#### Abstract

We study a class of linear matrix differential equations (regular case) of higher order whose coefficients are square constant matrices. By using matrix pencil theory and the Weierstrass canonical form of the pencil we obtain formulas for the solutions and we show that the solution is unique for consistent initial conditions and infinite for nonconsistent initial conditions. Moreover we provide some numerical examples. These kinds of systems are inherent in many physical and engineering phenomena.

#### 1. Introduction

Linear matrix differential equations (LMDEs) are inherent in many physical, engineering, mechanical, and financial/actuarial models. Having in mind such applications, for instance in finance, we provide the well-known input-output Leondief model and its several important extensions, see [1, 2]. In this paper, our long-term purpose is to study the solution of LMDEs of higher order (1.1) into the mainstream of matrix pencil theory. This effort is significant, since there are numerous applications. Thus, we consider where , , , (i.e., the algebra of square matrices with elements in the field ) with . For the sake of simplicity we set and . In the sequel we adopt the following notations: Or in Matrix form where (where is the transpose tensor) and the coefficient matrices , are given by with corresponding dimension of , , and , and , respectively. Matrix pencil theory has been extensively used for the study of linear differential equations (LDEs) with time invariant coefficients, see for instance [1–5]. Systems of type (1.1) are more general, including the special case when , where is the identity matrix of , since the well-known class of higher-order linear matrix differential equations of Apostol-Kolodner type is derived straightforwardly, see [6] for , [7, 8].

The paper is organized as follows: in Section 2, some notations and the necessary preliminary concepts from matrix pencil theory are presented. Section 3 contains the case that system (1.1) has consistent initial conditions. In Section 4, the nonconsistent initial condition case is fully discussed. In this case, the arbitrarily chosen initial conditions which have physical meaning for (regular) systems, in some sense, can be created or structurally changed at a fixed time . Hence, it is derived that (1.1) should adopt a generalized solution, in the sense of Dirac -solutions.

#### 2. Mathematical Background and Notation

This brief section introduces some preliminary concepts and definitions from matrix pencil theory, which are being used throughout the paper. Linear systems of type (1.1) are closely related to matrix pencil theory, since the algebraic geometric and dynamic properties stem from the structure by the associated pencil .

*Definition 2.1. * Given and an indeterminate , the matrix pencil is called regular when and . In any other case, the pencil will be called singular.

*Definition 2.2. * The pencil is said to be *strictly equivalent* to the pencil if and only if there exist nonsingular and such as

In this paper, we consider the case that pencil is *regular*. Thus, the strict equivalence relation can be defined rigorously on the set of regular pencils as follows. Here, we regard (2.2) as the set of pair of nonsingular elements of
and a composition rule defined on as follows:
It can be easily verified that forms a *nonabelian group*. Furthermore, an action of the group on the set of *regular* matrix pencils is defined as such that
This group has the following properties: (a) for every nonsingular and .(b), where is the *identity element* of the group on the set of defines a transformation group denoted by , see [9].

For , the subset
will be called the orbit of at . Also defines an equivalence relation on which is called a *strict-equivalence relation* and is denoted by .

So, if and only if , where are nonsingular elements of algebra .

The class of is characterized by a uniquely defined element, known as a complex Weierstrass canonical form, , see [9], specified by the complete set of invariants of .

This is the set of *elementary divisors* (e.d.) obtained by factorizing the invariant polynomials into powers of homogeneous polynomials irreducible over field . In the case where is a regular, we have e.d. of the following type:(i) e.d. of the type *are called zero finite elementary divisors* (z. f.e.d.), (ii) e.d. of the type , *are called nonzero finite elementary divisors* (nz. f.e.d.), (iii) e.d. of the type are called *infinite elementary divisors* (i.e.d.).

Let be elements of . The direct sum of them denoted by is the block diag.

Then, the complex Weierstrass form of the regular pencil is defined by , where the first normal Jordan type element is uniquely defined by the set of f.e.d. of and has the form And also the blocks of the second uniquely defined block correspond to the i.e.d. of and has the form Thus, is a nilpotent element of with index , where and , , are defined as In the last part of this section, some elements for the analytic computation of , are provided. To perform this computation, many theoretical and numerical methods have been developed. Thus, the interesting readers might consult papers [7, 8, 10–12] and the references therein. In order to have computational formulas, see the following Sections 3 and 4, the following known results should firstly be mentioned.

Lemma 2.3 (see [10]). *, where
*

Another expression for the exponential matrix of Jordan block, see (2.11), is provided by the following lemma.

Lemma 2.4 (see [12]). *
where the 's are given analytically by the following equations:
**
where
**
where
*

#### 3. Solution Space Form of Consistent Initial Conditions

In this section, the main results for consistent initial conditions are analytically presented for the regular case. Moreover, it should be stressed out that these results offer the necessary mathematical framework for interesting applications; see also Introduction. Now, in order to obtain a unique solution, we deal with consistent initial value problem. More analytically, we consider the system with known initial conditions Analytically, we consider the system From the regularity of , there exist nonsingular matrices and such that (see also Section 2), such as where , , , and are given by (2.11) where Note that and , where .

Lemma 3.1. * System (3.1) is divided into two subsystems: the so-called slow subsystem
**
and the relative fast subsystem
*

*Proof. * Consider the transformation
Substituting the previous expression into (3.1) we obtain
Whereby, multiplying by , we arrive at
Moreover, we can write as . Taking into account the above expressions, we arrive easily at (3.6) and (3.7).

Proposition 3.2. *The sub-system (3.6) has the unique solution
**
where .*

Proposition 3.3. * The fast subsystem (3.7) has only the zero solution. *

*Proof. * Let be the index of the nilpotent matrix , that is, , we obtain the following equations:
where is by definition the Laplace transform of . It is easy to show that and that , while for
where is by definition the Dirac function
The conclusion, that is, , is obtained by repetitive substitution of each equation in the next one, and using the fact that .

Theorem 3.4. * Consider the system (3.1)-(3.2).Then the solution is unique if and only if the initial conditions are consistent. Moreover the analytic solution of (3.1)-(3.2) is given by
*

*Proof. *Let where and ; combining (3.8) and (3.17), we obtain
solution that exists if and only if or . The columns of are the eigenvectors of the finite elementary divisors (eigenvalues) of the pencil . In that case the system has the unique solution
where is defined as
and .

#### 4. Form on Nonconsistent Initial Condition

In this short section, we describe the impulse behavior of the original system (1.1), at time . In that case, we reformulate Theorem 3.4, so the impulse solution is finally obtained.

Proposition 4.1. * Consider the system (3.3). Then for nonconsistent initial conditions () the system has infinite solutions. *

*Proof. * Let , be the matrices defined in Theorem 3.4. If the initial conditions are nonconsistent then and 0. Moreover .

This means (3.3) is defined for because if then and which is a contradiction. Let be the Heaviside function and
Then the system can be written as
This is a linear matrix differential equation of first order with being the solution of the homogeneous and a partial solution. And we obtain the general solution
where is constant vector and the dimension of the solution vector space is .

Theorem 4.2. * Consider the system (3.1)-(3.2) with nonconsistent initial conditions. Then the system has infinite solutions. *

*Proof. * We rewrite the system (3.1) in the following form:
where and are the functions defined in Theorem 4.2. Combining the results of Theorem 4.2 and the above discussion the solution of the system is
The dimension of the solution vector space is .

*Remark 4.3. * For , it is obvious that (4.5) is satisfied. Thus, we should stress out that the system (3.1)-(3.2) has the above impulse behaviour at time instant where a non-consistent initial value is assumed, while it returns to smooth behaviour at any subsequent time instant.

#### 5. Numerical Example

Let where . We adopt the following notations: Or in Matrix form where (where is the transpose tensor) and the coefficient matrices , are given by , , finite elementary divisors and the infinte elementary divisor of degree 3 of the pencil . There exist matrices nonsingular , such that and , where Let then For consistent initial conditions the solution is and for nonconsistent initial conditions the solution is where The columns of are the eigenvectors of the eigenvalues 1,2,3 Let the initial values of the system be Then (consistent initial conditions) and the solution of the system is and by calculating we get and the solution of the system is Next assume the initial conditions Then nonconsistent initial conditions and the solution is The dimension of the domain that describes the solutions of the system is 3.

#### 6. Conclusions

In this paper we investigate systems of the form suggested in [1–3, 5, 7], but from another point of view. By taking into consideration that the relevant pencil is regular, we use the Weierstrass canonical form in order to decompose the differential system into two sub-systems. Afterwards, we give necessary and sufficient conditions for existence and uniqness of solutions for that general class of linear matrix differential equations of higher order and we provide analytical formulas when we have consistent and non-consistent initial conditions. Moreover, as a further extension of the present paper, we can discuss the case where the pencil is singular. Thus, the Kronecker canonical form is required. The non-homogeneous case has also a special interest, since it appears often in applications. For all these, there is some research in progress.

#### Acknowledgment

The author would like to express his sincere gratitude to my supervisor Professor G. I. Kalogeropoulos for making for him available personal notes and notations that he used in Sections 1 and 2 and for his fruitful discusion that improved the paper.