Prime decomposition of quadratic matrix polynomials

1.   Introduction

The linear second-order differential equation

where $A, B\in \mathbb{C}^{n\times n}$ and $q(t)$ is an $n$th-order vector, frequently arise in the fields of mechanical and electrical oscillation ^[16]. The study of the solutions of the Eq (1.1) lead to the research of a quadratic matrix polynomial

where $A, B\in M_n(\mathbb{C})$ and $I$ is the identity matrix of order $n$ ^[7]. In this work we investigate the prime decomposition of quadratic matrix polynomial (1.2). By the prime decomposition of (1.2), we represent the general solution of Eq (1.1).

Consider the solution of the homogeneous equation (1.1) with $n = 1$ and $f = 0$, rewritten in the form

We knows that the general solution of Eq (1.3) with $a^2-4b\neq0$ is

where $c_1, c_2$ are constants and $\lambda_1, \lambda_2$ are the roots of $\lambda^2+a\lambda+b = 0.$ In analogy with Formula (1.4) we ask whether Eq (1.1) has the formula of the general solution.

The quadratic matrix polynomial (1.2) is called factorizable if it can be factorized into a product of two linear matrix polynomial, i.e.

where $C, D\in \mathbb{C}^{n\times n},$ and $I\lambda+D$ is called a right divisor of $L(\lambda)$. Right divisors $I\lambda+D_1$, $I\lambda+D_2$ for $L(\lambda)$ are said to form a complete pair if $D_1-D_2$ is invertible.

There have been extensive study and application of the matrix polynomial factorization (see ^{[4,9,10,11,14,15]}). The quadratic eigenvalue problems (see ^[13,16]) received much attention because of its applications in the dynamic analysis of mechanical systems in acoustics and linear stability of flows in fluid mechanics. A solution of a quadratic matrix equation can be obtained by the fraction of a quadratic matrix polynomial ^[7]. A system of second-order differential equation with self-adjoint coefficients may describe the ubiquitous problem of damped oscillatory systems with a finite number of degrees of freedom. This leads to the study of Hermitian quadratic matrix polynomials^[1,12].

Gohberg, Lancaster and Rodman analyzed some properties of linear second-order differential equation (1.1) and quadratic matrix polynomial (1.2) in ^[7]. It was shown that Eq (1.1) has a formula of the general solution similar to Formula (1.4) if the quadratic matrix polynomial (1.2) has a complete pair ^[7]. Motivated by results in ^[7], we propose a concept of prime decomposition to generalized this result.

This paper is organized as follows. In section 2, we give the definition of prime decomposition of a quadratic monic matrix polynomial $L(\lambda)$ and some properties. By the prime decomposition, we get an integral formula for the corresponding second order matrix differential equation. In section 3, we investigate the prime decomposition of a class of quadratic matrix polynomial. A sufficient condition for $L(\lambda)$ having a prime decomposition is presented in Theorem 3.1.

2.   Prime decomposition and its applications

First we give the definition of prime decomposability of a quadratic matrix polynomial $L(\lambda)$. Some equivalent conditions of prime decomposability are given in Remark 2.2.

Definition 2.1. The matrix polynomial $L(\lambda)$ has a prime decomposition if there exist $C, D\in M_n(\mathbb{C})$ such that

and there are $U, V\in M_n(\mathbb{C})$ such that

Remark 2.2. Our definition of prime decomposition of matrix polynomials is different from the definition of coprime factorization of matrix polynomials which was studied in many papers (e.g. Definition 3.1 in ^[6]). By Lemma 3.1 in ^[2], it is easy to verify that condition (2.2) is equivalent to each of the following statements:

(1) There is $H\in M_n(\mathbb{C})$ such that

(2) There are matrix polynomials $U(\lambda), V(\lambda)$ such that

Motivated by Lemma 3.1 in ^[8], we have the following theorem. It shows the significance of prime decomposability in solving equations.

Theorem 2.3. Let $R$ be a ring with identity $1$. Suppose that $f, g, a, b\in R$ satisfy the condition

Let $m$ be in a left $R$ module $M$. If $y, z\in M$ and

then $x = by+z$ is a solution of the equation $fg(x) = m.$ Conversely, any solution $x\in M$ of the equation $fg(x) = m$ can be written as the following form, $x = by+z,$ where $y, z\in M$ and

In particular, by taking $m = 0$ we get $\ker(fg) = b\ker(f)+\ker(g).$

Proof. Suppose that

Then we have

So $x = by+z$ is a solution to $fg(x) = m$.

Conversely, if $fg(x) = m$, then we take

We will get

The proof is complete.

The following result, whose proof is obvious, presents an immediate application of the the above theorem to differential equations.

Corollary 2.4. Suppose that $L(\lambda) = \lambda^2I+\lambda A+B$ is prime decomposable, i.e., there are $C, D, H\in M_n(\mathbb{C})$ such that $L(\lambda) = (\lambda I+C)(\lambda I+D)$ and $DH-HC = I.$ Then the solution of $L(d/dt)u = f$ has the form

for some $\alpha_1, \alpha_2\in \mathbb{C}^{n}$. In particular, the solution of $L(d/dt)u = 0$ has the form

for some $\alpha_1, \alpha_2\in \mathbb{C}^{n}$.

Remark 2.5. Suppose that

is a complete pair for $L(\lambda)$ (see Section 2.5 in ^[7]). Then $S_2-S_1$ is some invertible matrix, say $P$. Let $C = PS_2P^{-1}, D = S_1.$ Then $P^{-1}C-DP^{-1} = I$. So

is a prime decomposition. Thus we can recover Theorem 2.16 in ^[7] from Corollary 2.4.

3.   The prime decomposability of a quadratic matrix polynomial

We first propose some notations used in this section. Let $R_1$, $R_2$ be $n\times n$ matrices with $R_1R_2 = R_2R_1$. The joint spectrum, denoted by $\sigma(R_1, R_2)$, is a subset of $\mathbb{C}^2$ defined by

Since $R_1R_2 = R_2R_1$, there exists an invertible matrix $T$ such that

and the joint spectrum $\sigma(R_1, R_2)$ can be read off from the diagonal elements in (3.1), namely,

The multiplicity of $\mathbf{k}\in\sigma(R_1, R_2)$ is the number of $\mathbf{k}_i, i = 1, \ldots, n$ which are same as $\mathbf{k}$. The matrix $M$ is called upper Toeplitz matrix if

The polynomial $F(\lambda)$ is called upper Toeplitz matrix polynomial if

where $f_1(\lambda) = \lambda^2+a\lambda+b$, and $\deg(f_i)\leq1, i = 2, \ldots, n.$

It is shown that a matrix polynomial with pairwise commuting coefficients of the simple structure can be represented in the form of a product of linear factors ^[15]. Motivated by ^[15], we investigate the prime decomposability of $L(\lambda)$ with pairwise commuting coefficients. The following theorem gives a sufficient condition for $L(\lambda)$ having a prime decomposition.

Theorem 3.1. If polynomial matrix $L(\lambda) = \lambda^2I+A\lambda+B$ satisfies the following conditions:

(i) $AB = BA$,

(ii) each nonlinear elementary divisor of $A$ and $B$ is coprime with the other elementary divisors of $A$ and $B$, respectively,

(iii) the degrees of elementary divisors of $L(\lambda)$ are not great than $2$,

(iv) the multiplicity of eigenvalue $\mathbf{k} = (a, b)\in\sigma(A_1, A_0)$ satisfying $a^2-4b = 0$ is even.

Then $L(\lambda)$ has a prime decomposition.

Prior to the proof of this theorem, we formulate several auxiliary statements. By the following two lemmas, the problem of prime decomposability of a quadratic monic matrix polynomials can be reduced in some sense.

Lemma 3.2. Suppose that $A, B, T\in M_n(\mathbb{C})$, where $T$ is invertible. Then $L(\lambda) = \lambda^2I+A\lambda+B$ is prime decomposable if and only if $TL(\lambda)T^{-1}$ is prime decomposable.

Proof. Note that $TL(\lambda)T^{-1} = (\lambda I+TCT^{-1})(\lambda I+TDT^{-1}),$ where $T$ is invertible. Furthermore, Eq (2.3) is equivalent to equation $(TDT^{-1})(THT^{-1})-(THT^{-1})(TCT^{-1}) = I.$ By Definition 2.1, the result follows. The proof is complete.

Lemma 3.3. Suppose $a\in\mathbb{C}$. Then $L(\lambda) = \lambda^2I+A\lambda+B$ is prime decomposable if and only if $\lambda^2I+\lambda\tilde{A}+\tilde{B}$ is prime decomposable, where $\tilde{A} = 2aI+A, \tilde{B} = a^2I+aA +B.$

Proof. If $L(\lambda)$ is decomposable, then there exist $C, D\in M_n({\mathbb{C}})$ such that

Let $\tilde{C} = aI+C$, and $\tilde{D} = aI+D$. We have

Furthermore, the condition that there exists $H$ such that $DH-HC = I$ is equivalent to the condition $\tilde{C}H-H\tilde{D} = I.$ By Definition 2.1, the result follows.

The following lemma is a well known result about Sylvester equation which was studied in many papers ^[3,5].

Lemma 3.4. ^[7] Let $C, D\in M_n(\mathbb{C})$. If $\sigma(C)\cap \sigma(D) = \emptyset$, where $\sigma(X)$ is the set of eigenvalues of the matrix $X$, then for any $U\in M_n(\mathbb{C})$, there exists a unique $H\in M_n(\mathbb{C})$ such that $DH-HC = U$.

Lemma 3.5. Suppose $F(\lambda)$ is upper Toeplitz matrix polynomial with $f_1(\lambda) = \lambda^2+a\lambda+b$ on the diagonal. If $a^2\neq4b$, then $F(\lambda)$ has a prime decomposition.

Proof. As shown in ^[15], $F(\lambda)$ can be decomposed into a product of linear factors which has only one (without regard for multiplicity) characteristic root. By Lemma 3.4, we know that the decomposition is a prime decomposition.

Lemma 3.6. Suppose $F(\lambda)$ is upper Toeplitz matrix polynomial with $f_1(\lambda) = \lambda^2+a\lambda+b$ on the diagonal, and $n$ is the order of $F(\lambda)$. If $a^2 = 4b$ and the degrees of elementary divisors of $F(\lambda)$ are not great than $2$, then $F(\lambda)$ has a prime decomposition if and only if $n$ is even.

Proof. Since the degrees of elementary divisors of $F(\lambda)$ are not great than $2$, we have

where $c_1 = \frac{a}{2}$. By Lemma 3.3, it suffices to prove the prime decomposability of $\lambda^2I_n$, i.e., there exists $C$ such that

where $C^2 = 0$ and there is $H$ satisfying

We prove the result in two cases.

Case 1. $n = 2k$, $k$ is an nonnegative integer.

Let

Then

satisfy Eqs (3.2) and (3.3).

Case 2. $n = 2k+1,$ $k$ is an nonnegative integer.

By Lemma 3.2, we suppose $C = diag(J_1, \ldots, J_s),$ where $J_i$ is a Jordan block with zero on the diagonal for $i = 1, \ldots, s$. Since $C^2 = 0$, the order of $J_i$ is not great than $2$ for $i = 1, \ldots, s$. Thus there exists $r$ such that $J_r = 0.$ Without loss of generality, we assume $J_1 = 0$. Then the $(1, 1)$ entry of $-CH-HC$ is $0$. Hence there does not exist $H$ satisfying Eq (3.3).

Now we are ready to give the proof of Theorem 3.1.

Proof of Theorem 3.1. Suppose that $L(\lambda) = \lambda^2I+\lambda A+B$ satisfies the conditions of Theorem 3.1. Hence there exists an invertible matrix $T$ such that

where $J_i$ is a Jordan block with $b_i$ on the diagonal for $i = l+1, \ldots, s$, and $b_i\neq b_j$ for $i\neq j$($i, j = 1, \ldots, s$). Since $AB = BA$, we have

where $M_{i}$($i > l$) is upper Toeplitz matrix(see Theorem S2.2 in ^[7]). Note that there exist invertible $T_i$ such that

where $J_{i_j}$ is a Jordan block with $a_{i_j}$ on the diagonal for $j = l, \ldots, m_i$.

Let $H = Tdiag(T_1, \ldots, T_l, I, \ldots, I).$ We have

where $h = \sum_{i = 0}^lm_i+s-l,$ and $L_i(\lambda)$ is upper Toeplitz matrix polynomial for $i = 1, \ldots, h$. Because $L_i(\lambda)(i = 1, \ldots, h$) satisfy the conditions of Theorem 3.1, from Lemmas 3.5 and 3.6, we know that $L_i(\lambda)(i = 1, \ldots, h$) has a prime decomposition, say

Hence $HL(\lambda)H^{-1}$ has a prime decomposition

By Lemma 3.2, we know that $L(\lambda)$ has a prime decomposition.

4.   Conclusions

In this paper, we have presented the prime decomposition of a quadratic monic matrix polynomial and the application in solving corresponding second-order differential equation. We have got a sufficient condition for the existence of a prime decomposition for a quadratic matrix polynomial with pairwise commuting coefficients. As has been said, a complete pair can be used to form a prime decomposition of the quadratic matrix polynomial. Thus, we expect that the relation between a prime decomposition and a complete pair can be studied more thoroughly. The prime decomposition of a matrix polynomial with degree $n$ will be investigated in a future paper.

Acknowledgments

The first author is supported by the Natural Science Foundation of Shandong Province, China (Grant No. ZR2018PA002).

Conflict of interest

The authors declare that they have no conflicts of interest.