An Analog of the Adjugate Matrix for the Outer Inverse

Abstract

We investigate the determinantal representation by exploiting the limiting expression for the generalized inverse . We show the equivalent relationship between the existence and limiting expression of and some limiting processes of matrices and deduce the new determinantal representations of , based on some analog of the classical adjoint matrix. Using the analog of the classical adjoint matrix, we present Cramer rules for the restricted matrix equation .

1. Introduction

Throughout this paper denotes the set of matrices over the complex number field , and denotes its subset in which every matrix has rank . stands for the identity matrix of appropriate order (dimension).

Let , and let and be Hermitian positive definite matrices of orders and , respectively. Consider the following equations: is called a - (or outer) inverse of if it satisfies (2) and denoted by . is called the Moore-Penrose inverse of if it satisfies (1), (2), (3), and (5) and denoted by . is called the weighted Moore-Penrose inverse of (with respect to ) if it satisfies (1), (2), (4), and (6) and denoted by (see, e.g., [1, 2]).

Let . Then a matrix satisfying where is some positive integer, is called the Drazin inverse of and denoted by . The smallest positive integer such that and satisfy (7),  (8), and (9), then it is called the Drazin index and denoted by . It is clear that is the smallest positive integer satisfying (see [3]). If , then is called the group inverse of and denoted by . As is well known, exists if and only if . The generalized inverses, and in particular Moore-Penrose, group and Drazin inverses, have also been studied in the context of semigroups, rings of Banach and algebras (see [4–8]).

In addition, if a matrix satisfies (1) and (5), then it is called a -inverse of and is denoted by .

Let , . Then the matrix satisfying where is some nonnegative integer, is called the -weighted Drazin inverse of , and is denoted by (see [9]). It is obvious that when and , is called the Drazin inverse of .

Lemma 1.1 (see [1, Theorem  2.14]). Let , and let and be subspaces of and , respectively, with . Then has a -inverse such that and if and only if in which case is unique and denoted by .

If exists and there exists a matrix such that and , then and .

It is well known that several important generalized inverses, such as the Moore-Penrose inverse , the weighted Moore-Penrose inverse , the Drazin inverse , and the group inverse , are outer inverses for some specific choice of and , are all the generalized inverse , - (or outer) inverse of with the prescribed range and null space (see [2, 10] in the context of complex matrices and [11] in the context of semigroups).

Determinantal representation of the generalized inverse was studied in [12, 13]. We will investigate further such representation by exploiting the limiting expression for . The paper is organized as follows. In Section 2, we investigate the equivalent relationship between the existence of and the limiting process of matrices or and deduce the new determinantal representations of , based on some analog of the classical adjoint matrix, by exploiting limiting expression. In Section 3, using the analog of the classical adjoint matrix in Section 2, we present Cramer rules for the restricted matrix equation . In Section 4, we give an example for solving the solution of the restricted matrix equation by using our expression. We introduce the following notations.

For , the symbol denotes the set . And , where .

Let . The symbols and stand for the th column and the th row of , respectively. In the same way, denote by and the th column and the th row of Hermitian adjoint matrix . The symbol (or ) denotes the matrix obtained from by replacing its th column (or row) with some vector (or ). We write the range of by and the null space of by . Let . We define the range of a pair of and as .

Let and , where . Then denotes a minor of determined by the row indexed by and the columns indexed by . When , the cofactor of in is denoted by .

2. Analogs of the Adjugate Matrix for

We start with the following theorem which reveals the intrinsic relation between the existence of and of or . Here means through any neighborhood of 0 in which excludes the nonzero eigenvalues of a square matrix. In [14], Wei pointed out that the existence of implies the existence of or . The following result will show that the converse is true under some condition.

Theorem 2.1. Let , and let and be subspaces of and , respectively, with . Let with and . Then the following statements are equivalent:(i) exists;(ii) exists and ;(iii) exists and .
In this case,

Proof. (i)⇔(ii) Assume that exists. By [14, Theorem  2.4], exists. Since , .
Conversely, assume that exists and . So exists. By [15, Theorem], exists. So exists, and then, by [13, Theorem  2], exists.
Similarly, we can show that (i)⇔(iii). Equation (2.1) comes from [14, equation (2.16)].

Lemma 2.2. Let and . Then , where , , and , where , .

Proof. Let be an matrix with in the entry, 1 in all diagonal entries, and 0 in others. It is an elementary matrix and It follows from the invertibility of , that .
Analogously, the inequation can be proved. So the proof is complete.

Recall that if is the characteristic polynomial of an matrix— over , then is the sum of all principal minors of , where (see, e.g., [16]).

Theorem 2.3. Let , and be the same as in Theorem 2.1. Write . Suppose that the generalized inverse of exists. Then can be represented as follows: where or where

Proof. We will only show the representation (2.5) since the proof of (2.7) is similar. If is not the eigenvalue of , then the matrix is invertible, and where , are cofactors of . It is easy to see that So, by (2.1),
We have the characteristic polynomial of where is a sum of principal minors of . Since , and Expanding , we have where , , for and .
By Lemma 2.2, and so , and , for all . Therefore, , , for all . Consequently,
Substituting (2.12) and (2.14) into (2.10) yields
Substituting for in the above equation, we reach (2.5).

Remark 2.4. The proofs of Lemma 2.2 and Theorem 2.3 are based on the general techniques and methods obtained previously by [17], respectively.

Remark 2.5. (i) By using (2.5), we can obtain (2.17) in [12, Theorem  2.3]. In fact, and, by the Binet-Cauchy formula, where . Note that if or . In addition, using the symbols in [13], we can rewrite (2.5) as [13, equation (13)] over .
(ii) This method is especially efficient when or is given (comparing with [12, Theorem  2]).

Observing the particular case from Theorem 2.3, , where and are Hermitian positive definite matrices, we obtain the following corollary in which the symbols and .

Corollary 2.6. Let and , where and are Hermitian positive definite matrices of order and , respectively, Then where or where

If and are identity matrices, then we can obtain the following result.

Corollary 2.7 (see [17, Theorem  2.2]). The Moore-Penrose inverse of can be represented as follows: where or where

Note that . Therefore, when in Theorem 2.3, we have the following corollary.

Corollary 2.8. Let , , and . If , , and , then where or where

When with in Theorem 2.3, we have the following corollary.

Corollary 2.9 (see [17, Theorem  3.3]). Let with and , and . Then where

Finally, we turn our attention to the two projectors and . The limiting expressions for in (2.1) bring us the following:

Corollary 2.10. Let , and be the same as in Theorem 2.1. Write and . Suppose that exists. Then of can be represented as follows: where where

3. Cramer Rules for the Solution of the Restricted Matrix Equation

The restricted matrix equation problem is mainly to find solution of a matrix equation or a system of matrix equations in a set of matrices which satisfy some constraint conditions. Such problems play an important role in applications in structural design, system identification, principal component analysis, exploration, remote sensing, biology, electricity, molecular spectroscopy, automatics control theory, vibration theory, finite elements, circuit theory, linear optimal, and so on. For example, the finite-element static model correction problem can be transformed to solve some constraint condition solution and its best approximation of the matrix equation . The undamped finite-element dynamic model correction problem can be attributed to solve some constraint condition solution and its best approximation of the matrix equation . These motivate the gradual development of theory in respect of the solution to the restricted matrix equation in recent years (see [18–27]).

In this section, we consider the restricted matrix equation where , , , , , and , satisfy Assume that there exist matrices and satisfying If and exist and , then the restricted matrix equation (3.1) has the unique solution (see [2,  Theorem  3.3.3] for the proof).

In particular, when is a vector and , the restricted matrix equation (3.1) becomes the restricted linear equation If , then is the unique solution of the restricted linear equation (3.5) (see also [10,  Theorem  2.1]).

Theorem 3.1. Given , and as above. If and exist and , then is the unique solution of the restricted matrix equation (3.1) and it can be represented as or where and , , and .

Proof. By the argument above, we have is the unique solution of the restricted matrix equation (3.1). Setting and using (2.7), we get that where . Since , by (2.5), Hence, we have (3.6).
We can obtain (3.7) in the same way.