An Analog of the Adjugate Matrix for the Outer Inverse 𝐴^(2)𝑇,𝑆

Abstract

We investigate the determinantal representation by exploiting the limiting expression for the generalized inverse 𝐴^(2)𝑇,𝑆. We show the equivalent relationship between the existence and limiting expression of 𝐴^(2)𝑇,𝑆 and some limiting processes of matrices and deduce the new determinantal representations of 𝐴^(2)𝑇,𝑆, 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 𝐴𝑋𝐵=𝐷,ℛ(𝑋)⊂𝑇,𝒩(𝑋)⊃𝑆.

1. Introduction

Throughout this paper ℂ^𝑚×𝑛 denotes the set of 𝑚×𝑛 matrices over the complex number field ℂ, and ℂ^{𝑟𝑚×𝑛} denotes its subset in which every matrix has rank 𝑟. 𝐼 stands for the identity matrix of appropriate order (dimension).

Let 𝐴∈ℂ^𝑚×𝑛, and let 𝑀 and 𝑁 be Hermitian positive definite matrices of orders 𝑚 and 𝑛, respectively. Consider the following equations:𝐴𝑋𝐴=𝐴,(1)𝑋𝐴𝑋=𝐴,(2)(𝐴𝑋)^∗=𝐴𝑋,(3)(𝑀𝐴𝑋)^∗=𝑀𝐴𝑋,(3𝑀)(𝑋𝐴)^∗=𝑋𝐴,(4)(𝑁𝑋𝐴)^∗=𝑁𝑋𝐴.(4𝑁)𝑋 is called a {2}- (or outer) inverse of 𝐴 if it satisfies (2) and denoted by 𝐴⁽²⁾. 𝑋 is called the Moore-Penrose inverse of 𝐴 if it satisfies (1), (2), (3), and (5) and denoted by 𝐴^†. 𝑋 is called the weighted Moore-Penrose inverse of 𝐴 (with respect to 𝑀,𝑁) if it satisfies (1), (2), (4), and (6) and denoted by 𝐴^+𝑀𝑁 (see, e.g., [1, 2]).

Let 𝐴∈ℂ^𝑛×𝑛. Then a matrix 𝑋 satisfying𝐴^𝑘𝑋𝐴=𝐴^𝑘,(1^𝑘)𝑋𝐴𝑋=𝑋,(2^∗)𝐴𝑋=𝑋𝐴,(5) where 𝑘 is some positive integer, is called the Drazin inverse of 𝐴 and denoted by 𝐴_𝑑. The smallest positive integer 𝑘 such that 𝑋 and 𝐴 satisfy (7),  (8), and (9), then it is called the Drazin index and denoted by 𝑘=Ind(𝐴). It is clear that Ind(𝐴) is the smallest positive integer 𝑘 satisfying rank(𝐴^𝑘)=rank(𝐴^𝑘+1) (see [3]). If 𝑘=1, then 𝑋 is called the group inverse of 𝐴 and denoted by 𝐴_𝑔. As is well known, 𝐴_𝑔 exists if and only if rank𝐴=rank𝐴². 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 𝐶^∗ algebras (see [4–8]).

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

Let 𝐴∈ℂ^𝑚×𝑛, 𝑊∈ℂ^𝑛×𝑚. Then the matrix 𝑋∈ℂ^𝑚×𝑛 satisfying(𝐴𝑊)^𝑘+1𝑋𝑊=(𝐴𝑊)^𝑘,(1^𝑘𝑊)𝑋𝑊𝐴𝑊𝑋=𝑋,(2𝑊)𝐴𝑊𝑋=𝑋𝑊𝐴,(5𝑊) where 𝑘 is some nonnegative integer, is called the 𝑊-weighted Drazin inverse of 𝐴, and is denoted by 𝑋=𝐴_𝑑,𝑊 (see [9]). It is obvious that when 𝑚=𝑛 and 𝑊=𝐼_𝑛, 𝑋 is called the Drazin inverse of 𝐴.

Lemma 1.1 (see [1, Theorem  2.14]). Let 𝐴∈ℂ^{𝑟𝑚×𝑛}, and let 𝑇 and 𝑆 be subspaces of ℂ^𝑛 and ℂ^𝑚, respectively, with dim𝑇=dim𝑆^⟂=𝑡≤𝑟. Then A has a {2}-inverse 𝑋 such that ℛ(𝑋)=𝑇 and 𝒩(𝑋)=𝑆 if and only if 𝐴𝑇⊕𝑆=ℂ^𝑚(1.1) in which case 𝑋 is unique and denoted by 𝐴^(2)𝑇,𝑆.

If 𝐴^(2)𝑇,𝑆 exists and there exists a matrix 𝐺 such that ℛ(𝐺)=𝑇 and 𝒩(𝐺)=𝑆, then 𝐺𝐴𝐴^(2)𝑇,𝑆=𝐺 and 𝐴^(2)𝑇,𝑆𝐴𝐺=𝐺.

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

Determinantal representation of the generalized inverse 𝐴^(2)𝑇,𝑆 was studied in [12, 13]. We will investigate further such representation by exploiting the limiting expression for 𝐴^(2)𝑇,𝑆. The paper is organized as follows. In Section 2, we investigate the equivalent relationship between the existence of 𝐴^(2)𝑇,𝑆 and the limiting process of matrices lim_𝜆→0𝐺(𝐴𝐺+𝜆𝐼)⁻¹ or lim_𝜆→0(𝐺𝐴+𝜆𝐼)⁻¹𝐺 and deduce the new determinantal representations of 𝐴^(2)𝑇,𝑆, 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 𝐴𝑋𝐵=𝐷,ℛ(𝑋)⊂𝑇,𝒩(𝑋)⊃𝑆. 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 1≤𝑘≤𝑛, the symbol 𝒬_𝑘,𝑛 denotes the set {𝛼∶𝛼=(𝛼₁,…,𝛼_𝑘),1≤𝛼₁<⋯<𝛼_𝑘≤𝑛,where𝛼_𝑖,𝑖=1,…,𝑘,areintegers}. And 𝒬_𝑘,𝑛{𝑗}∶={𝛽∶𝛽∈𝒬_𝑘,𝑛,𝑗∈𝛽}, where 𝑗∈{1,…,𝑛}.

Let 𝐴=(𝑎_𝑖𝑗)∈ℂ^𝑚×𝑛. The symbols 𝑎_.𝑗 and 𝑎_𝑖. stand for the 𝑗th column and the 𝑖th row of 𝐴, respectively. In the same way, denote by 𝑎^∗.𝑗 and 𝑎^∗𝑖. the 𝑗th column and the 𝑖th row of Hermitian adjoint matrix 𝐴^∗. The symbol 𝐴_.𝑗(𝑏) (or 𝐴_𝑗.(𝑏)) denotes the matrix obtained from 𝐴 by replacing its 𝑗th column (or row) with some vector 𝑏 (or 𝑏^𝑇). We write the range of 𝐴 by ℛ(𝐴)={𝐴𝑥∶𝑥∈ℂ^𝑛} and the null space of 𝐴 by 𝒩(𝐴)={𝑥∈ℂ^𝑛∶𝐴𝑥=0}. Let 𝐵∈ℂ^𝑝×𝑞. We define the range of a pair of 𝐴 and 𝐵 as ℛ(𝐴,𝐵)={𝐴𝑊𝐵∶𝑊∈ℂ^𝑛×𝑝}.

Let 𝛼∈𝒬_𝑘,𝑚 and 𝛽∈𝒬_𝑘,𝑛, where 1≤𝑘≤min{𝑚,𝑛}. Then |𝐴^𝛼𝛽| denotes a minor of 𝐴 determined by the row indexed by 𝛼 and the columns indexed by 𝛽. When 𝑚=𝑛, the cofactor of 𝑎_𝑖𝑗 in 𝐴 is denoted by 𝜕|𝐴|/𝜕𝑎_𝑖𝑗.

2. Analogs of the Adjugate Matrix for 𝐴^(2)𝑇,𝑆

We start with the following theorem which reveals the intrinsic relation between the existence of 𝐴^(2)𝑇,𝑆 and of lim_𝜆→0𝐺(𝐴𝐺+𝜆𝐼)⁻¹or lim_𝜆→0(𝐺𝐴+𝜆𝐼)⁻¹𝐺. Here 𝜆→0 means 𝜆→0 through any neighborhood of 0 in ℂ which excludes the nonzero eigenvalues of a square matrix. In [14], Wei pointed out that the existence of 𝐴^(2)𝑇,𝑆 implies the existence of lim_𝜆→0𝐺(𝐴𝐺+𝜆𝐼)⁻¹ or lim_𝜆→0(𝐺𝐴+𝜆𝐼)⁻¹𝐺. The following result will show that the converse is true under some condition.

Theorem 2.1. Let 𝐴∈ℂ^{𝑟𝑚×𝑛}, and let 𝑇 and 𝑆 be subspaces of ℂ^𝑛 and ℂ^𝑚, respectively, with dim𝑇=dim𝑆^⟂=𝑡≤𝑟. Let 𝐺∈ℂ^{𝑟𝑛×𝑚} with ℛ(𝐺)=𝑇 and 𝒩(𝐺)=𝑆. Then the following statements are equivalent:(i)𝐴^(2)𝑇,𝑆 exists;(ii)lim_𝜆→0𝐺(𝐴𝐺+𝜆𝐼)⁻¹ exists and rank(𝐴𝐺)=rank(𝐺);(iii)lim_𝜆→0(𝐺𝐴+𝜆𝐼)⁻¹𝐺 exists and rank(𝐺𝐴)=rank(𝐺).
In this case, 𝐴^(2)𝑇,𝑆=lim_𝜆→0(𝐺𝐴+𝜆𝐼)⁻¹𝐺=lim_𝜆→0𝐺(𝐴𝐺+𝜆𝐼)⁻¹.(2.1)

Proof. (i)⇔(ii) Assume that 𝐴^(2)𝑇,𝑆 exists. By [14, Theorem  2.4], lim_𝜆→0𝐺(𝐴𝐺+𝜆𝐼)⁻¹ exists. Since 𝐺=𝐴^(2)𝑇,𝑆𝐴𝐺, rank(𝐴𝐺)=rank(𝐺).
Conversely, assume that lim_𝜆→0𝐺(𝐴𝐺+𝜆𝐼)⁻¹ exists and rank(𝐴𝐺)=rank(𝐺). So lim_𝜆→0(𝐴𝐺+𝜆𝐼)⁻¹𝐴𝐺=lim_𝜆→0𝐴𝐺(𝐴𝐺+𝜆𝐼)⁻¹(2.2) exists. By [15, Theorem], (𝐴𝐺)_𝑔 exists. So (𝐴𝐺)^(1,5) exists, and then, by [13, Theorem  2], 𝐴^(2)𝑇,𝑆 exists.
Similarly, we can show that (i)⇔(iii). Equation (2.1) comes from [14, equation (2.16)].

Lemma 2.2. Let 𝐴=(𝑎_𝑖𝑗)∈ℂ^𝑚×𝑛 and 𝐺=(𝑔_𝑖𝑗)∈ℂ^{𝑡𝑛×𝑚}. Then rank(𝐺𝐴)_.𝑖(𝑔_.𝑗)≤𝑡, where 1≤𝑖≤𝑛, 1≤𝑗≤𝑚, and rank(𝐴𝐺)_𝑖.(𝑔_𝑗.)≤𝑡, where 1≤𝑖≤𝑚, 1≤𝑗≤𝑛.

Proof. Let 𝑃_𝑖𝑘(𝑎) be an 𝑛×𝑛 matrix with 𝑎 in the (𝑖,𝑘) entry, 1 in all diagonal entries, and 0 in others. It is an elementary matrix and (𝐺𝐴)_.𝑖𝑔_.𝑗_𝑘≠𝑖𝑃_𝑖𝑘−𝑎_𝑗𝑘=⎛⎜⎜⎜⎜⎜⎜⎝^𝑘≠𝑗𝑔_1𝑘𝑎_𝑘1⋯𝑔_1𝑗⋯_𝑘≠𝑗𝑔_1𝑘𝑎_𝑘𝑛⋮⋮⋮⋮⋮_𝑘≠𝑗𝑔_𝑛𝑘𝑎_𝑘1⋯𝑔_𝑛𝑗⋯_𝑘≠𝑗𝑔_𝑛𝑘𝑎_𝑘𝑛𝑖⎞⎟⎟⎟⎟⎟⎟⎠th=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑔¹¹⋯𝑔_1𝑗⋯𝑔_1𝑚𝑔⋮⋮⋮⋮⋮_𝑖1⋯𝑔_𝑖𝑗⋯𝑔_𝑖𝑚𝑔⋮⋮⋮⋮⋮_𝑛1⋯𝑔_𝑛𝑗⋯𝑔_𝑛𝑚⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑎¹¹⋯0⋯𝑎_1𝑛𝑎⋮⋮0⋯⋮⋮⋮⋮1⋯⋮⋮⋮⋮_𝑚10⋯0⋯𝑎_𝑚𝑚⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠𝑗th.𝑖th(2.3) It follows from the invertibility of 𝑃_𝑖𝑘(𝑎),𝑖≠𝑘, that rank(𝐺𝐴)_.𝑖(𝑔_.𝑗)≤𝑡.
Analogously, the inequation rank(𝐴𝐺)_𝑖.(𝑔_𝑗.)≤𝑡 can be proved. So the proof is complete.

Recall that if 𝑓_𝐴(𝜆)=det(𝜆𝐼+𝐴)=𝜆^𝑛+𝑑₁𝜆^𝑛−1⋯+𝑑_𝑛−1𝜆+𝑑_𝑛 is the characteristic polynomial of an 𝑛×𝑛 matrix—𝐴 over ℂ, then 𝑑_𝑖 is the sum of all 𝑖×𝑖 principal minors of 𝐴, where 𝑖=1,…,𝑛 (see, e.g., [16]).

Theorem 2.3. Let 𝐴,𝑇,𝑆, and 𝐺 be the same as in Theorem 2.1. Write 𝐺=(𝑔_𝑖𝑗). Suppose that the generalized inverse 𝐴^(2)𝑇,𝑆 of 𝐴 exists. Then 𝐴^(2)𝑇,𝑆 can be represented as follows: 𝐴^(2)𝑇,𝑆=𝑥^𝑖𝑗𝑑_𝑡(𝐺𝐴)_𝑛×𝑚,(2.4) where 𝑥_𝑖𝑗=_{𝛽∈𝒬𝑡,𝑛{𝑖}}|||(𝐺𝐴)_.𝑖(𝑔_.𝑗)^𝛽𝛽|||,𝑑_𝑡(𝐺𝐴)=_{𝛽∈𝒬𝑡,𝑛}|||(𝐺𝐴)_𝛽𝛽|||,(2.5) or 𝐴^(2)𝑇,𝑆=𝑦^𝑖𝑗𝑑_𝑡(𝐴𝐺)_𝑛×𝑚,(2.6) where 𝑦_𝑖𝑗=_{𝛼∈𝒬𝑡,𝑚{𝑗}}||(𝐴𝐺)_𝑗.(𝑔_𝑖.)^𝛼𝛼||,𝑑_𝑡(𝐴𝐺)=_{𝛼∈𝒬𝑡,𝑚}||(𝐴𝐺)^𝛼𝛼||.(2.7)

Proof. We will only show the representation (2.5) since the proof of (2.7) is similar. If −𝜆 is not the eigenvalue of 𝐺𝐴, then the matrix 𝜆𝐼+𝐺𝐴 is invertible, and (𝜆𝐼+𝐺𝐴)⁻¹=1det(𝜆𝐼+𝐺𝐴)⎛⎜⎜⎜⎜⎜⎜⎝𝑋¹¹𝑋₂₁⋯𝑋_𝑛1𝑋₁₂𝑋₂₂⋯𝑋_𝑛2𝑋⋮⋮⋮⋮_1𝑛𝑋_2𝑛⋯𝑋_𝑛𝑛⎞⎟⎟⎟⎟⎟⎟⎠,(2.8) where 𝑋_𝑖𝑗,𝑖,𝑗=1,…,𝑛, are cofactors of 𝜆𝐼+𝐺𝐴. It is easy to see that ^𝑛_𝑙=1𝑋_𝑖𝑙𝑔_𝑙𝑗=det(𝜆𝐼+𝐺𝐴)_.𝑖𝑔_.𝑗.(2.9) So, by (2.1), 𝐴^(2)𝑇,𝑆=lim_𝜆→0⎛⎜⎜⎜⎜⎜⎜⎝det(𝜆𝐼+𝐺𝐴)_.1𝑔_.1det(𝜆𝐼+𝐺𝐴)⋯det(𝜆𝐼+𝐺𝐴)^.1𝑔_.𝑚det(𝜆𝐼+⋮𝐺𝐴)_.𝑛⋮det(𝜆𝐼⋮+𝐺𝐴)𝑔_.1det(𝜆𝐼+𝐺𝐴)⋯det(𝜆𝐼+𝐺𝐴)^.𝑛𝑔_.𝑚det(𝜆𝐼+𝐺𝐴)⎞⎟⎟⎟⎟⎟⎟⎠.(2.10)
We have the characteristic polynomial of 𝐺𝐴𝑓_𝐺𝐴(𝜆)=det(𝜆𝐼+𝐺𝐴)=𝜆^𝑛+𝑑₁𝜆^𝑛−1+𝑑₂𝜆^𝑛−2+⋯+𝑑_𝑛,(2.11) where 𝑑_𝑖(1≤𝑖≤𝑛) is a sum of 𝑖×𝑖 principal minors of 𝐺𝐴. Since rank(𝐺𝐴)≤rank(𝐺)=𝑡, 𝑑_𝑛=𝑑_𝑛−1=⋯=𝑑_𝑡+1=0 and det(𝜆𝐼+𝐺𝐴)=𝜆^𝑛+𝑑₁𝜆^𝑛−1+𝑑₂𝜆^𝑛−2+⋯+𝑑_𝑡𝜆^𝑛−𝑡.(2.12) Expanding det(𝜆𝐼+𝐺𝐴)_.𝑖(𝑔_.𝑗), we have det(𝜆𝐼+𝐺𝐴)_.𝑖𝑔_.𝑗=𝑥^1(𝑖𝑗)𝜆^𝑛−1+𝑥^2(𝑖𝑗)𝜆^𝑛−2+⋯+𝑥^{𝑛(𝑖𝑗)},(2.13) where 𝑥^{𝑘(𝑖𝑗)}=∑_{𝛽∈𝒬𝑘,𝑛{𝑖}}|((𝐺𝐴)_.𝑖(𝑔_.𝑗))^𝛽𝛽|, 1≤𝑘≤𝑛, for 1≤𝑖≤𝑛 and 1≤𝑗≤𝑚.
By Lemma 2.2, rank(𝐺𝐴)_.𝑖(𝑔_.𝑗)≤𝑡 and so |((𝐺𝐴)_.𝑖(𝑔_.𝑗))^𝛽𝛽|=0, 𝑘>𝑡 and 𝛽∈𝒬_𝑘,𝑛{𝑖}, for all 𝑖,𝑗. Therefore, 𝑥^{𝑘(𝑖𝑗)}=0, 𝑘≤𝑛, for all 𝑖,𝑗. Consequently, det(𝜆𝐼+𝐺𝐴)_.𝑖𝑔_.𝑗=𝑥^1(𝑖𝑗)𝜆^𝑛−1+𝑥^2(𝑖𝑗)𝜆^𝑛−2+⋯+𝑥^{𝑡(𝑖𝑗)}𝜆^𝑛−𝑡.(2.14)
Substituting (2.12) and (2.14) into (2.10) yields 𝐴^(2)𝑇,𝑆=lim_𝜆→0⎛⎜⎜⎜⎜⎜⎜⎝𝑥¹⁽¹¹⁾𝜆^𝑛−1+⋯+𝑥^𝑡(11)𝜆^𝑛−𝑡𝜆^𝑛+𝑑₁𝜆^𝑛−1+⋯+𝑑_𝑡𝜆^𝑛−𝑡⋯𝑥^1(1𝑚)𝜆^𝑛−1+⋯+𝑥^𝑡(1𝑚)𝜆^𝑛−𝑡𝜆^𝑛+𝑑₁𝜆^𝑛−1+⋯+𝑑_𝑡𝜆^𝑛−𝑡𝑥⋮⋮⋮_1(𝑛1)𝜆^𝑛−1+⋯+𝑥^𝑡(𝑛1)𝜆^𝑛−𝑡𝜆^𝑛+𝑑₁𝜆^𝑛−1+⋯+𝑑_𝑡𝜆^𝑛−𝑡⋯𝑥^1(𝑛𝑚)𝜆^𝑛−1+⋯+𝑥^{𝑡(𝑛𝑚)}𝜆^𝑛−𝑡𝜆^𝑛+𝑑₁𝜆^𝑛−1+⋯+𝑑_𝑡𝜆^𝑛−𝑡=⎛⎜⎜⎜⎜⎜⎜⎝𝑥_𝑡(11)⎞⎟⎟⎟⎟⎟⎟⎠𝑑_𝑡⋯𝑥^𝑡(1𝑚)𝑑_𝑡𝑥⋮⋮⋮_𝑡(𝑛1)𝑑_𝑡⋯𝑥^{𝑡(𝑛𝑚)}𝑑_𝑡⎞⎟⎟⎟⎟⎟⎟⎠.(2.15)
Substituting 𝑥_𝑖𝑗 for 𝑥^{𝑡(𝑖𝑗)} 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, 𝑢=𝑑_𝑡(𝐺𝐴) and, by the Binet-Cauchy formula, 𝑥_𝑖𝑗=_{𝛽∈𝒬𝑡,𝑛{𝑖}}|||(𝐺𝐴)_.𝑖(𝑔_.𝑗)^𝛽𝛽|||=_{𝛽∈𝒬𝑡,𝑛{𝑖}}_𝑘𝑔_𝑘𝑗𝜕|||(𝐺𝐴)^𝛽𝛽|||𝜕𝑠_𝑘𝑖=_{𝛽∈𝒬𝑡,𝑛,𝛼∈𝒬𝑡,𝑚}_𝑘𝑔_𝑘𝑗𝜕||𝐺^𝛼𝛽||𝜕𝑔_𝑘𝑗𝜕||𝐴^𝛽𝛼||𝜕𝑎_𝑗𝑖=_{𝛼∈𝒬𝑡,𝑚,𝛽∈𝒬𝑡,𝑛}det𝐺^𝛼𝛽𝜕||𝐴^𝛽𝛼||𝜕𝑎_𝑗𝑖,(2.16) where 𝑠_𝑘𝑗=(𝐺𝐴)_𝑘𝑗. Note that 𝜕|𝐴^𝛽𝛼|/𝜕𝑎_𝑖𝑗=0 if 𝑖∉𝛼 or 𝑗∉𝛽. In addition, using the symbols in [13], we can rewrite (2.5) as [13, equation (13)] over ℂ.
(ii) This method is especially efficient when 𝐺𝐴 or 𝐴𝐺 is given (comparing with [12, Theorem  2]).

Observing the particular case from Theorem 2.3, 𝐺=(𝑔_𝑖𝑗)=𝑁⁻¹𝐴^∗𝑀, where 𝑀 and 𝑁 are Hermitian positive definite matrices, we obtain the following corollary in which the symbols 𝑔_.𝑗∶=(𝑔)_.𝑗 and 𝑔_𝑖.∶=(𝑔)_𝑖..

Corollary 2.6. Let 𝐴∈ℂ^{𝑟𝑚×𝑛} and 𝐺=𝑁⁻¹𝐴^∗𝑀, where 𝑀 and 𝑁 are Hermitian positive definite matrices of order 𝑚 and 𝑛, respectively, Then 𝐴^†𝑀𝑁=𝑥^𝑖𝑗𝑑_𝑟(𝐺𝐴)_𝑛×𝑚,(2.17) where 𝑥_𝑖𝑗=_{𝛽∈𝒬𝑟,𝑛{𝑖}}|||(𝐺𝐴)_.𝑖(𝑔_.𝑗)^𝛽𝛽|||,𝑑_𝑟(𝐺𝐴)=_{𝛽∈𝒬𝑟,𝑛}|||(𝐺𝐴)_𝛽𝛽|||,(2.18) or 𝐴^†𝑀𝑁=𝑦^𝑖𝑗𝑑_𝑟(𝐴𝐺)_𝑛×𝑚,(2.19) where 𝑦_𝑖𝑗=_{𝛼∈𝒬𝑟,𝑚{𝑗}}||(𝐴𝐺)_𝑗.(𝑔_𝑖.)^𝛼𝛼||,𝑑_𝑟(𝐴𝐺)=_{𝛼∈𝒬𝑟,𝑚}||(𝐴𝐺)^𝛼𝛼||.(2.20)

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

Corollary 2.7 (see [17, Theorem  2.2]). The Moore-Penrose inverse 𝐴^† of 𝐴=(𝑎_𝑖𝑗)∈ℂ^{𝑟𝑚×𝑛} can be represented as follows: 𝐴^†=𝑥^𝑖𝑗𝑑_𝑟(𝐴^∗𝐴)_𝑛×𝑚,(2.21) where 𝑥_𝑖𝑗=_{𝛽∈𝒬𝑟,𝑛{𝑖}}|||𝐴_∗𝐴_.𝑖𝑎^∗.𝑗^𝛽𝛽|||,𝑑_𝑟𝐴^∗𝐴=_{𝛽∈𝒬𝑟,𝑛}|||𝐴_∗𝐴^𝛽𝛽|||,(2.22) or 𝐴^†=𝑦^𝑖𝑗𝑑_𝑟(𝐴𝐴^∗)_𝑛×𝑚,(2.23) where 𝑦_𝑖𝑗=_{𝛼∈𝒬𝑟,𝑚{𝑗}}|||𝐴𝐴_∗_𝑗.(𝑎^∗𝑖.)^𝛼𝛼|||,𝑑_𝑟𝐴𝐴^∗=_{𝛼∈𝒬𝑟,𝑚}||𝐴𝐴^∗^𝛼𝛼||.(2.24)

Note that 𝐴_𝑑,𝑊=(𝑊𝐴𝑊)^{(2)ℛ((𝐴𝑊)}_{𝑘𝐴),𝒩((𝐴𝑊)𝑘𝐴)}. Therefore, when 𝐺=(𝐴𝑊)^𝑘𝐴 in Theorem 2.3, we have the following corollary.

Corollary 2.8. Let 𝐴∈ℂ^𝑚×𝑛, 𝑊∈ℂ^𝑛×𝑚, and 𝑘=max{Ind(𝐴𝑊),Ind(𝑊𝐴)}. If rank(𝐴𝑊)^𝑘=𝑡, rank(𝑊𝐴)^𝑘=𝑟, and (𝐴𝑊)^𝑘𝐴=(𝑐_𝑖𝑗)_𝑚×𝑛, then 𝐴_𝑑,𝑊=𝑥^𝑖𝑗𝑑_𝑡(𝐴𝑊)^𝑘+2_𝑚×𝑛,(2.25) where 𝑥_𝑖𝑗=_{𝛽∈𝒬𝑡,𝑚{𝑖}}|||(𝐴𝑊)_𝑘+2_.𝑖𝑐_.𝑗^𝛽𝛽|||,𝑑_𝑡(𝑊𝐴)^𝑘+2=_{𝛽∈𝒬𝑡,m}|||(𝐴𝑊)_𝑘+2^𝛽𝛽|||,(2.26) or 𝐴_𝑑,𝑊=𝑦^𝑖𝑗𝑑_𝑟(𝑊𝐴)^𝑘+2_𝑚×𝑛,(2.27) where 𝑦_𝑖𝑗=_{𝛼∈𝒬𝑟,𝑛{𝑗}}|||(𝑊𝐴)_𝑘+2_𝑗.𝑐_𝑖.^𝛼𝛼|||,𝑑_𝑟(𝑊𝐴)^𝑘+2=_{𝛼∈𝒬𝑟,𝑛}||(𝑊𝐴)^𝑘+2^𝛼𝛼||.(2.28)

When 𝐺=𝐴^𝑘 with 𝑘=Ind(𝐴) in Theorem 2.3, we have the following corollary.

Corollary 2.9 (see [17, Theorem  3.3]). Let 𝐴∈ℂ^𝑛×𝑛 with Ind𝐴=𝑘 and rank𝐴^𝑘=𝑟, and 𝐴^𝑘=(𝑎^{(𝑘)𝑖𝑗})_𝑛×𝑛. Then 𝐴_𝑑=𝑥^𝑖𝑗𝑑_𝑟𝐴^𝑘+1_𝑛×𝑛,(2.29) where 𝑑_𝑖𝑗=_{𝛽∈𝒬𝑟,𝑛{𝑖}}||||𝐴_𝑘+1.𝑖𝑎^(𝑘).𝑗^𝛽𝛽||||,𝑑_𝑟𝐴^𝑘+1=_{𝛽∈𝒬𝑟,𝑛}|||𝐴_𝑘+1^𝛽𝛽|||.(2.30)

Finally, we turn our attention to the two projectors 𝐴^(2)𝑇,𝑆𝐴 and 𝐴𝐴^(2)𝑇,𝑆. The limiting expressions for 𝐴^(2)𝑇,𝑆 in (2.1) bring us the following:𝐴𝐴𝐴^(2)𝑇,𝑆𝐴_(2)𝑇,𝑆=lim_𝜆→0(𝐺𝐴+𝜆𝐼)⁻¹𝐺𝐴,=lim_𝜆→0𝐴𝐺(𝐴𝐺+𝜆𝐼)⁻¹.(2.31)

Corollary 2.10. Let 𝐴,𝑇,𝑆, and 𝐺 be the same as in Theorem 2.1. Write 𝐺𝐴=(𝑠_𝑖𝑗) and 𝐴𝐺=(ℎ_𝑖𝑗). Suppose that 𝐴^(2)𝑇,_S exists. Then 𝐴^(2)𝑇,𝑆𝐴 of 𝐴𝐴^(2)𝑇,𝑆 can be represented as follows: 𝐴^(2)𝑇,𝑆𝐴=𝑥^𝑖𝑗𝑑_𝑡(𝐺𝐴)_𝑛×𝑛,(2.32) where 𝑥_𝑖𝑗=_{𝛽∈𝒬𝑡,𝑛{𝑖}}|||(𝐺𝐴)_.𝑖𝑠_.𝑗^𝛽𝛽|||,𝑑_𝑡(𝐺𝐴)=_{𝛽∈𝒬𝑡,𝑛}𝐴𝐴_(2)𝑇,𝑆|||(𝐺𝐴)_𝛽𝛽|||,=𝑦^𝑖𝑗𝑑_𝑡(𝐴𝐺)_𝑚×𝑚,(2.33) where 𝑦_𝑖𝑗=_{𝛼∈𝒬𝑡,𝑚{𝑗}}||(𝐴𝐺)_𝑗.(ℎ_𝑖.)^𝛼𝛼||,𝑑_𝑡(𝐴𝐺)=_{𝛼∈𝒬𝑡,𝑚}||(𝐴𝐺)^𝛼𝛼||.(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 𝐴𝑋=𝐵. 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 𝐴^𝑇𝑋𝐴=𝐵. 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 equation𝐴𝑋𝐵=𝐷,ℛ(𝑋)⊂𝑇,𝒩(𝑋)⊃𝑆,(3.1) where 𝐴∈ℂ^{𝑟𝑚×𝑛}, 𝐵∈ℂ^{𝑝×𝑞̃𝑟}, 𝐷∈ℂ^𝑚×𝑞, 𝑇⊂ℂ^𝑛, 𝑆⊂ℂ^𝑚, 𝑇⊂ℂ^𝑞 and 𝑆⊂ℂ^𝑝, satisfydim(𝑇)=dim𝑆^⟂=𝑡≤𝑟,dim𝑇=dim𝑆^⟂=𝑡̃≤𝑟.̃(3.2) Assume that there exist matrices 𝐺∈ℂ^𝑛×𝑚 and 𝐺∈ℂ^𝑞×𝑝 satisfyingℛ(𝐺)=𝑇,𝒩(𝐺)=𝑆,ℛ𝐺=𝑇,𝒩𝐺=𝑆.(3.3) If 𝐴^(2)𝑇,𝑆 and 𝐵^(2)𝑇,𝑆 exist and 𝐷∈ℛ(𝐴𝐺,𝐺𝐵), then the restricted matrix equation (3.1) has the unique solution𝑋=𝐴^(2)𝑇,𝑆𝐷𝐵^(2)𝑇,𝑆(3.4) (see [2,  Theorem  3.3.3] for the proof).

In particular, when 𝐷 is a vector 𝑏 and 𝐵=𝐺=𝐼₁, the restricted matrix equation (3.1) becomes the restricted linear equation𝐴𝑥=𝑏,𝑥∈𝑇.(3.5) If 𝑏∈𝐴ℛ(𝐺), then 𝑥=𝐴^(2)𝑇,𝑆𝑏 is the unique solution of the restricted linear equation (3.5) (see also [10,  Theorem  2.1]).

Theorem 3.1. Given 𝐴,𝐵,𝐷=(𝑑_𝑖𝑗),𝐺=(𝑔_𝑖𝑗),𝐺=(𝑔̃_𝑖𝑗),𝑇,𝑆,𝑇, and 𝑆 as above. If 𝐴^(2)𝑇,𝑆 and 𝐵^(2)𝑇,𝑆 exist and 𝐷∈ℛ(𝐴𝐺,𝐺𝐵), then 𝑋=𝐴^(2)𝑇,𝑆𝐷𝐵^(2)𝑇,𝑆 is the unique solution of the restricted matrix equation (3.1) and it can be represented as 𝑥_𝑖𝑗=∑^𝑚𝑘=1∑_{𝛽∈𝒬𝑡,𝑛{𝑖},𝛼∈𝒬̃𝑡,𝑝{𝑗}}|||(𝐺𝐴)_.𝑖𝑔_.𝑘^𝛽𝛽|||||||𝐵𝐺_𝑗.𝑓^𝑘.^𝛼𝛼||||𝑑_𝑡(𝐺𝐴)𝑑_̃𝑡𝐵𝐺(3.6) or 𝑥_𝑖𝑗=∑^𝑞𝑘=1∑_{𝛽∈𝒬𝑡,𝑛{𝑖},𝛼∈𝒬̃𝑡,𝑝{𝑗}}|||(𝐺𝐴)_.𝑖(𝑓_.𝑘)^𝛽𝛽|||||||𝐵𝐺_𝑗.(𝑔̃_𝑘.)^𝛼𝛼||||𝑑_𝑡(𝐺𝐴)𝑑_̃𝑡𝐵𝐺,(3.7) where 𝑓_𝑘.=𝑑_𝑘.𝐺 and 𝑓_.𝑘=𝐺𝑑_.𝑘, 𝑖=1,…,𝑛, and 𝑗=1,…,𝑝.

Proof. By the argument above, we have 𝑋=𝐴^(2)𝑇,𝑆𝐷𝐵^(2)𝑇,𝑆 is the unique solution of the restricted matrix equation (3.1). Setting 𝑌=𝐷𝐵^(2)𝑇,𝑆 and using (2.7), we get that 𝑦_𝑘𝑗=^𝑞_ℎ=1𝑑_𝑘ℎ𝐵^(2)𝑇,𝑆_ℎ𝑗=^𝑞_ℎ=1𝑑_𝑘ℎ∑_{𝛼∈𝒬̃𝑡,𝑝{𝑗}}||||𝐵𝐺_𝑗.𝑔̃_ℎ.^𝛼𝛼||||=∑_{𝛼∈𝒬̃𝑡,𝑝{𝑗}}𝑑_̃𝑡𝐵𝐺||||𝐵𝐺_𝑗.∑^𝑞ℎ=1𝑑_𝑘ℎ𝑔̃_ℎ.^𝛼𝛼||||𝑑_̃𝑡𝐵𝐺=∑^{𝛼∈𝒬̃𝑡,𝑝{𝑗}}||||𝐵𝐺_𝑗.𝑓^𝑘.^𝛼𝛼||||𝑑_̃𝑡𝐵𝐺,(3.8) where 𝑓_𝑘.=𝑑_𝑘.𝐺. Since 𝑋=𝐴^(2)𝑇,𝑆𝑌, by (2.5), 𝑥_𝑖𝑗=^𝑚_𝑘=1𝐴^(2)𝑇,𝑆_𝑖𝑘𝑦_𝑘𝑗=^𝑚_𝑘=1∑_{𝛽∈𝒬𝑡,𝑛{𝑖}}|||(𝐺𝐴)_.𝑖𝑔_.𝑘^𝛽𝛽|||𝑑_𝑡(𝐺𝐴)∑^{𝛼∈𝒬̃𝑡,𝑝{𝑗}}||||𝐵𝐺_𝑗.𝑓^𝑘.^𝛼𝛼||||𝑑_̃𝑡𝐵𝐺.(3.9) Hence, we have (3.6).
We can obtain (3.7) in the same way.

In particular, when 𝐷 is a vector 𝑏 and 𝐵=𝐺=𝐼₁ in the above theorem, we have the following result from (3.7).

Theorem 3.2. Given 𝐴,𝐺,𝑇, and 𝑆 as above. If 𝑏∈𝐴ℛ(𝐺), then 𝑥=𝐴^(2)𝑇,𝑆𝑏 is the unique solution of the restricted linear equation 𝐴𝑥=𝑏, 𝑥∈𝑇, and it can be represented as 𝑥_𝑖=∑^{𝛽∈𝒬𝑡,𝑛{𝑖}}|||(𝐺𝐴)_.𝑖(𝑓)^𝛽𝛽|||𝑑_𝑡(𝐺𝐴),𝑗=1,…,𝑛,(3.10) where 𝑓=𝐺𝑏.

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

4. Example

Let𝐴=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝10000220002142021210⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝐵=⎛⎜⎜⎜⎜⎝210101⎞⎟⎟⎟⎟⎠𝐺,=⎛⎜⎜⎝10𝐷02=00⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎠−0000.101000⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝐺=⎛⎜⎜⎜⎜⎜⎜⎝1000−2001210000000000⎞⎟⎟⎟⎟⎟⎟⎠,(4.1) Obviously, rank𝐴=3, dim𝐴𝑇=dim𝑇=2, and𝑇=ℛ(𝐺)⊂ℂ⁴,𝑆=𝒩(𝐺)⊂ℂ⁵,𝑇=ℛ𝐺⊂ℂ²,𝑆=𝒩𝐺⊂ℂ³.(4.2) It is easy to verify that 𝐴𝑇⊕𝑆=ℂ⁵ and 𝐵𝑇⊕𝑆=ℂ³. Thus, 𝐴^(2)𝑇,𝑆 and 𝐵^(2)𝑇,𝑆 exist by Lemma 1.1.

Now consider the restricted matrix equation𝐴𝑋𝐵=𝐷,ℛ(𝑋)⊂𝑇,𝒩(𝑋)⊃𝑆.(4.3) Clearly,𝐴𝐺=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝1000034000420000000000000⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝐺𝐵=⎛⎜⎜⎝2210⎞⎟⎟⎠,(4.4) and it is easy to verify that ℛ(𝐷)⊂ℛ(𝐴𝐺) and 𝒩(𝐷)⊃𝒩(𝐺𝐵) hold.

Note that ℛ(𝐷)⊂ℛ(𝐴𝐺) and 𝒩(𝐷)⊃𝒩(𝐺𝐵) if and only if 𝐷∈ℛ(𝐴𝐺,𝐺𝐵). So, by Theorem 3.1, the unique solution of (4.3) exists.

Computing𝑑𝐺𝐴=⎛⎜⎜⎜⎜⎜⎜⎝1000040096005400⎞⎟⎟⎟⎟⎟⎟⎠,𝐵𝐺=⎛⎜⎜⎜⎜⎝210202000₂⎞⎟⎟⎟⎟⎠,(𝐺𝐴)=𝑑||||||1004||||||+||||||1090||||||+||||||1050||||||+||||||4060||||||+||||||4040||||||+||||||0000||||||=4,₂𝑓=𝐵⎛⎜⎜⎜⎜⎜⎜⎝1000𝐺−2001=2100||||||21000020||||||0000+⎞⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝||||||20−0000100||||||01000+⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠||||||02=00⎛⎜⎜⎜⎜⎜⎜⎝−||||||100=2−01002⎞⎟⎟⎟⎟⎟⎟⎠,,(4.5) and setting 𝑦_𝑖𝑘=∑_{𝛽∈𝒬𝑡,𝑛{𝑖}}|((𝐺𝐴)_.𝑖(𝑓_.𝑘))^𝛽𝛽|, we have Table 1.

Table 1 

Similarly, setting 𝑧_𝑘𝑗=∑_{𝛼∈𝒬̃𝑡,𝑝{𝑗}}|((𝐵𝐺)_𝑗.(𝑔̃_𝑘.))^𝛼𝛼|, we have Table 2.

Table 2 

So, by (3.7), we have𝑋=𝑥_𝑖𝑗=⎛⎜⎜⎜⎜⎜⎜⎝1000−−001120000⎞⎟⎟⎟⎟⎟⎟⎠.(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.