1 −Ψ2; then Ψ satisfies (5) with the initial conditions. Therefore, Ψ (t) = 0,and so Ψ1 = Ψ2.

Theorem 4.3 Let A ∈Mn

(M

′

l (K))

be MMs with characteristic matrix poly-nomial

φ (N) = (−1)n [INn + E1Nn−1 + E2N

n−2 + ....+ En−1N + En],

thenexp(At) = X1 (t) I+X2 (t)A+ ....+Xn (t)An−1,

where the Xi (t) , 1 ≤ i ≤ n, are the solutions to the nth order scalar differentialmatrix equation

X(n) + E1X(n−1) + ...+ En−1X

′+ En = O.


satisfying the following initial conditions:

X1 (0) = I X2 (0) = O · · · Xn (0) = OX

′1 (0) = O X

′2 (0) = I · · · X

′n (0) = O

...... · · · ...

X(n−1)1 (0) = O X

(n−1)2 (0) = O · · · X

(n−1)n (0) = I

.

Proof. Assume that A is a constant n× n MMs, and let

φ (N) = (−1)n [INn + E1Nn−1 + E2N

n−2 + ....+ En−1N + En],

be characteristic matrix polynomial.Define

Ψ (t) = X1 (t) I+X2 (t)A+ ....+Xn (t)An−1,where the Xi (t) , 1 ≤ i ≤ n, are the solutions to the nth order scalar differentialmatrix equation

X(n) + E1X(n−1) + ...+ En−1X

′+ En = O,

satisfying the initial conditions stated in the Theorem 4.3, then

Ψ(n) + E1Ψ(n−1) + ....+ En−1Ψ

′ + EnΨ

=(X

(n)1 + E1X

(n−1)1 + ...+ En−1X1 + En

)I

+(X

(n)2 + E1X

(n−1)2 + ...+ En−1X2 + En

)A

...

+(X(n)n + E1X

(n−1)n + ...+ En−1Xn + En

)An−1

= O,

thereforeΨ (t) = X1 (t) I+X2 (t)A+ ....+Xn (t)An−1,

satisfies the initial value problem

Ψ(n) + E1Ψ(n−1) + ....+ En−1Ψ

′ + EnΨ = O,

Ψ (0) = I, Ψ′(0) = A,..., Ψ(n−1) (0) = An−1.

Then the uniqueness of the solution gives

exp(At) = X1 (t) I+X2 (t)A+ ....+Xn (t)An−1,

for all t ∈ K.

Remark 4.4 As similar to usual ODEs, if N is a root of the characteristicmatrix polynomial of A, with multiplicity m, then the general solution of thescalar nth order matrix equation is of the form(

C0 + C1t+ C2t2 + ....+ Cm−1t

m−1) exp(Nt).


5 Examples and applications

In this section, we shall introduce an application to illustrate the results ofthis work in ODEs which real-life applications.

Predator-prey is one of the most famous real-life applications of ODEs,which describe the competition between several types of carnivore and vege-tarian (see [3]).As a result of conflicts, wars among nations, loss of life and loss of economy, theresearchers studied these models and ways to solve these problems and reducethese losses. Several pioneers such as Richardson (cf. [22, 23, 24]), Lanchester(see [17]), Morse (see [20]), Kimball (see [20]), provided mathematical modelsof combats for military or commercial purposes.

Here, we focus on the extension of the study of model Richardson, whichdescribes the relationship between two countries that deem war to be immi-nent. Let X = (xi,j[t]) be the armaments for the alliance of several countriesX and Y = (yi,j[t]) be the armaments for the alliance of several countries Yat time t for all i, j. The rate of change of the armaments on one side dependson the number of armaments. On the opposing side, if one the alliance ofcountries increases its armaments, the other will follow suit i.e., dX

dtand dY

dt

are proportional to X and Y, respectively. We are appointing constant matri-ces of proportionally A and B to X and Y , respectively, which represent theefficiency of increasing armaments, where X, Y ∈M ′

l (K[t]).Now, we established a system of differential matrix equations in the follow-

ing form:dX

dt= AY,

dY

dt= BX,

(9)

where X, Y ∈M ′

l (K[t]) and A, B ∈M ′

l (K).This system can be used to describe the relationship between two alliance of

several countries. One of them decides to defend itself against possible attackby the other.

Since

A =

(O AB O

),

then the characteristic matrix equation of A

det(A−NI) = O,

i.e.,N2 = AB,

by Cayley-Hamilton theorem


N2 − Tr(N)N + det(N)I = O,

Tr(N)N = AB + det(N)I,

moreover, since (det(N))2 = det(N2) = det (AB), then det(N) = ±√

det(AB),also we can solve for Tr(N) by taking traces on both sides of the above equa-tion to get

[Tr(N)]2 = Tr (AB)± l√

det(AB).

Therefore, the eigenvalues of A are N1 = 1Tr(N)

[AB +

√det(AB)I

]and N2 =

1Tr(N)

[AB −

√det(AB)I

], and the corresponding eigenvectors are

(Tr(N)B

[AB +

√det(AB)I

]−1I

)and

(Tr(N)

[AB −

√det(AB)I

]B−1

I

).

The solution of system (9) is given by:(XY

)= C1

(Tr(N)B

[AB +

√det(AB)I

]−1I

)exp

(1

Tr(N)

[AB +

√det(AB)I

])t+

C2

(Tr(N)B

[AB −

√det(AB)I

]−1I

)exp

(1

Tr(N)

[AB −

√det(AB)I

])t.

(10)

Given the initial conditions,

X (0) = X0, Y (0) = Y0,

result in

C1 =[X0 − Y0Tr(N)B

[AB −

√det(AB)I

]−1]

[Tr(N)B[AB +

√det(AB)I

]−1− Tr(N)B

[AB −

√det(AB)I

]−1],

C2 = Y0 −[X0 − Y0Tr(N)B

[AB −

√det(AB)I

]−1]

[Tr(N)B[AB +

√det(AB)I

]−1− Tr(N)B

[AB −

√det(AB)I

]−1].

(11)

Remark 5.1 (i) Note that, the values of A and B in equation (11) arethe main role of the behavior of the system. For instance, when Y is aconstant matrix C, it follows from (9) that

dX

dt= AC, (12)


solving (12), we obtain1

AX = Ct+D, (13)

assuming X(0) = O; it follows from (10) that D = O and

1

AX = Ct.

Hence when X has caught up to Y ; which means X = C; we have A−1 =It, i.e., A−1 is the time required for the alliance of several countries; Xto catch up with the armaments of Y provided that Y remains constantmatrix. Similar to Richardson we can observed that A is proportional tothe amount of industry in the alliance of several countries.

(ii) Assume that the relationship between two alliance of four countries andthe efficiency of increasing armaments for each the alliance is equal, andtaking

A = B =

(1 22 1

),

leads to

A =

(O AA O

).

Then the eigenvalues ofA areN1 =

(92

22 9

2

)andN1 =

(3√2

2√

2

2√

2 3√2

);

corresponding to the following eigenvectors 652

(92−2

−2 92

)I

,

−72

(3√2

−2√

2

−2√

2 3√2

)I

.

It follows from (10) and (11), with the initial condition

X (O) = 20I, Y (0) = O,

that

C1 =

20[584− 21√

2]

27188164− 12285

8√2+455

√2

20[ 652−7√2]

27188164− 12285

8√2+455

√2

20[ 652−7√2]

27188164− 12285

8√2+455

√2

20[584− 21√2]

27188164− 12285

8√2+455

√2

= −C2,


thus (XY

)= C1

652

(92−2

−2 92

)I

exp

(92

22 9

2

)t−

C1

−72

(3√2

−2√

2

−2√

2 3√2

)I

exp

(3√2

2√

2

2√

2 3√2

)t.

For the convenient of the reader, figure 1 was presented to show therelationship between the two alliance of four countries with several valuesof t. From figures 1 we note that the values of the X and Y are increasingto infinity, which means that there is a race in armaments, possiblyleading to a war between these two alliances.

Figuares 1. Solutions for the model of an arms race (9)(X-red and blue colors, Y - the rest of the colors)

Example 5.2 Given the system of matrix equations

dU

dt=

(1 22 1

)U + I2Y,

dY

dt= 2I2U + I2Y,


where U =

(u1 u2u3 u4

), Y =

(y1 y2y3 y4

).

Since

A =

(1 22 1

)I2

2I2 I2

,

we can use the previous theorem to calculate eAt as follows. The eigenvalues

of A are N1 =

(1 +√

3 1

1 1 +√

3

)and N2 =

(1−√

3 1

1 1−√

3

). The

general solution toX

′′+ E1X

′+ E0 = O,

is given by

X1 (t) = C1exp(

(1 +√

3 1

1 1 +√

3

)t) + C2exp(

(1−√

3 1

1 1−√

3

)t),

and form this we find the following:(i) The solution satisfying the initial conditions X (0) = I2 and X

′(0) = O2 is

X1 (t) =

(16

(3−√

3) −1

2√3

−12√3

16

(3−√

3) ) exp(( 1 +

√3 1

1 1 +√

3

)t) +(

16

(3 +√

3)

12√3

12√3

16

(3 +√

3) ) exp(( 1−

√3 1

1 1−√

3

)t).

(ii) Finally, the solution satisfying the initial conditionsX (0) = O2 andX′(0) =

I2 is

X2 (t) =

(1

2√3

0

0 12√3

)exp(

(1 +√

3 1

1 1 +√

3

)t) +(

−12√3

0

0 −12√3

)exp(

(1−√

3 1

1 1−√

3

)t),

thereforeexp(At) = X1 (t) I+X2 (t)A.

Example 5.3 In this example we consider the system of matrix equations

dU

dt= AU + I2Y +O2Z,

dY

dt= O2U + AY + I2Z,

dZ

dt= O2U +O2Y + AZ,


where, U =

(u1 u2u3 u4

), Y =

(y1 y2y3 y4

)and Z =

(z1 z2z3 z4

)∈ M2 (K[t])

and A ∈M ′2 (K) . The coefficient MMs is now given as

A =

A I2 O2

O2 A I2O2 O2 A

,

and the eigenvalues of A are A, A and A. The general solution to X′′′

+E2X′′+

E1X′+ E0X = O is given by

X (t) = C1exp(At) + C2texp(At) + C2t2exp(At),

where, the coefficients can be determined as;(i) The solution satisfying the initial conditions X (0) = I2, X

′(0) = O2 and

X′′

(0) = O2 is

X1 (t) =

(I2 − At+

A2t2

2

)exp(At).

(ii) While the solution satisfying the initial conditions X (0) = O2, X′(0) = I2

and X′′

(0) = O2 isX2 (t) =

(I2t− At2

)exp(At).

(iii) Finally, the solution satisfying the initial conditions X (0) = O2, X′(0) =

O2 and X′′

(0) = I2 is

X3 (t) =1

2I2t

2exp(At),

hence, we get

exp(At) = X1 (t) I+X2 (t)A+X3 (t)A2 = X1 (t) +X2 (t)A+X3 (t)A2 X2 (t) + 2X3 (t)A X3 (t)O2 X1 (t) +X2 (t)A+X3 (t)A2 X2 (t) + 2X3 (t)AO2 O2 X1 (t) +X2 (t)A+X3 (t)A2

.

In the above system if we take A =

(1 22 1

)and the initial condition

X (0) =

I2I2I

,

that

C =

I2I2I

,


thus UYZ

=

I + 2tI + t2I2I + 2tI

2I

exp

(1 22 1

)t.

Figure 2 was presented to show the soulation of the above system with severalvalues of t.

Figure 2. Soulation of the above system(U − 1 and 2, Y − 3 and 4, Z − 5 and 6)

Remark 5.4 It is worthy to ensure that when we write the above system ofhomogeneous linear matrix of first-order differential equations in the case ofsystem of homogeneous of linear first-order differential equations whose coeffi-cients are the scalar number instead of matrices and solve any system by usingthe exponential matrix we shall obtain the same results but we shall need alot of papers and time.

6 Conclusion

In this work, we have introduced definitions for eigenvalues, eigenvector, andexponential of MMs. Also, the system of homogeneous linear matrix of first-order differential equations is solved and existence and uniqueness of a solution


have been proven for that system. Furthermore, explain some examples andapplications of a solution the system of homogeneous linear matrix of first-order differential equations. Finally, it is worth to mention that, dealing withthe above system in an ordinary way, leads to the same results as using MMsrather than the used time and effort which makes the impacts of the MMsconcept.

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Received: March 5, 2019; Published: March 24, 2019

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