The Scientific World Journal

Volume 2014 (2014), Article ID 493606, 6 pages

http://dx.doi.org/10.1155/2014/493606

Research Article

Extending an Almost Complete Pair of Matrices to a Complete Triple

Sheng Chen and Chao Xia

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 18 February 2014; Accepted 3 April 2014; Published 27 April 2014

Academic Editor: Lingju Kong

Copyright © 2014 Sheng Chen and Chao Xia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Motivated by the concept of complete pairs, which was introduced by Krein and Langer, we present the concepts of an almost complete pair of matrices and a complete triple of matrices. It is proved that an almost complete pair of matrices can be extended to a complete triple. An application of the result to differential equations is also given.

1. Introduction

Let

be a monic matrix polynomial, where for . If there exist a matrix and a polynomial of degree such that , then is said to be a monic right divisor of . As is well known, is a monic right divisor of if and only if is a root of ; that is,

Let be the companion matrix of ; that is,

Then we have . If are the different roots of , it is clear that the Vandermonde matrix

plays a role in the investigation of the spectral analysis of [1].

Definition 1. We say that the -tuple of complex square matrices of order is complete if the Vandermonde matrix is invertible. When and , we call a complete -tuple of matrices a complete pair and a complete triple, respectively.

Suppose that is complete. Take . Then is invertible. If we take

then , , form a complete pair of right divisors of (cf. Lemma 2.4 in [2]). The concept of complete pair was introduced in [3, 4]. In [5], Lancaster gave some important applications of complete pairs to solutions of differential equations. Motivated by the results in [5], we are interested in the following question.

Question 1. Suppose that is not a complete pair. Is it possible for us to find and a cubic matrix polynomial such that is a complete triple and

are right divisors of ?

If is a complete triple, then

is of full column rank. Note that if and the rank of is less than , we cannot always find such that is a complete triple. For example, let be defined as follows:

where , are square matrices of orders and 2, respectively, and is in the form

Then

If , then the last two columns of are linearly dependent. For any ,

cannot be invertible.

If , then

is of full column rank. From the condition that

is of full column rank, we cannot conclude that .

After the above discussion, we give the following definition.

Definition 2. For two matrices , one says they form an almost complete pair if the matrix

is of full column rank and the rank of is . For simplification, one says is an almost complete pair of matrices of order .

The following is the main result of this paper.

Theorem 3. Suppose that , , and is an almost complete pair. Then one can find a matrix such that is a complete triple; that is,

is invertible.

We prove this theorem in Section 2. In Section 3, we partially answer the question mentioned before and give an application of the main result to differential equations.

2. Proof of the Main Theorem

In this section, we prove Theorem 3. Note that there is an invertible matrix such that is the Jordan normal form of . Let . Then the condition that

is invertible is equivalent to that

is invertible. So from the beginning, we can assume that is in Jordan normal form.

First we prove a lemma.

Lemma 4. Assume and the pair is an almost complete pair with . Write

where . Then is not a zero vector.

Proof. By some computations, we have

The last column of

is , which is a -dimensional column vector. By the definition of almost complete pair, is not a zero vector.

Now we prove the main theorem.

Proof. From now on, we write . We denote by the matrix which is obtained by exchanging the columns of and of identity matrix.

Without loss of generality, we can assume is in the Jordan normal form.

Case  1. Firstly, we consider the case that

If , then we take

By some computation, the last columns of and are and , respectively, and the th () column of is

and the th column of is , where and . It is clear that

is a polynomial in . If , then the degree of this polynomial is and the leading term of this polynomial is . If is sufficiently large, we can get that

is invertible. If , by Lemma 4, then there must be some

where . So

where the first entry of the vector is . Take such that

Then

is invertible.

Case  2. Now we consider the case that has the Jordan normal form

where , . If , then we take

If and , then we take such that

In any case when is sufficiently large, we can get that

is invertible.

Case  3. Now we consider the case that 0 is not a single root of characteristic polynomial of ; that is, has the Jordan normal form:

If , then

By Lemma 4, . Take

Then

We can conclude that

is invertible.

Firstly, we prove the case that the normal form itself is

If , then we take

So

By some computation,

is a polynomial in , where the leading term of it is . So if , then is invertible when is sufficiently large. If , then there must be an such that . If , take such that

When is sufficiently large, we have being invertible. If , that is to say, the last column of is , then we take

We have

where the first entry of the last column of is . So the leading term of the polynomial

is . Thus

is invertible when is sufficiently large.

Now we prove the general case that has the Jordan normal form:

where , . If there is some , , we assume that is a matrix of order . For any square matrix of order , we can construct a square matrix of order such that

is invertible. Take

If , for all , there must be some such that , . Let , where is the matrix which is obtained by exchanging the columns and of identity matrix. Then and

where the square matrix of order has the form

Let be a matrix such that has the Jordan normal form . Then we can consider the matrix . Note that has the form

Denote . For any square matrix , we can construct a square matrix of order such that

is invertible. Take . Then we can conclude that

is invertible. The proof is complete.

3. Applications

For given monic matrix polynomials , the construction of a common multiple of them was presented in [2].

Suppose that is an almost complete pair. By Theorem 3 in Section 1, we can find such that is a complete triple; that is,

is invertible. Let

Then we can conclude that

By Theorem 9.11 of [2], there exists a monic matrix polynomial with degree 3 which is a common multiple of , , .

As is well known, spectral method for the analysis of monic matrix polynomials was important (cf. [6–13]). The spectral method basically uses the construction of standard pairs and standard triples (e.g., [14, 15]) to solve differential equations.

Now we recall definitions of standard pair and standard triple for monic polynomials.

Definition 5 (see [2]). A pair of matrices , where is and is , is called a standard pair for the monic matrix polynomial if the following conditions are satisfied:

(i) is nonsingular, where

(ii).

If is in Jordan normal form, we call a Jordan pair for the monic matrix polynomial .

Definition 6 (see [2]). A triple of matrices , where is , is , and is , is called a standard triple for the monic matrix polynomial if is a standard pair for , and .

Now we recall the explicit formulas for solutions of the following differential equation:

Lemma 7 (Theorem 2.9 in [2]). The general solution of (60) is given by the formula

where is a standard triple of and is arbitrary. In particular, the general solution of the homogeneous equation

is given by the formula

Before we give the main result of this section, we introduce some notations. Suppose

are the Jordan forms of , , , respectively. Write

Theorem 8. Let be a complete triple of , where . Then every solution of

is given by

for some , where

In particular, every solution of

has the form

where .

Proof. Using Lemma 7, we take

Then is a standard triple; it is easy to get the conclusion.

Remark 9. The above theorem and its proof are motivated by those in [5] (cf. also Sections 2.4 and 2.5 in [2]).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referees for their helpful comments and suggestions. The research was supported by National Natural Science Foundation of China (Grant no. 11001064), by the Fundamental Research Funds for the Central Universities (Grant no. HIT. NSRIF. 2014085), and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars.

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