正规矩阵的广义逆Generalized inverse of a normal matrix

Any decomposition of a matrix A, as describe in Lemma 1.2(a), is called a unitary diagonal decomposition of a normal matrix. In this paper, we assume that the matrix A ∈ C n×n is a normal matrix with a unitary diagonal decomposition (1.1) and denote by Σ = diag (λ1 , · · · , λr ), where λi = λi + iλi , (i = 1, · · · , r) are the nonzero eigenvalues of A.
X21 K = X21 K from which we conclude that K −1 ΣX = Y O s r−s
s , X21 K = (Z
r−s O) .
12
Therefore X12 = Σ−1 K
Y O
,
X21 = (Z
O ) K −1 ,
(2.7)
where Y, Z are arbitrary submatrices of suitable size. Substituting in (2.5), the expressions for X21 , X12 given in (2.7), we get X22 = ZY .
n×n be a normal matrix. Then any {1, 2}-inverse is given by Corollary 2.1 Let A ∈ Cr
V ∗ XV =
(a) Using the unitary diagonal decomposition of A, we have that X ∈ A{1} if and only if Σ O O O X11 X21 X12 X22 Σ O O O = Σ O O O .
Hence, X11 = Σ−1 and X12 , X21 , X22 are arbitrary matrices of suitable size. (b) Similarly, X satisfies XAX = X if and only if X11 ΣX11 X21 ΣX11 i.e. X11 ΣX11 = X11 X11 ΣX12 = X12 X21 ΣX11 = X21 X21 ΣX12 = X22 3 (2.2) (2.3) (2.4) (2.5) X11 ΣX12 X21 ΣX12 = X11 X21 X12 X22 ,
by its Schur decomposition.
Keywords:
Schur decomposition; normal matrix; generalized inverse.
1
Introduction
Let C n×m denote the set of all complex n × m matrices. By A∗ we denote the conjugate

Σ− 1 X21 Σ− 1 K
X12 X22 Is
V ∗, O Y O
V ∗,
(b)
A(2) = V
O O ( Z O ) K −1
K −1
Σ− 1 K ZY
where X12 ∈ C r×(n−r) , X21 ∈ C (n−r)×r , X22 ∈ C (n−r)×(n−r) , Y ∈ C s×(n−r) and Z ∈ C (n−r)×s are arbitrary submatrices and 0 ≤ s ≤ r. Proof. Let X ∈ C n×n be given by r n−r X11 X12 X21 X22 r n−r (2.1)
n×n the set of all matrices A ∈ C n×n such that transpose matrix of A ∈ C n×m and by Cr
rank (A) = r. In denotes the unit matrix of order n. Let us recall that the Moore-Penrose inverse of A ∈ C n×m is the unique matrix A† ∈ C m×n which satisfies (1) AA† A = A, (2) A† AA† = A† , (3) (AA† )∗ = AA† , (4) (A† A)∗ = A† A.
gular matrix K ∈ C r×r such that B=K Is O O O K −1 .
n×n be a normal matrix. Then all matrices A(1) , A(2) are given by Theorem 2.1 Let A ∈ Cr
(a)
A(1) = V
For any A ∈ C n×m , let A{i, j, ..., l} denote the set of matrices X ∈ C m×n which satisfy equations (i), (j ), ..., (l) from among the equations (1), (2), (3), (4). A matrix X ∈ A{i, j, ..., l} is called an {i, j, ..., l}-inverse of A, and also denoted by A(i,j,...,l) . All of these matrices are called the generalized inverse of A. Generalized inverses of matrices are widely applied in many fields, for example, in numerical analysis, network theory, mathematical statistics, optimization problems. The Drazin inverse of A ∈ C n×n is the matrix X , denoted by AD ∈ C n×n which satisfies Ak+1 X = Ak , XAX = X, AX = XA, for some nonnegative integer k . The least such k is the index of A, denoted by ind(A). Also, the index of A can be defined as the least nonnegative integer k such that rank (Ak ) = rank (Ak+1 ).
Generalized inverse of a normal matrix
Bing Zhenga , Lijuan Yea , Dragana S. Cvetkovic-Ilifor generalized inverses of a normal matrix are discussed
Multiplying equation (2.2) with Σ from the left side, we get (ΣX11 )2 = ΣX11 . Now, by Lemma 2.1, it follows that there exists a nonsingular matrix K ∈ C r×r such that ΣX11 = K where 0 ≤ s = rank (X11 ) ≤ r. Hence, X11 = Σ−1 K Now, equations (2.3) and (2.4) have the form Is O O O K −1 ΣX12 = K −1 ΣX12 Is O O O (2.6) Is O O O K −1 . Is O O O K −1 ,
1
When k = 1, X is called the group inverse of A. Drazin inverses [5] provide solutions for systems of linear differential equations and linear difference equations. In [1] singular value decompositions are used to obtain formulae for the generalized inverse of a matrix A. Recently, [2] studies expressions for generalized inverses of a real symmetric matrix by means of congruence decompositions. In this paper, we discuss expressions for generalized inverses of a special class of matrices, normal matrices, using their Schur decomposition. Remark that the class of normal matrices is very important in matrix analysis. First, we will state the definition of normal matrices and the Schur decomposition theorem. Definition 1.1 A ∈ C n×n is a normal matrix if A∗ A = AA∗ .
(1) (2)
2
2
General expressions for {1}, {2}, {1, 2}−inverses