Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 591256 | https://doi.org/10.1155/2012/591256

Xiaoji Liu, Guangyan Zhu, Guangping Zhou, Yaoming Yu, "An Analog of the Adjugate Matrix for the Outer Inverse ๐ด(2)๐‘‡,๐‘†", Mathematical Problems in Engineering, vol. 2012, Article ID 591256, 14 pages, 2012. https://doi.org/10.1155/2012/591256

An Analog of the Adjugate Matrix for the Outer Inverse ๐ด(2)๐‘‡,๐‘†

Academic Editor: Kui Fu Chen
Received05 Aug 2011
Revised24 Nov 2011
Accepted07 Dec 2011
Published19 Feb 2012

Abstract

We investigate the determinantal representation by exploiting the limiting expression for the generalized inverse ๐ด(2)๐‘‡,๐‘†. We show the equivalent relationship between the existence and limiting expression of ๐ด(2)๐‘‡,๐‘† and some limiting processes of matrices and deduce the new determinantal representations of ๐ด(2)๐‘‡,๐‘†, 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 ๎‚๐‘†๐ด๐‘‹๐ต=๐ท,โ„›(๐‘‹)โŠ‚๐‘‡,๐’ฉ(๐‘‹)โŠƒ.

1. Introduction

Throughout this paper โ„‚๐‘šร—๐‘› denotes the set of ๐‘šร—๐‘› matrices over the complex number field โ„‚, and โ„‚๐‘Ÿ๐‘šร—๐‘› denotes its subset in which every matrix has rank ๐‘Ÿ. ๐ผ stands for the identity matrix of appropriate order (dimension).

Let ๐ดโˆˆโ„‚๐‘šร—๐‘›, and let ๐‘€ and ๐‘ be Hermitian positive definite matrices of orders ๐‘š and ๐‘›, respectively. Consider the following equations:๐ด๐‘‹๐ด=๐ด,(1)๐‘‹๐ด๐‘‹=๐ด,(2)(๐ด๐‘‹)โˆ—=๐ด๐‘‹,(3)(๐‘€๐ด๐‘‹)โˆ—=๐‘€๐ด๐‘‹,(3๐‘€)(๐‘‹๐ด)โˆ—=๐‘‹๐ด,(4)(๐‘๐‘‹๐ด)โˆ—=๐‘๐‘‹๐ด.(4๐‘)๐‘‹ is called a {2}- (or outer) inverse of ๐ด if it satisfies (2) and denoted by ๐ด(2). ๐‘‹ is called the Moore-Penrose inverse of ๐ด if it satisfies (1), (2), (3), and (5) and denoted by ๐ดโ€ . ๐‘‹ is called the weighted Moore-Penrose inverse of ๐ด (with respect to ๐‘€,๐‘) if it satisfies (1), (2), (4), and (6) and denoted by ๐ด+๐‘€๐‘ (see, e.g., [1, 2]).

Let ๐ดโˆˆโ„‚๐‘›ร—๐‘›. Then a matrix ๐‘‹ satisfying๐ด๐‘˜๐‘‹๐ด=๐ด๐‘˜,(1๐‘˜)๐‘‹๐ด๐‘‹=๐‘‹,(2โˆ—)๐ด๐‘‹=๐‘‹๐ด,(5) where ๐‘˜ is some positive integer, is called the Drazin inverse of ๐ด and denoted by ๐ด๐‘‘. The smallest positive integer ๐‘˜ such that ๐‘‹ and ๐ด satisfy (7),โ€‰โ€‰(8), and (9), then it is called the Drazin index and denoted by ๐‘˜=Ind(๐ด). It is clear that Ind(๐ด) is the smallest positive integer ๐‘˜ satisfying rank(๐ด๐‘˜)=rank(๐ด๐‘˜+1) (see [3]). If ๐‘˜=1, then ๐‘‹ is called the group inverse of ๐ด and denoted by ๐ด๐‘”. As is well known, ๐ด๐‘” exists if and only if rank๐ด=rank๐ด2. 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 ๐ถโˆ— algebras (see [4โ€“8]).

In addition, if a matrix ๐‘‹ satisfies (1) and (5), then it is called a {1,5}-inverse of ๐ด and is denoted by ๐ด(1,5).

Let ๐ดโˆˆโ„‚๐‘šร—๐‘›, ๐‘Šโˆˆโ„‚๐‘›ร—๐‘š. Then the matrix ๐‘‹โˆˆโ„‚๐‘šร—๐‘› satisfying(๐ด๐‘Š)๐‘˜+1๐‘‹๐‘Š=(๐ด๐‘Š)๐‘˜,(1๐‘˜๐‘Š)๐‘‹๐‘Š๐ด๐‘Š๐‘‹=๐‘‹,(2๐‘Š)๐ด๐‘Š๐‘‹=๐‘‹๐‘Š๐ด,(5๐‘Š) where ๐‘˜ is some nonnegative integer, is called the ๐‘Š-weighted Drazin inverse of ๐ด, and is denoted by ๐‘‹=๐ด๐‘‘,๐‘Š (see [9]). It is obvious that when ๐‘š=๐‘› and ๐‘Š=๐ผ๐‘›, ๐‘‹ is called the Drazin inverse of ๐ด.

Lemma 1.1 (see [1, Theoremโ€‰โ€‰2.14]). Let ๐ดโˆˆโ„‚๐‘Ÿ๐‘šร—๐‘›, and let ๐‘‡ and ๐‘† be subspaces of โ„‚๐‘› and โ„‚๐‘š, respectively, with dim๐‘‡=dim๐‘†โŸ‚=๐‘กโ‰ค๐‘Ÿ. Then A has a {2}-inverse ๐‘‹ such that โ„›(๐‘‹)=๐‘‡ and ๐’ฉ(๐‘‹)=๐‘† if and only if ๐ด๐‘‡โŠ•๐‘†=โ„‚๐‘š(1.1) in which case ๐‘‹ is unique and denoted by ๐ด(2)๐‘‡,๐‘†.

If ๐ด(2)๐‘‡,๐‘† exists and there exists a matrix ๐บ such that โ„›(๐บ)=๐‘‡ and ๐’ฉ(๐บ)=๐‘†, then ๐บ๐ด๐ด(2)๐‘‡,๐‘†=๐บ and ๐ด(2)๐‘‡,๐‘†๐ด๐บ=๐บ.

It is well known that several important generalized inverses, such as the Moore-Penrose inverse ๐ดโ€ , the weighted Moore-Penrose inverse ๐ด+๐‘€,๐‘, the Drazin inverse ๐ด๐‘‘, and the group inverse ๐ด๐‘”, are outer inverses ๐ด(2)๐‘‡,๐‘† for some specific choice of ๐‘‡ and ๐‘†, are all the generalized inverse ๐ด(2)๐‘‡,๐‘†, {2}- (or outer) inverse of ๐ด with the prescribed range ๐‘‡ and null space ๐‘† (see [2, 10] in the context of complex matrices and [11] in the context of semigroups).

Determinantal representation of the generalized inverse ๐ด(2)๐‘‡,๐‘† was studied in [12, 13]. We will investigate further such representation by exploiting the limiting expression for ๐ด(2)๐‘‡,๐‘†. The paper is organized as follows. In Section 2, we investigate the equivalent relationship between the existence of ๐ด(2)๐‘‡,๐‘† and the limiting process of matrices lim๐œ†โ†’0๐บ(๐ด๐บ+๐œ†๐ผ)โˆ’1 or lim๐œ†โ†’0(๐บ๐ด+๐œ†๐ผ)โˆ’1๐บ and deduce the new determinantal representations of ๐ด(2)๐‘‡,๐‘†, 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 ๎‚๐‘†๐ด๐‘‹๐ต=๐ท,โ„›(๐‘‹)โŠ‚๐‘‡,๐’ฉ(๐‘‹)โŠƒ. 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 1โ‰ค๐‘˜โ‰ค๐‘›, the symbol ๐’ฌ๐‘˜,๐‘› denotes the set {๐›ผโˆถ๐›ผ=(๐›ผ1,โ€ฆ,๐›ผ๐‘˜),1โ‰ค๐›ผ1<โ‹ฏ<๐›ผ๐‘˜โ‰ค๐‘›,where๐›ผ๐‘–,๐‘–=1,โ€ฆ,๐‘˜,areintegers}. And ๐’ฌ๐‘˜,๐‘›{๐‘—}โˆถ={๐›ฝโˆถ๐›ฝโˆˆ๐’ฌ๐‘˜,๐‘›,๐‘—โˆˆ๐›ฝ}, where ๐‘—โˆˆ{1,โ€ฆ,๐‘›}.

Let ๐ด=(๐‘Ž๐‘–๐‘—)โˆˆโ„‚๐‘šร—๐‘›. The symbols ๐‘Ž.๐‘— and ๐‘Ž๐‘–. stand for the ๐‘—th column and the ๐‘–th row of ๐ด, respectively. In the same way, denote by ๐‘Žโˆ—.๐‘— and ๐‘Žโˆ—๐‘–. the ๐‘—th column and the ๐‘–th row of Hermitian adjoint matrix ๐ดโˆ—. The symbol ๐ด.๐‘—(๐‘) (or ๐ด๐‘—.(๐‘)) denotes the matrix obtained from ๐ด by replacing its ๐‘—th column (or row) with some vector ๐‘ (or ๐‘๐‘‡). We write the range of ๐ด by โ„›(๐ด)={๐ด๐‘ฅโˆถ๐‘ฅโˆˆโ„‚๐‘›} and the null space of ๐ด by ๐’ฉ(๐ด)={๐‘ฅโˆˆโ„‚๐‘›โˆถ๐ด๐‘ฅ=0}. Let ๐ตโˆˆโ„‚๐‘ร—๐‘ž. We define the range of a pair of ๐ด and ๐ต as โ„›(๐ด,๐ต)={๐ด๐‘Š๐ตโˆถ๐‘Šโˆˆโ„‚๐‘›ร—๐‘}.

Let ๐›ผโˆˆ๐’ฌ๐‘˜,๐‘š and ๐›ฝโˆˆ๐’ฌ๐‘˜,๐‘›, where 1โ‰ค๐‘˜โ‰คmin{๐‘š,๐‘›}. Then |๐ด๐›ผ๐›ฝ| denotes a minor of ๐ด determined by the row indexed by ๐›ผ and the columns indexed by ๐›ฝ. When ๐‘š=๐‘›, the cofactor of ๐‘Ž๐‘–๐‘— in ๐ด is denoted by ๐œ•|๐ด|/๐œ•๐‘Ž๐‘–๐‘—.

2. Analogs of the Adjugate Matrix for ๐ด(2)๐‘‡,๐‘†

We start with the following theorem which reveals the intrinsic relation between the existence of ๐ด(2)๐‘‡,๐‘† and of lim๐œ†โ†’0๐บ(๐ด๐บ+๐œ†๐ผ)โˆ’1or lim๐œ†โ†’0(๐บ๐ด+๐œ†๐ผ)โˆ’1๐บ. Here ๐œ†โ†’0 means ๐œ†โ†’0 through any neighborhood of 0 in โ„‚ which excludes the nonzero eigenvalues of a square matrix. In [14], Wei pointed out that the existence of ๐ด(2)๐‘‡,๐‘† implies the existence of lim๐œ†โ†’0๐บ(๐ด๐บ+๐œ†๐ผ)โˆ’1 or lim๐œ†โ†’0(๐บ๐ด+๐œ†๐ผ)โˆ’1๐บ. The following result will show that the converse is true under some condition.

Theorem 2.1. Let ๐ดโˆˆโ„‚๐‘Ÿ๐‘šร—๐‘›, and let ๐‘‡ and ๐‘† be subspaces of โ„‚๐‘› and โ„‚๐‘š, respectively, with dim๐‘‡=dim๐‘†โŸ‚=๐‘กโ‰ค๐‘Ÿ. Let ๐บโˆˆโ„‚๐‘Ÿ๐‘›ร—๐‘š with โ„›(๐บ)=๐‘‡ and ๐’ฉ(๐บ)=๐‘†. Then the following statements are equivalent:(i)๐ด(2)๐‘‡,๐‘† exists;(ii)lim๐œ†โ†’0๐บ(๐ด๐บ+๐œ†๐ผ)โˆ’1 exists and rank(๐ด๐บ)=rank(๐บ);(iii)lim๐œ†โ†’0(๐บ๐ด+๐œ†๐ผ)โˆ’1๐บ exists and rank(๐บ๐ด)=rank(๐บ).
In this case, ๐ด(2)๐‘‡,๐‘†=lim๐œ†โ†’0(๐บ๐ด+๐œ†๐ผ)โˆ’1๐บ=lim๐œ†โ†’0๐บ(๐ด๐บ+๐œ†๐ผ)โˆ’1.(2.1)

Proof. (i)โ‡”(ii) Assume that ๐ด(2)๐‘‡,๐‘† exists. By [14, Theoremโ€‰โ€‰2.4], lim๐œ†โ†’0๐บ(๐ด๐บ+๐œ†๐ผ)โˆ’1 exists. Since ๐บ=๐ด(2)๐‘‡,๐‘†๐ด๐บ, rank(๐ด๐บ)=rank(๐บ).
Conversely, assume that lim๐œ†โ†’0๐บ(๐ด๐บ+๐œ†๐ผ)โˆ’1 exists and rank(๐ด๐บ)=rank(๐บ). So lim๐œ†โ†’0(๐ด๐บ+๐œ†๐ผ)โˆ’1๐ด๐บ=lim๐œ†โ†’0๐ด๐บ(๐ด๐บ+๐œ†๐ผ)โˆ’1(2.2) exists. By [15, Theorem], (๐ด๐บ)๐‘” exists. So (๐ด๐บ)(1,5) exists, and then, by [13, Theoremโ€‰โ€‰2], ๐ด(2)๐‘‡,๐‘† exists.
Similarly, we can show that (i)โ‡”(iii). Equation (2.1) comes from [14, equation (2.16)].

Lemma 2.2. Let ๐ด=(๐‘Ž๐‘–๐‘—)โˆˆโ„‚๐‘šร—๐‘› and ๐บ=(๐‘”๐‘–๐‘—)โˆˆโ„‚๐‘ก๐‘›ร—๐‘š. Then rank(๐บ๐ด).๐‘–(๐‘”.๐‘—)โ‰ค๐‘ก, where 1โ‰ค๐‘–โ‰ค๐‘›, 1โ‰ค๐‘—โ‰ค๐‘š, and rank(๐ด๐บ)๐‘–.(๐‘”๐‘—.)โ‰ค๐‘ก, where 1โ‰ค๐‘–โ‰ค๐‘š, 1โ‰ค๐‘—โ‰ค๐‘›.

Proof. Let ๐‘ƒ๐‘–๐‘˜(๐‘Ž) be an ๐‘›ร—๐‘› matrix with ๐‘Ž in the (๐‘–,๐‘˜) entry, 1 in all diagonal entries, and 0 in others. It is an elementary matrix and (๐บ๐ด).๐‘–๎€ท๐‘”.๐‘—๎€ธ๎‘๐‘˜โ‰ ๐‘–๐‘ƒ๐‘–๐‘˜๎€ทโˆ’๐‘Ž๐‘—๐‘˜๎€ธ=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๎“๐‘˜โ‰ ๐‘—๐‘”1๐‘˜๐‘Ž๐‘˜1โ‹ฏ๐‘”1๐‘—โ‹ฏ๎“๐‘˜โ‰ ๐‘—๐‘”1๐‘˜๐‘Ž๐‘˜๐‘›๎“โ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฎ๐‘˜โ‰ ๐‘—๐‘”๐‘›๐‘˜๐‘Ž๐‘˜1โ‹ฏ๐‘”๐‘›๐‘—โ‹ฏ๎“๐‘˜โ‰ ๐‘—๐‘”๐‘›๐‘˜๐‘Ž๐‘˜๐‘›โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ๐‘–th=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘”11โ‹ฏ๐‘”1๐‘—โ‹ฏ๐‘”1๐‘š๐‘”โ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฎ๐‘–1โ‹ฏ๐‘”๐‘–๐‘—โ‹ฏ๐‘”๐‘–๐‘š๐‘”โ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฎ๐‘›1โ‹ฏ๐‘”๐‘›๐‘—โ‹ฏ๐‘”๐‘›๐‘šโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘Ž11โ‹ฏ0โ‹ฏ๐‘Ž1๐‘›๐‘Žโ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฎ0โ‹ฏ1โ‹ฏ0โ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฎ๐‘š1โ‹ฏ0โ‹ฏ๐‘Ž๐‘š๐‘šโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ๐‘—th.๐‘–th(2.3) It follows from the invertibility of ๐‘ƒ๐‘–๐‘˜(๐‘Ž),๐‘–โ‰ ๐‘˜, that rank(๐บ๐ด).๐‘–(๐‘”.๐‘—)โ‰ค๐‘ก.
Analogously, the inequation rank(๐ด๐บ)๐‘–.(๐‘”๐‘—.)โ‰ค๐‘ก can be proved. So the proof is complete.

Recall that if ๐‘“๐ด(๐œ†)=det(๐œ†๐ผ+๐ด)=๐œ†๐‘›+๐‘‘1๐œ†๐‘›โˆ’1โ‹ฏ+๐‘‘๐‘›โˆ’1๐œ†+๐‘‘๐‘› is the characteristic polynomial of an ๐‘›ร—๐‘› matrixโ€”๐ด over โ„‚, then ๐‘‘๐‘– is the sum of all ๐‘–ร—๐‘– principal minors of ๐ด, where ๐‘–=1,โ€ฆ,๐‘› (see, e.g., [16]).

Theorem 2.3. Let ๐ด,๐‘‡,๐‘†, and ๐บ be the same as in Theorem 2.1. Write ๐บ=(๐‘”๐‘–๐‘—). Suppose that the generalized inverse ๐ด(2)๐‘‡,๐‘† of ๐ด exists. Then ๐ด(2)๐‘‡,๐‘† can be represented as follows: ๐ด(2)๐‘‡,๐‘†=๎‚ต๐‘ฅ๐‘–๐‘—๐‘‘๐‘ก๎‚ถ(๐บ๐ด)๐‘›ร—๐‘š,(2.4) where ๐‘ฅ๐‘–๐‘—=๎“๐›ฝโˆˆ๐’ฌ๐‘ก,๐‘›{๐‘–}|||๎€ท(๐บ๐ด).๐‘–(๐‘”.๐‘—)๎€ธ๐›ฝ๐›ฝ|||,๐‘‘๐‘ก๎“(๐บ๐ด)=๐›ฝโˆˆ๐’ฌ๐‘ก,๐‘›|||(๐บ๐ด)๐›ฝ๐›ฝ|||,(2.5) or ๐ด(2)๐‘‡,๐‘†=๎‚ต๐‘ฆ๐‘–๐‘—๐‘‘๐‘ก๎‚ถ(๐ด๐บ)๐‘›ร—๐‘š,(2.6) where ๐‘ฆ๐‘–๐‘—=๎“๐›ผโˆˆ๐’ฌ๐‘ก,๐‘š{๐‘—}||๎€ท(๐ด๐บ)๐‘—.(๐‘”๐‘–.)๎€ธ๐›ผ๐›ผ||,๐‘‘๐‘ก(๎“๐ด๐บ)=๐›ผโˆˆ๐’ฌ๐‘ก,๐‘š||(๐ด๐บ)๐›ผ๐›ผ||.(2.7)

Proof. We will only show the representation (2.5) since the proof of (2.7) is similar. If โˆ’๐œ† is not the eigenvalue of ๐บ๐ด, then the matrix ๐œ†๐ผ+๐บ๐ด is invertible, and (๐œ†๐ผ+๐บ๐ด)โˆ’1=1โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘‹det(๐œ†๐ผ+๐บ๐ด)11๐‘‹21โ‹ฏ๐‘‹๐‘›1๐‘‹12๐‘‹22โ‹ฏ๐‘‹๐‘›2๐‘‹โ‹ฎโ‹ฎโ‹ฎโ‹ฎ1๐‘›๐‘‹2๐‘›โ‹ฏ๐‘‹๐‘›๐‘›โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,(2.8) where ๐‘‹๐‘–๐‘—,๐‘–,๐‘—=1,โ€ฆ,๐‘›, are cofactors of ๐œ†๐ผ+๐บ๐ด. It is easy to see that ๐‘›๎“๐‘™=1๐‘‹๐‘–๐‘™๐‘”๐‘™๐‘—=det(๐œ†๐ผ+๐บ๐ด).๐‘–๎€ท๐‘”.๐‘—๎€ธ.(2.9) So, by (2.1), ๐ด(2)๐‘‡,๐‘†=lim๐œ†โ†’0โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽdet(๐œ†๐ผ+๐บ๐ด).1๎€ท๐‘”.1๎€ธโ‹ฏdet(๐œ†๐ผ+๐บ๐ด)det(๐œ†๐ผ+๐บ๐ด).1๎€ท๐‘”.๐‘š๎€ธdet(๐œ†๐ผ+๐บ๐ด)โ‹ฎโ‹ฎโ‹ฎdet(๐œ†๐ผ+๐บ๐ด).๐‘›๎€ท๐‘”.1๎€ธโ‹ฏdet(๐œ†๐ผ+๐บ๐ด)det(๐œ†๐ผ+๐บ๐ด).๐‘›๎€ท๐‘”.๐‘š๎€ธโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .det(๐œ†๐ผ+๐บ๐ด)(2.10)
We have the characteristic polynomial of ๐บ๐ด๐‘“๐บ๐ด(๐œ†)=det(๐œ†๐ผ+๐บ๐ด)=๐œ†๐‘›+๐‘‘1๐œ†๐‘›โˆ’1+๐‘‘2๐œ†๐‘›โˆ’2+โ‹ฏ+๐‘‘๐‘›,(2.11) where ๐‘‘๐‘–(1โ‰ค๐‘–โ‰ค๐‘›) is a sum of ๐‘–ร—๐‘– principal minors of ๐บ๐ด. Since rank(๐บ๐ด)โ‰คrank(๐บ)=๐‘ก, ๐‘‘๐‘›=๐‘‘๐‘›โˆ’1=โ‹ฏ=๐‘‘๐‘ก+1=0 and det(๐œ†๐ผ+๐บ๐ด)=๐œ†๐‘›+๐‘‘1๐œ†๐‘›โˆ’1+๐‘‘2๐œ†๐‘›โˆ’2+โ‹ฏ+๐‘‘๐‘ก๐œ†๐‘›โˆ’๐‘ก.(2.12) Expanding det(๐œ†๐ผ+๐บ๐ด).๐‘–(๐‘”.๐‘—), we have det(๐œ†๐ผ+๐บ๐ด).๐‘–๎€ท๐‘”.๐‘—๎€ธ=๐‘ฅ1(๐‘–๐‘—)๐œ†๐‘›โˆ’1+๐‘ฅ2(๐‘–๐‘—)๐œ†๐‘›โˆ’2+โ‹ฏ+๐‘ฅ๐‘›(๐‘–๐‘—),(2.13) where ๐‘ฅ๐‘˜(๐‘–๐‘—)=โˆ‘๐›ฝโˆˆ๐’ฌ๐‘˜,๐‘›{๐‘–}|((๐บ๐ด).๐‘–(๐‘”.๐‘—))๐›ฝ๐›ฝ|, 1โ‰ค๐‘˜โ‰ค๐‘›, for 1โ‰ค๐‘–โ‰ค๐‘› and 1โ‰ค๐‘—โ‰ค๐‘š.
By Lemma 2.2, rank(๐บ๐ด).๐‘–(๐‘”.๐‘—)โ‰ค๐‘ก and so |((๐บ๐ด).๐‘–(๐‘”.๐‘—))๐›ฝ๐›ฝ|=0, ๐‘˜>๐‘ก and ๐›ฝโˆˆ๐’ฌ๐‘˜,๐‘›{๐‘–}, for all ๐‘–,๐‘—. Therefore, ๐‘ฅ๐‘˜(๐‘–๐‘—)=0, ๐‘˜โ‰ค๐‘›, for all ๐‘–,๐‘—. Consequently, det(๐œ†๐ผ+๐บ๐ด).๐‘–๎€ท๐‘”.๐‘—๎€ธ=๐‘ฅ1(๐‘–๐‘—)๐œ†๐‘›โˆ’1+๐‘ฅ2(๐‘–๐‘—)๐œ†๐‘›โˆ’2+โ‹ฏ+๐‘ฅ๐‘ก(๐‘–๐‘—)๐œ†๐‘›โˆ’๐‘ก.(2.14)
Substituting (2.12) and (2.14) into (2.10) yields ๐ด(2)๐‘‡,๐‘†=lim๐œ†โ†’0โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘ฅ1(11)๐œ†๐‘›โˆ’1+โ‹ฏ+๐‘ฅ๐‘ก(11)๐œ†๐‘›โˆ’๐‘ก๐œ†๐‘›+๐‘‘1๐œ†๐‘›โˆ’1+โ‹ฏ+๐‘‘๐‘ก๐œ†๐‘›โˆ’๐‘กโ‹ฏ๐‘ฅ1(1๐‘š)๐œ†๐‘›โˆ’1+โ‹ฏ+๐‘ฅ๐‘ก(1๐‘š)๐œ†๐‘›โˆ’๐‘ก๐œ†๐‘›+๐‘‘1๐œ†๐‘›โˆ’1+โ‹ฏ+๐‘‘๐‘ก๐œ†๐‘›โˆ’๐‘ก๐‘ฅโ‹ฎโ‹ฎโ‹ฎ1(๐‘›1)๐œ†๐‘›โˆ’1+โ‹ฏ+๐‘ฅ๐‘ก(๐‘›1)๐œ†๐‘›โˆ’๐‘ก๐œ†๐‘›+๐‘‘1๐œ†๐‘›โˆ’1+โ‹ฏ+๐‘‘๐‘ก๐œ†๐‘›โˆ’๐‘กโ‹ฏ๐‘ฅ1(๐‘›๐‘š)๐œ†๐‘›โˆ’1+โ‹ฏ+๐‘ฅ๐‘ก(๐‘›๐‘š)๐œ†๐‘›โˆ’๐‘ก๐œ†๐‘›+๐‘‘1๐œ†๐‘›โˆ’1+โ‹ฏ+๐‘‘๐‘ก๐œ†๐‘›โˆ’๐‘กโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘ฅ๐‘ก(11)๐‘‘๐‘กโ‹ฏ๐‘ฅ๐‘ก(1๐‘š)๐‘‘๐‘ก๐‘ฅโ‹ฎโ‹ฎโ‹ฎ๐‘ก(๐‘›1)๐‘‘๐‘กโ‹ฏ๐‘ฅ๐‘ก(๐‘›๐‘š)๐‘‘๐‘กโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .(2.15)
Substituting ๐‘ฅ๐‘–๐‘— for ๐‘ฅ๐‘ก(๐‘–๐‘—) 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, Theoremโ€‰โ€‰2.3]. In fact, ๐‘ข=๐‘‘๐‘ก(๐บ๐ด) and, by the Binet-Cauchy formula, ๐‘ฅ๐‘–๐‘—=๎“๐›ฝโˆˆ๐’ฌ๐‘ก,๐‘›{๐‘–}|||๎€ท(๐บ๐ด).๐‘–(๐‘”.๐‘—)๎€ธ๐›ฝ๐›ฝ|||=๎“๐›ฝโˆˆ๐’ฌ๐‘ก,๐‘›{๐‘–}๎“๐‘˜๐‘”๐‘˜๐‘—๐œ•|||(๐บ๐ด)๐›ฝ๐›ฝ|||๐œ•๐‘ ๐‘˜๐‘–=๎“๐›ฝโˆˆ๐’ฌ๐‘ก,๐‘›,๐›ผโˆˆ๐’ฌ๐‘ก,๐‘š๎“๐‘˜๐‘”๐‘˜๐‘—๐œ•||๐บ๐›ผ๐›ฝ||๐œ•๐‘”๐‘˜๐‘—๐œ•||๐ด๐›ฝ๐›ผ||๐œ•๐‘Ž๐‘—๐‘–=๎“๐›ผโˆˆ๐’ฌ๐‘ก,๐‘š,๐›ฝโˆˆ๐’ฌ๐‘ก,๐‘›๎‚€๐บdet๐›ผ๐›ฝ๎‚๐œ•||๐ด๐›ฝ๐›ผ||๐œ•๐‘Ž๐‘—๐‘–,(2.16) where ๐‘ ๐‘˜๐‘—=(๐บ๐ด)๐‘˜๐‘—. Note that ๐œ•|๐ด๐›ฝ๐›ผ|/๐œ•๐‘Ž๐‘–๐‘—=0 if ๐‘–โˆ‰๐›ผ or ๐‘—โˆ‰๐›ฝ. In addition, using the symbols in [13], we can rewrite (2.5) as [13, equation (13)] over โ„‚.
(ii) This method is especially efficient when ๐บ๐ด or ๐ด๐บ is given (comparing with [12, Theoremโ€‰โ€‰2]).

Observing the particular case from Theorem 2.3, ๐บ=(๐‘”๐‘–๐‘—)=๐‘โˆ’1๐ดโˆ—๐‘€, where ๐‘€ and ๐‘ are Hermitian positive definite matrices, we obtain the following corollary in which the symbols ๐‘”.๐‘—โˆถ=(๐‘”).๐‘— and ๐‘”๐‘–.โˆถ=(๐‘”)๐‘–..

Corollary 2.6. Let ๐ดโˆˆโ„‚๐‘Ÿ๐‘šร—๐‘› and ๐บ=๐‘โˆ’1๐ดโˆ—๐‘€, where ๐‘€ and ๐‘ are Hermitian positive definite matrices of order ๐‘š and ๐‘›, respectively, Then ๐ดโ€ ๐‘€๐‘=๎‚ต๐‘ฅ๐‘–๐‘—๐‘‘๐‘Ÿ๎‚ถ(๐บ๐ด)๐‘›ร—๐‘š,(2.17) where ๐‘ฅ๐‘–๐‘—=๎“๐›ฝโˆˆ๐’ฌ๐‘Ÿ,๐‘›{๐‘–}|||๎€ท(๐บ๐ด).๐‘–(๐‘”.๐‘—)๎€ธ๐›ฝ๐›ฝ|||,๐‘‘๐‘Ÿ๎“(๐บ๐ด)=๐›ฝโˆˆ๐’ฌ๐‘Ÿ,๐‘›|||(๐บ๐ด)๐›ฝ๐›ฝ|||,(2.18) or ๐ดโ€ ๐‘€๐‘=๎‚ต๐‘ฆ๐‘–๐‘—๐‘‘๐‘Ÿ๎‚ถ(๐ด๐บ)๐‘›ร—๐‘š,(2.19) where ๐‘ฆ๐‘–๐‘—=๎“๐›ผโˆˆ๐’ฌ๐‘Ÿ,๐‘š{๐‘—}||๎€ท(๐ด๐บ)๐‘—.(๐‘”๐‘–.)๎€ธ๐›ผ๐›ผ||,๐‘‘๐‘Ÿ(๎“๐ด๐บ)=๐›ผโˆˆ๐’ฌ๐‘Ÿ,๐‘š||(๐ด๐บ)๐›ผ๐›ผ||.(2.20)

If ๐‘€ and ๐‘ are identity matrices, then we can obtain the following result.

Corollary 2.7 (see [17, Theoremโ€‰โ€‰2.2]). The Moore-Penrose inverse ๐ดโ€  of ๐ด=(๐‘Ž๐‘–๐‘—)โˆˆโ„‚๐‘Ÿ๐‘šร—๐‘› can be represented as follows: ๐ดโ€ =๎‚ต๐‘ฅ๐‘–๐‘—๐‘‘๐‘Ÿ(๐ดโˆ—๎‚ถ๐ด)๐‘›ร—๐‘š,(2.21) where ๐‘ฅ๐‘–๐‘—=๎“๐›ฝโˆˆ๐’ฌ๐‘Ÿ,๐‘›{๐‘–}|||๐ด๎€ท๎€ทโˆ—๐ด๎€ธ.๐‘–๎€ท๐‘Žโˆ—.๐‘—๎€ธ๎€ธ๐›ฝ๐›ฝ|||,๐‘‘๐‘Ÿ๎€ท๐ดโˆ—๐ด๎€ธ=๎“๐›ฝโˆˆ๐’ฌ๐‘Ÿ,๐‘›|||๎€ท๐ดโˆ—๐ด๎€ธ๐›ฝ๐›ฝ|||,(2.22) or ๐ดโ€ =๎‚ต๐‘ฆ๐‘–๐‘—๐‘‘๐‘Ÿ(๐ด๐ดโˆ—)๎‚ถ๐‘›ร—๐‘š,(2.23) where ๐‘ฆ๐‘–๐‘—=๎“๐›ผโˆˆ๐’ฌ๐‘Ÿ,๐‘š{๐‘—}|||๎‚€๎€ท๐ด๐ดโˆ—๎€ธ๐‘—.(๐‘Žโˆ—๐‘–.)๎‚๐›ผ๐›ผ|||,๐‘‘๐‘Ÿ๎€ท๐ด๐ดโˆ—๎€ธ=๎“๐›ผโˆˆ๐’ฌ๐‘Ÿ,๐‘š||๎€ท๐ด๐ดโˆ—๎€ธ๐›ผ๐›ผ||.(2.24)

Note that ๐ด๐‘‘,๐‘Š=(๐‘Š๐ด๐‘Š)(2)โ„›((๐ด๐‘Š)๐‘˜๐ด),๐’ฉ((๐ด๐‘Š)๐‘˜๐ด). Therefore, when ๐บ=(๐ด๐‘Š)๐‘˜๐ด in Theorem 2.3, we have the following corollary.

Corollary 2.8. Let ๐ดโˆˆโ„‚๐‘šร—๐‘›, ๐‘Šโˆˆโ„‚๐‘›ร—๐‘š, and ๐‘˜=max{Ind(๐ด๐‘Š),Ind(๐‘Š๐ด)}. If rank(๐ด๐‘Š)๐‘˜=๐‘ก, rank(๐‘Š๐ด)๐‘˜=๐‘Ÿ, and (๐ด๐‘Š)๐‘˜๐ด=(๐‘๐‘–๐‘—)๐‘šร—๐‘›, then ๐ด๐‘‘,๐‘Š=๎ƒฉ๐‘ฅ๐‘–๐‘—๐‘‘๐‘ก๎€ท(๐ด๐‘Š)๐‘˜+2๎€ธ๎ƒช๐‘šร—๐‘›,(2.25) where ๐‘ฅ๐‘–๐‘—=๎“๐›ฝโˆˆ๐’ฌ๐‘ก,๐‘š{๐‘–}|||๎€ท๎€ท(๐ด๐‘Š)๐‘˜+2๎€ธ.๐‘–๎€ท๐‘.๐‘—๎€ธ๎€ธ๐›ฝ๐›ฝ|||,๐‘‘๐‘ก๎€ท(๐‘Š๐ด)๐‘˜+2๎€ธ=๎“๐›ฝโˆˆ๐’ฌ๐‘ก,m|||๎€ท(๐ด๐‘Š)๐‘˜+2๎€ธ๐›ฝ๐›ฝ|||,(2.26) or ๐ด๐‘‘,๐‘Š=๎ƒฉ๐‘ฆ๐‘–๐‘—๐‘‘๐‘Ÿ๎€ท(๐‘Š๐ด)๐‘˜+2๎€ธ๎ƒช๐‘šร—๐‘›,(2.27) where ๐‘ฆ๐‘–๐‘—=๎“๐›ผโˆˆ๐’ฌ๐‘Ÿ,๐‘›{๐‘—}|||๎‚€๎€ท(๐‘Š๐ด)๐‘˜+2๎€ธ๐‘—.๎€ท๐‘๐‘–.๎€ธ๎‚๐›ผ๐›ผ|||,๐‘‘๐‘Ÿ๎€ท(๐‘Š๐ด)๐‘˜+2๎€ธ=๎“๐›ผโˆˆ๐’ฌ๐‘Ÿ,๐‘›||๎€ท(๐‘Š๐ด)๐‘˜+2๎€ธ๐›ผ๐›ผ||.(2.28)

When ๐บ=๐ด๐‘˜ with ๐‘˜=Ind(๐ด) in Theorem 2.3, we have the following corollary.

Corollary 2.9 (see [17, Theoremโ€‰โ€‰3.3]). Let ๐ดโˆˆโ„‚๐‘›ร—๐‘› with Ind๐ด=๐‘˜ and rank๐ด๐‘˜=๐‘Ÿ, and ๐ด๐‘˜=(๐‘Ž(๐‘˜)๐‘–๐‘—)๐‘›ร—๐‘›. Then ๐ด๐‘‘=๎ƒฉ๐‘ฅ๐‘–๐‘—๐‘‘๐‘Ÿ๎€ท๐ด๐‘˜+1๎€ธ๎ƒช๐‘›ร—๐‘›,(2.29) where ๐‘‘๐‘–๐‘—=๎“๐›ฝโˆˆ๐’ฌ๐‘Ÿ,๐‘›{๐‘–}||||๎‚€๎€ท๐ด๐‘˜+1.๐‘–๎€ธ๎‚€๐‘Ž(๐‘˜).๐‘—๎‚๎‚๐›ฝ๐›ฝ||||,๐‘‘๐‘Ÿ๎€ท๐ด๐‘˜+1๎€ธ=๎“๐›ฝโˆˆ๐’ฌ๐‘Ÿ,๐‘›|||๎€ท๐ด๐‘˜+1๎€ธ๐›ฝ๐›ฝ|||.(2.30)

Finally, we turn our attention to the two projectors ๐ด(2)๐‘‡,๐‘†๐ด and ๐ด๐ด(2)๐‘‡,๐‘†. The limiting expressions for ๐ด(2)๐‘‡,๐‘† in (2.1) bring us the following:๐ด(2)๐‘‡,๐‘†๐ด=lim๐œ†โ†’0(๐บ๐ด+๐œ†๐ผ)โˆ’1๐บ๐ด,๐ด๐ด(2)๐‘‡,๐‘†=lim๐œ†โ†’0๐ด๐บ(๐ด๐บ+๐œ†๐ผ)โˆ’1.(2.31)

Corollary 2.10. Let ๐ด,๐‘‡,๐‘†, and ๐บ be the same as in Theorem 2.1. Write ๐บ๐ด=(๐‘ ๐‘–๐‘—) and ๐ด๐บ=(โ„Ž๐‘–๐‘—). Suppose that ๐ด(2)๐‘‡,S exists. Then ๐ด(2)๐‘‡,๐‘†๐ด of ๐ด๐ด(2)๐‘‡,๐‘† can be represented as follows: ๐ด(2)๐‘‡,๐‘†๎‚ต๐‘ฅ๐ด=๐‘–๐‘—๐‘‘๐‘ก๎‚ถ(๐บ๐ด)๐‘›ร—๐‘›,(2.32) where ๐‘ฅ๐‘–๐‘—=๎“๐›ฝโˆˆ๐’ฌ๐‘ก,๐‘›{๐‘–}|||๎€ท(๐บ๐ด).๐‘–๎€ท๐‘ .๐‘—๎€ธ๎€ธ๐›ฝ๐›ฝ|||,๐‘‘๐‘ก๎“(๐บ๐ด)=๐›ฝโˆˆ๐’ฌ๐‘ก,๐‘›|||(๐บ๐ด)๐›ฝ๐›ฝ|||,๐ด๐ด(2)๐‘‡,๐‘†=๎‚ต๐‘ฆ๐‘–๐‘—๐‘‘๐‘ก๎‚ถ(๐ด๐บ)๐‘šร—๐‘š,(2.33) where ๐‘ฆ๐‘–๐‘—=๎“๐›ผโˆˆ๐’ฌ๐‘ก,๐‘š{๐‘—}||๎€ท(๐ด๐บ)๐‘—.(โ„Ž๐‘–.)๎€ธ๐›ผ๐›ผ||,๐‘‘๐‘ก(๎“๐ด๐บ)=๐›ผโˆˆ๐’ฌ๐‘ก,๐‘š||(๐ด๐บ)๐›ผ๐›ผ||.(2.34)

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 ๐ด๐‘‹=๐ต. 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 ๐ด๐‘‡๐‘‹๐ด=๐ต. These motivate the gradual development of theory in respect of the solution to the restricted matrix equation in recent years (see [18โ€“27]).

In this section, we consider the restricted matrix equation๎‚๐ด๐‘‹๐ต=๐ท,โ„›(๐‘‹)โŠ‚๐‘‡,๐’ฉ(๐‘‹)โŠƒ๐‘†,(3.1) where ๐ดโˆˆโ„‚๐‘Ÿ๐‘šร—๐‘›, ๐ตโˆˆโ„‚๐‘ร—๐‘žฬƒ๐‘Ÿ, ๐ทโˆˆโ„‚๐‘šร—๐‘ž, ๐‘‡โŠ‚โ„‚๐‘›, ๐‘†โŠ‚โ„‚๐‘š, ๎‚๐‘‡โŠ‚โ„‚๐‘ž and ๎‚๐‘†โŠ‚โ„‚๐‘, satisfy๎€ท๐‘†dim(๐‘‡)=dimโŸ‚๎€ธ๎‚€๎‚๐‘‡๎‚๎‚€๎‚๐‘†=๐‘กโ‰ค๐‘Ÿ,dim=dimโŸ‚๎‚=ฬƒ๐‘กโ‰คฬƒ๐‘Ÿ.(3.2) Assume that there exist matrices ๐บโˆˆโ„‚๐‘›ร—๐‘š and ๎‚๐บโˆˆโ„‚๐‘žร—๐‘ satisfying๎‚€๎‚๐บ๎‚=๎‚๎‚€๎‚๐บ๎‚=๎‚โ„›(๐บ)=๐‘‡,๐’ฉ(๐บ)=๐‘†,โ„›๐‘‡,๐’ฉ๐‘†.(3.3) If ๐ด(2)๐‘‡,๐‘† and ๐ต๎‚(2)๐‘‡,๎‚๐‘† exist and ๎‚๐ทโˆˆโ„›(๐ด๐บ,๐บ๐ต), then the restricted matrix equation (3.1) has the unique solution๐‘‹=๐ด(2)๐‘‡,๐‘†๐ท๐ต๎‚(2)๐‘‡,๎‚๐‘†(3.4) (see [2,โ€‰โ€‰Theoremโ€‰โ€‰3.3.3] for the proof).

In particular, when ๐ท is a vector ๐‘ and ๎‚๐ต=๐บ=๐ผ1, the restricted matrix equation (3.1) becomes the restricted linear equation๐ด๐‘ฅ=๐‘,๐‘ฅโˆˆ๐‘‡.(3.5) If ๐‘โˆˆ๐ดโ„›(๐บ), then ๐‘ฅ=๐ด(2)๐‘‡,๐‘†๐‘ is the unique solution of the restricted linear equation (3.5) (see also [10,โ€‰โ€‰Theoremโ€‰โ€‰2.1]).

Theorem 3.1. Given ๐ด,๐ต,๐ท=(๐‘‘๐‘–๐‘—),๐บ=(๐‘”๐‘–๐‘—๎‚),๐บ=(ฬƒ๐‘”๐‘–๐‘—๎‚๐‘‡),๐‘‡,๐‘†,, and ๎‚๐‘† as above. If ๐ด(2)๐‘‡,๐‘† and ๐ต๎‚(2)๐‘‡,๎‚๐‘† exist and ๎‚๐ทโˆˆโ„›(๐ด๐บ,๐บ๐ต), then ๐‘‹=๐ด(2)๐‘‡,๐‘†๐ท๐ต๎‚(2)๐‘‡,๎‚๐‘† is the unique solution of the restricted matrix equation (3.1) and it can be represented as ๐‘ฅ๐‘–๐‘—=โˆ‘๐‘š๐‘˜=1โˆ‘๐›ฝโˆˆ๐’ฌ๐‘ก,๐‘›{๐‘–},๐›ผโˆˆ๐’ฌฬƒ๐‘ก,๐‘{๐‘—}|||๎€ท(๐บ๐ด).๐‘–๎€ท๐‘”.๐‘˜๎€ธ๎€ธ๐›ฝ๐›ฝ|||||||๎‚ต๎‚€๐ต๎‚๐บ๎‚๐‘—.๎‚€๎‚๐‘“๐‘˜.๎‚๎‚ถ๐›ผ๐›ผ||||๐‘‘๐‘ก(๐บ๐ด)๐‘‘ฬƒ๐‘ก๎‚€๐ต๎‚๐บ๎‚(3.6) or ๐‘ฅ๐‘–๐‘—=โˆ‘๐‘ž๐‘˜=1โˆ‘๐›ฝโˆˆ๐’ฌ๐‘ก,๐‘›{๐‘–},๐›ผโˆˆ๐’ฌฬƒ๐‘ก,๐‘{๐‘—}|||๎€ท(๐บ๐ด).๐‘–(๐‘“.๐‘˜)๎€ธ๐›ฝ๐›ฝ|||||||๎‚ต๎‚€๐ต๎‚๐บ๎‚๐‘—.(ฬƒ๐‘”๐‘˜.)๎‚ถ๐›ผ๐›ผ||||๐‘‘๐‘ก(๐บ๐ด)๐‘‘ฬƒ๐‘ก๎‚€๐ต๎‚๐บ๎‚,(3.7) where ๎‚๐‘“๐‘˜.=๐‘‘๐‘˜.๎‚๐บ and ๐‘“.๐‘˜=๐บ๐‘‘.๐‘˜, ๐‘–=1,โ€ฆ,๐‘›, and ๐‘—=1,โ€ฆ,๐‘.

Proof. By the argument above, we have ๐‘‹=๐ด(2)๐‘‡,๐‘†๐ท๐ต๎‚(2)๐‘‡,๎‚๐‘† is the unique solution of the restricted matrix equation (3.1). Setting ๐‘Œ=๐ท๐ต๎‚(2)๐‘‡,๎‚๐‘† and using (2.7), we get that ๐‘ฆ๐‘˜๐‘—=๐‘ž๎“โ„Ž=1๐‘‘๐‘˜โ„Ž๎‚ต๐ต๎‚(2)๐‘‡,๎‚๐‘†๎‚ถโ„Ž๐‘—=๐‘ž๎“โ„Ž=1๐‘‘๐‘˜โ„Žโˆ‘๐›ผโˆˆ๐’ฌฬƒ๐‘ก,๐‘{๐‘—}||||๎‚ต๎‚€๐ต๎‚๐บ๎‚๐‘—.๎€ทฬƒ๐‘”โ„Ž.๎€ธ๎‚ถ๐›ผ๐›ผ||||๐‘‘ฬƒ๐‘ก๎‚€๐ต๎‚๐บ๎‚=โˆ‘๐›ผโˆˆ๐’ฌฬƒ๐‘ก,๐‘{๐‘—}||||๎‚ต๎‚€๐ต๎‚๐บ๎‚๐‘—.๎€ทโˆ‘๐‘žโ„Ž=1๐‘‘๐‘˜โ„Žฬƒ๐‘”โ„Ž.๎€ธ๎‚ถ๐›ผ๐›ผ||||๐‘‘ฬƒ๐‘ก๎‚€๐ต๎‚๐บ๎‚=โˆ‘๐›ผโˆˆ๐’ฌฬƒ๐‘ก,๐‘{๐‘—}||||๎‚ต๎‚€๐ต๎‚๐บ๎‚๐‘—.๎‚€๎‚๐‘“๐‘˜.๎‚๎‚ถ๐›ผ๐›ผ||||๐‘‘ฬƒ๐‘ก๎‚€๐ต๎‚๐บ๎‚,(3.8) where ๎‚๐‘“๐‘˜.=๐‘‘๐‘˜.๎‚๐บ. Since ๐‘‹=๐ด(2)๐‘‡,๐‘†๐‘Œ, by (2.5), ๐‘ฅ๐‘–๐‘—=๐‘š๎“๐‘˜=1๎‚€๐ด(2)๐‘‡,๐‘†๎‚๐‘–๐‘˜๐‘ฆ๐‘˜๐‘—=๐‘š๎“๐‘˜=1โˆ‘๐›ฝโˆˆ๐’ฌ๐‘ก,๐‘›{๐‘–}|||๎€ท(๐บ๐ด).๐‘–๎€ท๐‘”.๐‘˜๎€ธ๎€ธ๐›ฝ๐›ฝ|||๐‘‘๐‘กโˆ‘(๐บ๐ด)๐›ผโˆˆ๐’ฌฬƒ๐‘ก,๐‘{๐‘—}||||๎‚ต๎‚€๐ต๎‚๐บ๎‚๐‘—.๎‚€๎‚๐‘“๐‘˜.๎‚๎‚ถ๐›ผ๐›ผ||||๐‘‘ฬƒ๐‘ก๎‚€๐ต๎‚๐บ๎‚.(3.9) Hence, we have (3.6).
We can obtain (3.7) in the same way.

In particular, when ๐ท is a vector ๐‘ and ๎‚๐ต=๐บ=๐ผ1 in the above theorem, we have the following result from (3.7).

Theorem 3.2. Given ๐ด,๐บ,๐‘‡, and ๐‘† as above. If ๐‘โˆˆ๐ดโ„›(๐บ), then ๐‘ฅ=๐ด(2)๐‘‡,๐‘†๐‘ is the unique solution of the restricted linear equation ๐ด๐‘ฅ=๐‘, ๐‘ฅโˆˆ๐‘‡, and it can be represented as ๐‘ฅ๐‘–=โˆ‘๐›ฝโˆˆ๐’ฌ๐‘ก,๐‘›{๐‘–}|||๎€ท(๐บ๐ด).๐‘–๎€ธ(๐‘“)๐›ฝ๐›ฝ|||๐‘‘๐‘ก(๐บ๐ด),๐‘—=1,โ€ฆ,๐‘›,(3.10) where ๐‘“=๐บ๐‘.

Remark 3.3. Using the symbols in [13], we can rewrite (3.10) as [13, equation (27)].

4. Example

LetโŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,๎‚โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ .๐ด=12220211004200210000,๐ต=211001,๐ท=01โˆ’10000000,๐บ=1โˆ’1200021000000000000๐บ=100020(4.1) Obviously, rank๐ด=3, dim๐ด๐‘‡=dim๐‘‡=2, and๐‘‡=โ„›(๐บ)โŠ‚โ„‚4,๐‘†=๐’ฉ(๐บ)โŠ‚โ„‚5,๎‚๎‚€๎‚๐บ๎‚๐‘‡=โ„›โŠ‚โ„‚2,๎‚๎‚€๎‚๐บ๎‚๐‘†=๐’ฉโŠ‚โ„‚3.(4.2) It is easy to verify that ๐ด๐‘‡โŠ•๐‘†=โ„‚5 and ๐ต๎‚๎‚๐‘‡โŠ•๐‘†=โ„‚3. Thus, ๐ด(2)๐‘‡,๐‘† and ๐ต๎‚(2)๐‘‡,๎‚๐‘† exist by Lemma 1.1.

Now consider the restricted matrix equation๎‚๐ด๐‘‹๐ต=๐ท,โ„›(๐‘‹)โŠ‚๐‘‡,๐’ฉ(๐‘‹)โŠƒ๐‘†.(4.3) Clearly,โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,๎‚โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ ,๐ด๐บ=1340004200000000000000000๐บ๐ต=2120(4.4) and it is easy to verify that โ„›(๐ท)โŠ‚โ„›(๐ด๐บ) and ๎‚๐’ฉ(๐ท)โŠƒ๐’ฉ(๐บ๐ต) hold.

Note that โ„›(๐ท)โŠ‚โ„›(๐ด๐บ) and ๎‚๐’ฉ(๐ท)โŠƒ๐’ฉ(๐บ๐ต) if and only if ๎‚๐ทโˆˆโ„›(๐ด๐บ,๐บ๐ต). So, by Theorem 3.1, the unique solution of (4.3) exists.

ComputingโŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ๎‚โŽ›โŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽ ,๐‘‘๐บ๐ด=1095046400000000,๐ต๐บ=2201000202||||||||||||+||||||||||||+||||||||||||+||||||||||||+||||||||||||+||||||||||||๐‘‘(๐บ๐ด)=100419001500460044000000=4,2๎‚€๐ต๎‚๐บ๎‚=||||||||||||+||||||||||||+||||||||||||๎‚โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,221020000020=โˆ’2,๐‘“=1โˆ’120002100000000000001โˆ’1000000011โˆ’200000(4.5) and setting ๐‘ฆ๐‘–๐‘˜=โˆ‘๐›ฝโˆˆ๐’ฌ๐‘ก,๐‘›{๐‘–}|((๐บ๐ด).๐‘–(๎‚๐‘“.๐‘˜))๐›ฝ๐›ฝ|, we have Table 1.


๐‘ฆ ๐‘– ๐‘˜ ๐‘– = 1 ๐‘– = 2 ๐‘– = 3 ๐‘– = 4

๐‘˜ = 1 4โˆ’200
๐‘˜ = 2 4000

Similarly, setting ๐‘ง๐‘˜๐‘—=โˆ‘๐›ผโˆˆ๐’ฌฬƒ๐‘ก,๐‘{๐‘—}๎‚|((๐ต๐บ)๐‘—.(ฬƒ๐‘”๐‘˜.))๐›ผ๐›ผ|, we have Table 2.


๐‘ง ๐‘˜ ๐‘— ๐‘— = 1 ๐‘— = 2 ๐‘— = 3

๐‘˜ = 1 0โˆ’20
๐‘˜ = 2 โˆ’240

So, by (3.7), we have๎€ท๐‘ฅ๐‘‹=๐‘–๐‘—๎€ธ=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ11โˆ’100โˆ’20โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .000000(4.6)

Acknowledgments

The authors would like to thank the referees for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China 11061005, the Ministry of Education Science and Technology Key Project under Grant 210164, and Grants HCIC201103 of Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis Open Fund.

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Copyright ยฉ 2012 Xiaoji Liu et al. 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.


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