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.