����������� ������ 46 (1994), 41–50

UDC 512.64

���������� ������ ��� research paper

DETERMINANTAL REPRESENTATION OF WEIGHTED MOORE-PENROSE INVERSE

Predrag Stanimirović and Miomir Stanković

Abstract. In this paper we introduce determinantal representation of weighted Moore-Penrose inverse of a rectangular matrix.

We generalize concept of generalized algebraic complement, introduced by Moore, Arghiriade, Dragomir and Gabriel. This extension is denoted as weighted generalized algebraic complement.

Moreover, we derive an explicit determinantal representation for the weighted least-squares minimum norm solution of a linear system and prove that this solution lies in the convex hull of the solutions to the square subsystems of the original system.

1. Introduction

Let C^{n} be the n-dimensional complex vector space, C^{m×n} the set of m × n complex matrices, and C_{r}^{m×n} = {X ∈ C^{m×n} : rank(X) = r}. We suppose that A ∈ C_{r}^{m×n}, unless indicated otherwise. The adjungate matrix of a square matrix B will be denoted as adj(B), and its determinant as |B|. Conjugate, transponsed and conjugate-transponsed matrix of A will be denoted as \overline{A}, A^{T} and A^{∗} respectively. Submatrix of A containing rows α_{1},…, α_{t} and columns β_{1},…, β_{t} is denoted as A [ \begin{matrix}α_{1} & … & α_{t} \\ β_{1} & … & β_{t} \end{matrix} ] . Also, minor of a rectangular matrix A ∈ C^{m×n} containing rows

α_{1},…, α_{t} and columns β_{1},…, β_{t} is denoted as A ( \begin{matrix}α_{1} & … & α_{t} \\ β_{1} & … & β_{t} \end{matrix} ) an its algebraic complement is defined as

A_{ij} ( \begin{matrix}α_{1} & ... & α_{p−1} & i & α_{p+1} & ... & α_{t} \\ β_{1} & ... & β_{q−1} & j & β_{q+1} & ... & β_{t} \end{matrix} ) = (−1)^{i+j}A ( \begin{matrix}α_{1} & ... & α_{p−1} & α_{p+1} & ... & α_{t} \\ β_{1} & ... & β_{q−1} & β_{q+1} & ... & β_{t} \end{matrix} ) .

For any A ∈ C^{m×n} , x ∈ C^{m} , j ∈ {1,…, n} , A(j → x) denotes the matrix obtained by replacing the jth column of A with x, and |A|(j → x) = |A(j → x)|.

Penrose [16] has shown the existence and uniqueness of a solution X ∈ C^{n×m} to the equations

(1) AXA = A , (2) XAX = X , (3) (AX)^{∗} = AX , (4) (XA)^{∗} = XA.

AMS Subject Classification (1991): 15A09, 15A60, 65F35

For a subset S of the set {1, 2, 3, 4} the set of matrices G obeying the conditions represented in S will be denoted by A{S}. A matrix G in A{S} is called an Sinverse of A and is denoted by A^{(S)}. In particular for any A ∈ C^{m×n} the set A{1, 2, 3, 4} consists of a single element, the Moore-Penrose inverse of A, denoted by A^{†} [2], [17].

In the following theorem general forms of the sets A{S} are described.

Theorem 1.1 [18] If A ∈ C_{r}^{m×n} has a full-rank factorization A = P Q , P ∈ C_{r}^{m×r} , Q ∈ C_{r}^{r×n}, W_{1} ∈ C^{n×r} and W_{2} ∈ C^{r×m} are some matrices such that rank(QW_{1}) = rank(W_{2}P) = rank(A), then

\begin{aligned}A^{†} & = Q^{†}P^{†} = Q^{∗}(QQ^{∗})^{−1}(P^{∗}P)^{−1}P^{∗} \\ A\{1, 2\} & = \{W_{1}(QW_{1})^{−1}(W_{2}P)^{−1}W_{2}\} \\ A\{1, 2, 3\} & = \{W_{1}(QW_{1})^{−1}(P^{∗}P)^{−1}P^{∗}\} \\ A\{1, 2, 4\} & = \{Q^{∗}(QQ^{∗})^{−1}(W_{2}P)^{−1}W_{2}\}. \end{aligned}

Concept of determinant i.e. algebraic complement is intimately related to the concept of generalized inversion of matrices. Determinantal representation of Moore-Penrose inverse is studied in [1], [3], [7], [8], [9], [15]. The main result is contained in the following theorem.

Theorem 1.2 Element lying on the i-row and j-column of the Moore-Penrose inverse of a given matrix A ∈ C_{r}^{m×n} can be represented in terms of determinants of square matrices, as follows:

a_{ij}^{(†,r)} = \frac{A_{ji}^{(†,r)}}{∥A∥_{r}} = \frac{∑_{\begin{matrix}1≤β_{1}<...<β_{r}≤n \\ 1≤α_{1}<...<α_{r}≤m \end{matrix}} \overline{A} ( \begin{matrix}α_{1} & ... & i & ... & α_{r} \\ β_{1} & ... & j & ... & β_{r} \end{matrix} ) A_{ji} ( \begin{matrix}α_{1} & ... & i & ... & α_{r} \\ β_{1} & ... & j & ... & β_{r} \end{matrix} )}{∑_{\begin{matrix}1≤δ_{1}<...<δ_{r}≤n \\ 1≤γ_{1}<...<γ_{r}≤m \end{matrix}} A ( \begin{matrix}γ_{1} & ... & γ_{r} \\ δ_{1} & ... & δ_{r} \end{matrix} ) \overline{A} ( \begin{matrix}γ_{1} & ... & γ_{r} \\ δ_{1} & ... & δ_{r} \end{matrix} )} , ( \begin{matrix}1≤i≤n \\ 1≤j≤m \end{matrix} ) .

The numerator of this expression represents generalized algebraic complement of the order r corresponding to a_{ij}, while the denominator expresses determinantal representation of the norm of A.

Weighted Moore-Penrose inverse is investaged in [2], [6], [12]. The main results are contained in the following three theorems.

Theorem 1.3 Let positive-definite (and hermitian) matrices M ∈ C^{m×m} and N ∈ C^{n×n} be given. For any matrix A ∈ C^{m×n} there exists a unique solution X = A_{M•,•N}^{†} ∈ A{1, 2} satisfying

(5)  (MAX)^{∗} = MAX  (6)  (XAN)^{∗} = XAN.

Similarly, we use the following notations:

A_{M•,N•}^{†} represents unique solution of the equations (1), (2), and

(7)  (MAX)^{∗} = MAX  (8)  (NXA)^{∗} = NXA;

A_{•M,N•}^{†} is unique solution of the equations (1), (2), and

(9)  (AXM)^{∗} = AXM  (10)  (NXA)^{∗} = NXA,

while A_{•M,•N}^{†} is unique solution of the equations (1), (2), and

(11)  (AXM)^{∗} = AXM  (12)  (XAN)^{∗} = XAN.

Theorem 1.4 [6] An equivalent of condition (5) is (AXM^{−1})^{∗} = AXM^{−1}, while the condition (6) can be expressed in the form (N^{−1}XA)^{∗} = N^{−1}XA.

Theorem 1.5 [6] If A = P Q is a full rank factorization of A, then: A_{M•,•N}^{†} = (QN)^{∗}(Q(QN)^{∗})^{−1}((MP)^{∗}P)^{−1}(MP)^{∗}.

Using these notions, Theorem 1.4. and Theorem 1.5. the following corollary can be proved.

Corollary 1.1 a) A_{M•,•N}^{†} = A_{•M^{−1},•N}^{†} = A_{M•,N^{−1}•}^{†} = A_{•M^{−1},N^{−1}•}^{†}.

b) A_{M•,N•}^{†} = (QN^{−1})^{∗}(Q(QN^{−1})^{∗})^{−1}((MP)^{∗}P)^{−1}(MP)^{∗} = A_{M•,•N^{−1}};

c) A_{•M,N•}^{†} = (QN^{−1})^{∗}(Q(QN^{−1})^{∗})^{−1}((M^{−1}P)^{∗}P)^{−1}(M^{−1}P)^{∗} = A_{M^{−1}•,•N^{−1}};

d) A_{•M,•N}^{†} = (QN)^{∗}(Q(QN)^{∗})^{−1}((M^{−1}P)^{∗}P)^{−1}(M^{−1}P)^{∗}.

One of indices of the form •M^{[−1]}, •N^{[−1]}; M^{[−1]}•, •N^{[−1]}; •M^{[−1]}, N^{[−1]}•; M^{[−1]}•, N^{[−1]}• , where M^{[−1]} denotes M^{−1} or M and N^{[−1]} denotes N^{−1} or N we formally denote as φ(M, N).

In this paper weighted Moore-Penrose inverse of a rectangular matrix is presented in terms of her own square minors and square minors of matrix product MAN. This determinantal representation is developed using two different methods. In the first method we develop the determinantal representation of {1, 2} inverse and weighted Moore-penrose inverse is treated as an {1, 2} inverse. In the second access we generalize concept of generalized algebraic complement.

Also, we introduce and investage determinantal representation of weighted least-squares minimum norm solution of a linear system.

2. Weighted Moore-Penrose inverse as an {1, 2} inverse

In the following two Theorems we develop determinantal representation of class of {1, 2} inverses, and derive determinantal representation of weighted Moore-Penrose inverse, which is treated as an {1, 2} inverse. The determinantal representation of the class of {1, 2} inverses is a significant result in itself.

Theorem 2.1 If A = P Q is a full rank factorization of A ∈ C_{r}^{m×n} and W_{1} ∈ C^{n×r}, W_{2} ∈ C^{r×m} are some matrices such that rank(QW_{1}) = rank(W_{2}P) = rank(W_{1}W_{2}) = rank(A), then an element a_{ij}^{(1,2)} ∈ A^{(1,2)} is given by

a_{ij}^{(1,2)} = ^{\frac{∑ ; \begin{matrix}1≤β_{1}<...<β_{r}≤n \\ 1≤α_{1}<...<α_{r}≤m \end{matrix}(W_{1}W_{2})^{T} ( \begin{matrix}α_{1} & ... & i & ... & α_{r} \\ β_{1} & ... & j & ... & β_{r} \end{matrix} )A_{ji}( \begin{matrix}α_{1} & ... & j & ... & α_{r} \\ β_{1} & ... & i & ... & β_{r} \end{matrix} )}{∑ ; \begin{matrix}1≤δ_{1}<...<δ_{r}≤n \\ 1≤γ_{1}<...<γ_{r}≤m \end{matrix} A( \begin{matrix}γ_{1} & ... & γ_{r} \\ δ_{1} & ... & δ_{r} \end{matrix} )(W_{1}W_{2})^{T} ( \begin{matrix}γ_{1} & ... & γ_{r} \\ δ_{1} & ... & δ_{r} \end{matrix} )}}.

Proof. Starting from A^{(1,2)} = W_{1}(QW_{1})^{−1}(W_{2}P)^{−1}W_{2}, it is easy to see that a_{ij}^{(1,2)} is equal to

\begin{aligned}∑_{k=1}^{r} & \frac{∑_{β1<...<βr} W_{1}^{T} ( \begin{matrix}1 & ... & k & ... & r \\ β1 & ... & i & ... & βr \end{matrix} ) Q_{ki} ( \begin{matrix}1 & ... & k & ... & r \\ β1 & ... & i & ... & βr \end{matrix} ) ∑_{α1<...<αr} W_{2}^{T} ( \begin{matrix}α1 & ... & j & ... & αr \\ 1 & ... & k & ... & r \end{matrix} ) P_{jk} ( \begin{matrix}α1 & ... & j & ... & αr \\ 1 & ... & k & ... & r \end{matrix} )}{∑_{δ1<...<δr} Q ( \begin{matrix}1 & ... & r \\ δ1 & ... & δr \end{matrix} ) W_{1}^{T} ( \begin{matrix}1 & ... & r \\ δ1 & ... & δr \end{matrix} ) ∑_{1≤γ1<...<γr ≤r} W_{2}^{T} ( \begin{matrix}γ1 & ... & γr \\ 1 & ... & r \end{matrix} ) P ( \begin{matrix}γ1 & ... & γr \\ 1 & ... & r \end{matrix} )} \\ & = \frac{∑_{\begin{matrix}β1<...<βr \\ α1<...<αn \end{matrix}}(W_{1}W_{2})^{T} ( \begin{matrix}α1 & ... & j & ... & αr \\ β1 & ... & i & ... & βr \end{matrix} ) [ ∑_{k=1}^{r} P_{jk} ( \begin{matrix}α1 & ... & j & ... & αr \\ 1 & ... & k & ... & r \end{matrix} ) Q_{ki} ( \begin{matrix}1 & ... & k & ... & r \\ β1 & ... & i & ... & βr \end{matrix} )]}{∑_{\begin{matrix}1≤δ1<...<δr ≤r \\ 1≤γ1<...<γr ≤m \end{matrix}} A ( \begin{matrix}γ1 & ... & γr \\ δ1 & ... & δr \end{matrix} ) (W_{1}W_{2})^{T} ( \begin{matrix}δ1 & ... & δr \\ γ1 & ... & γr \end{matrix} )} . \end{aligned}

Using the Cauchy-Binet formula, we can show

∑_{k=1}^{r} P_{jk} ( \begin{matrix}α_{1} & ... & j & ... & α_{r} \\ 1 & ... & k & ... & r \end{matrix} ) Q_{ki} ( \begin{matrix}1 & ... & k & ... & r \\ β_{1} & ... & i & ... & β_{r} \end{matrix} ) = A_{ji} ( \begin{matrix}α_{1} & ... & j & ... & α_{r} \\ β_{1} & ... & i & ... & β_{r} \end{matrix} )

and the proof is completed.

Theorem 2.2 Let M ∈ C^{m×m} , N ∈ C^{n×n} be positive definite, and suppose that A = P Q is a full rank factorization of A, such that rank(P^{∗}MP) = rank(QNQ^{∗}) = rank(MAN) = r. Element of the weighted Moore-Penrose inverse A_{M•,•N}^{†}, lying on the ith row and jth column, can be represented in terms of square minors as follows:

(a_{M•,•N}^{†})_{ij} = ^{\frac{∑ ; \begin{matrix}1≤β_{1}<...<β_{r}≤n \\ 1≤α_{1}<...<α_{r}≤m \end{matrix} \overline{(MAN)}( \begin{matrix}α_{1} & ... & j & ... & α_{r} \\ β_{1} & ... & i & ... & β_{r} \end{matrix} )A_{ji}( \begin{matrix}α_{1} & ... & j & ... & α_{r} \\ β_{1} & ... & i & ... & β_{r} \end{matrix} )}{∑ ; \begin{matrix}1≤δ_{1}<...<δ_{r}≤n \\ 1≤γ_{1}<...<γ_{r}≤m \end{matrix} A( \begin{matrix}γ_{1} & ... & γ_{r} \\ δ_{1} & ... & δ_{r} \end{matrix} )\overline{(MAN)}( \begin{matrix}γ_{1} & ... & γ_{r} \\ δ_{1} & ... & δ_{r} \end{matrix} )}}

Proof. According to Theorem 1.1 and Theorem 1.5, weighted Moore-Penrose inverse of a matrix A can be obtained as an element from the class of A{1, 2} inverses satisfying relations W_{1} = (QN)^{∗} , W_{2} = (MP)^{∗}. Applying these substitutions in formula which represent determinantal representation of the class of {1, 2} inverses, the proof can be elementary obtained.

From Theorem 2.2., and Corollary 1.1. it follows:

Corollary 2.1 Let M ∈ C^{m×m} , N ∈ C^{n×n} be positive definite and A = P Q is a full rank factorization of A. Then

(A_{φ(M,N)}^{†})_{ij} = ^{\frac{∑ ; \begin{matrix}1≤β_{1}<...<β_{r}≤n \\ 1≤α_{1}<...<α_{r}≤m \end{matrix} \overline{(ω(M,N))}( \begin{matrix}α_{1} & ... & j & ... & α_{r} \\ β_{1} & ... & i & ... & β_{r} \end{matrix} )A_{ji}( \begin{matrix}α_{1} & ... & j & ... & α_{r} \\ β_{1} & ... & i & ... & β_{r} \end{matrix} )}{∑ ; \begin{matrix}1≤δ_{1}<...<δ_{r}≤n \\ 1≤γ_{1}<...<γ_{n}≤r \end{matrix} A( \begin{matrix}γ_{1} & ... & γ_{r} \\ δ_{1} & ... & δ_{r} \end{matrix} )\overline{(ω(M,N))}( \begin{matrix}γ_{1} & ... & γ_{r} \\ δ_{1} & ... & δ_{r} \end{matrix} )}},

where the expression ω(M, N) represents a matrix such that rank(ω(M, N)) = rank(A) and

ω(M, N) = \{ \begin{matrix}MAN, & φ(M, N) = M•, •N \\ MAN^{−1}, & φ(M, N) = M•, N• \\ M^{−1}AN, & φ(M, N) = •M, •N \\ M^{−1}AN^{−1}, & φ(M, N) = •M, N• \end{matrix} .

3. Weighted generalized algebraic complement and weighted matrix norm

In this section we define weighted generalized algebraic complement and weighted norm of rectangular complex matrices, and using these notions we find the known determinantal representation of the weighted Moore-Penrose inverse.

Definition 3.1 Weighted norm of A ∈ C_{r}^{m×n}, denoted as ∥A∥_{φ(M,N)}^{r} is equal to

| (MP)^{∗}P | | Q(QN)^{∗} |

while the weighted adjoint matrix of A, denoted as adj (A_{M•,•N}^{(†,r)}), is

(QN)^{∗} adj(Q(QN)^{∗}) · adj((MP)^{∗}P)(MP)^{∗}.

Theorem 3.1 Weighted norm of A has the following determinantal representation

∥A∥_{φ(M,N)}^{r} = ∑_{\begin{matrix}1≤i_{1}<...<i_{r}≤m \\ 1≤j_{1}<...<j_{r}≤n \end{matrix}} A ( \begin{matrix}j_{1} & ... & j_{r} \\ i_{1} & ... & i_{r} \end{matrix} ) \overline{(ω(M, N))} ( \begin{matrix}j_{1} & ... & j_{r} \\ i_{1} & ... & i_{r} \end{matrix} ) .

Proof. Suppose that φ(M, N) = M•, •N and A = P Q is a full rank factorization of A.

∥A∥_{M•,•N}^{r} = | (MP)^{∗}P | | Q(QN)^{∗} |

= [ ∑_{i_{1}<...<i_{r}} P ( \begin{matrix}i_{1} & ... & i_{r} \\ 1 & ... & r \end{matrix} ) \overline{(MP)} ( \begin{matrix}i_{1} & ... & i_{r} \\ 1 & ... & r \end{matrix} )] [ ∑_{j_{1}<...<j_{r}} Q ( \begin{matrix}1 & ... & r \\ j_{1} & ... & j_{r} \end{matrix} ) \overline{(QN)} ( \begin{matrix}1 & ... & r \\ j_{1} & ... & j_{r} \end{matrix} )]

= ∑_{\begin{matrix}1≤i_{1}<...<i_{r}≤r \\ 1≤j_{1}<...<j_{r}≤s \end{matrix}} A ( \begin{matrix}i_{1} & ... & i_{r} \\ j_{1} & ... & j_{r} \end{matrix} ) \overline{(MAN)} ( \begin{matrix}i_{1} & ... & i_{r} \\ j_{1} & ... & j_{r} \end{matrix} ) .

Theorem 3.2 Element lying on ith row and jth column of the weighted adjoint matrix of A, denoted as adj (A_{M•,•N}^{(†,r)})_{ij} can be represented in terms of square

minors as follows:

adj (A_{φ(M,N)}^{(†,r)})_{ij} = ∑_{\begin{matrix}1≤α_{1}<...<α_{r}≤m \\ 1≤β_{1}<...<β_{r}≤n \end{matrix}} \overline{(ω(M, N))} ( \begin{matrix}α_{1} & ... & j & ... & α_{r} \\ β_{1} & ... & i & ... & β_{r} \end{matrix} ) A_{ji} ( \begin{matrix}α_{1} & ... & j & ... & α_{r} \\ β_{1} & ... & i & ... & β_{r} \end{matrix} ) .

Proof. Let φ(M, N) = M•, •N and consider a full rank factorization A = P Q. Element on ith row and jth column of (QN)^{∗} adj(Q(QN)^{∗}) is equal to

∑_{k=1}^{r}(QN)_{ik}^{∗}(adj(Q(QN)^{∗}))_{kj}

= ∑_{k=1}^{r} \overline{(QN)}_{ki} [(−1)^{k+j} ∑_{j1<...<jr−1} Q ( \begin{matrix}j1 & ... & ... & ... & ... & jr−1 \\ 1 & ... & j−1 & j+1 & ... & r \end{matrix} ) \overline{(QN)} ( \begin{matrix}j1 & ... & ... & ... & ... & jr−1 \\ 1 & ... & k−1 & k+1 & ... & r \end{matrix} )]

= ∑_{j1<...<jr−1}(−1)^{j}Q ( \begin{matrix}j1 & ... & ... & ... & ... & jr−1 \\ 1 & ... & j−1 & j+1 & ... & r \end{matrix} ) [∑_{k=1}^{r}(−1)^{k}\overline{(QN)}_{ki}\overline{(QN)} ( \begin{matrix}j1 & ... & ... & ... & ... & jr−1 \\ 1 & ... & k−1 & k+1 & ... & r \end{matrix} )] .

If i is contained in combination j_{1},…, j_{r−1}, then

∑_{k=1}^{r}(−1)^{k}\overline{(QN)}_{ki}\overline{(QN)} ( \begin{matrix}j_{1} & ... & ... & ... & ... & j_{r−1} \\ 1 & ... & k−1 & k+1 & ... & r \end{matrix} ) = 0.

If the set {j_{1},…, j_{r−1}} does not contain i, then i = j_{p} and the system is denoted as j_{1},…, j_{p−1}, j_{p+1},…, j_{r}. Now we get the following representation for

∑_{k=1}^{r}(QN)_{ik}^{∗}(adj(Q(QN)^{∗}))_{kj}:

\begin{aligned}∑_{j1<...<jp−1<jp+1<...<jr} & (−1)^{j}Q ( \begin{matrix}j1 & ... & jp−1 & jp+1 & ... & jr−1 \\ 1 & ... & j−1 & j+1 & ... & r \end{matrix} ) (−1)^{p}\overline{(QN)} ( \begin{matrix}j1 & ... & ... & i=jp & ... & ... & jr \\ 1 & ... & k−1 & k & k+1 & ... & r \end{matrix} ) \\ & = ∑_{j1<...<i<...<jr} \overline{(QN)} ( \begin{matrix}j1 & ... & i & ... & jr \\ 1 & ... & ... & ... & r \end{matrix} ) Q_{ji} ( \begin{matrix}j1 & ... & i & ... & jr \\ 1 & ... & ... & ... & r \end{matrix} ) . \end{aligned}

Similarly, element on ith row and jth column of adj((MP)^{∗}P)(MP)^{∗}) is equal to

∑_{1≤α_{1}<...<α_{r}≤m} \overline{(MP)} ( \begin{matrix}1 & ... & ... & ... & r \\ α_{1} & ... & j & ... & α_{r} \end{matrix} ) P_{jk} ( \begin{matrix}1 & ... & ... & ... & r \\ α_{1} & ... & j & ... & α_{r} \end{matrix} ) .

Now, element lying on the ith row an jth column of weighted adjoint matrix, denoted as adj(A_{M•,•N}^{(†,r)})_{ij} is equal to

∑_{k=1}^{r} [  ∑_{1≤β1<...<βr ≤n} \overline{(QN)} ( \begin{matrix}β1 & ... & i & ... & βr \\ β1 & ... & i & ... & βr \end{matrix} ) Q_{ki} ( \begin{matrix}1 & ... & ... & ... & r \\ β1 & ... & i & ... & βr \end{matrix} )]

× [ ∑_{1≤α1<...<αr ≤m} \overline{(MP)} ( \begin{matrix}α1 & ... & j & ... & αr \\ 1 & ... & ... & ... & r \end{matrix} ) P_{jk} ( \begin{matrix}α1 & ... & j & ... & αr \\ 1 & ... & ... & ... & r \end{matrix} )]

= [ ∑_{\begin{matrix}1α1<...<αr ≤m \\ 1≤β1<...<βr ≤n \end{matrix}} \overline{(MAN)} ( \begin{matrix}α1 & ... & j & ... & αr \\ β1 & ... & i & ... & βr \end{matrix} )] [∑_{k=1}^{r} P_{jk} \binom{1 ... ... ... rα1 ... j ... αr}{1 ... ... ... r} Q_{ki} ( \begin{matrix}1 & ... & ... & ... & r \\ β1 & ... & i & ... & βr \end{matrix} )]

= ∑_{\begin{matrix}1≤α1<...<αr ≤m \\ 1≤β1<...<βr ≤n \end{matrix}} \overline{(MAN)} ( \begin{matrix}α1 & ... & j & ... & αr \\ β1 & ... & i & ... & βr \end{matrix} ) A_{ji} ( \begin{matrix}α1 & ... & j & ... & αr \\ β1 & ... & i & ... & βr \end{matrix} ) .

Theorem 3.3 Element on the ith row and jth column of the weighted Moore-Penrose inverse is equal to

(A_{φ(M,N)}^{(†,r)})_{ij} = \frac{adj(A_{φ(M,N)}^{(†,r)})_{ji}}{∥A∥_{φ(M,N)}^{r}} .

Proof. Follows from A_{M•,•N}^{†} = (QN)^{∗}(Q(QN)^{∗})^{−1} · ((MP)^{∗}P)^{−1}(MP)^{∗} and Corollary 1.1.

Theorem 3.3 is an equivalent of Theorem 2.2.

4. Representation of the weighted Moore-Penrose solution of a system of linear equations

In [5] an explicit determinanatal representation of the Moore-Penrose solution of an arbitrary system of linear equations is derived. Using this representation it is proved that the Moore-Penrose solution is a convex combination of all uniquely solvable partial subsystems. In [4] an equivalent determinantal representation for the least-squares solution of an overdetermined linear system is derived. From this fromula it is proved that the least-squares solution lies in the convex hull of the solutions to the square subsystems of the original system. Also, in [4] this result is extended, and it is proved that this geometric property holds for a more general class of problems which includes the weighted least-squares and l_{p} norm ninimization problems.

In the following theorem we derive determinantal representation of the weighted Moore-Penrose solution of a system of linear equations.

Theorem 4.1 The ith component of the weighted Moore-Penrose solution x_{φ(M,N)}^{†} = A_{φ(M,N)}^{†}z of a linear system Ax = z, A ∈ C_{r}^{m×n}, x ∈ C^{n} , z ∈ C^{m} can be represented in the following determinant representation:

(x_{φ(M,N)}^{†})_{i} = ^{\frac{∑ ; \begin{matrix}1≤q_{1}<...<q_{r}≤n \\ 1≤p_{1}<...<p_{r}≤m \end{matrix} \overline{(ω(M,N))}( \begin{matrix}p_{1} & ... & ... & ... & p_{r} \\ q_{1} & ... & i & ... & q_{r} \end{matrix} )A( \begin{matrix}p_{1} & ... & ... & ... & p_{r} \\ q_{1} & ... & i & ... & q_{r} \end{matrix} )(i→_{p}z)}{∥A∥_{φ(M,N)}^{r}}},

where _{p}z denotes the vector {z_{p_{1}},…, z_{p_{r}}}.

Proof. If φ(M, N) = M•, •N and A = BC is a full-rank factorization of A, then

x_{M•,•N}^{†} = (CN)^{∗}(C(CN)^{∗})^{−1}((MB)^{∗}B)^{−1}(MB)^{∗}z = C_{M•,•N}^{†}B_{M•,•N}^{†}z.

In this manner, the starting system splits up into two equivalent systems. First we calculate y_{M•,•N}^{†} = B_{M•,•N}^{†}z , y ∈ C^{n}. In view of B_{M•,•N}^{†} = ((MB)^{∗}B)^{−1} (MB)^{∗}, we get ((MB)^{∗}B)y_{M•,•N}^{†} = (MB)^{∗}z. The ith component of y_{M•,•N}^{†} is

(y_{M•,•N}^{†})_{i} = \begin{matrix}| & ((MB)^{∗}B)(i → (MB)^{∗}z) & | \\ | & (MB)^{∗}B & | \end{matrix} = \begin{matrix}| & (MB)^{∗} · B(i → z) & | \\ | & (MB)^{∗}B & | \end{matrix} , 1 ≤ i ≤ r.

Applying Cauchy-Binet Theorem, we obtain

(y_{M•,•N}^{†})_{i} = \frac{∑_{1≤p_{1}<...<pr ≤m} \overline{(MB)}( \begin{matrix}p_{1} & ... & p_{r} \\ 1 & ... & r \end{matrix} )B( \begin{matrix}p_{1} & ... & p_{r} \\ 1 & ... & r \end{matrix} )(i→_{p}z)}{| (MB)^{∗}B |} , 1 ≤ i ≤ r.

Also, using x_{M•,•N}^{†} = C_{M•,•N}^{†}y_{M•,•N}^{†} = (CN)^{∗}(C(CN)^{∗})^{−1}y_{M•,•N}^{†}, it is easy to see that

(x_{M•,•N}^{†})_{i} = \frac{1}{| C(CN)^{∗} |} · ( ∑_{k=1}^{r}((CN)^{∗} adj(C(CN)^{∗}))_{ik}(y_{M•,•N}^{†})_{k}) .

Element on the ith row and jth column of the matrix (CN)^{∗} adj(C(CN)^{∗}) is (see Theorem 3.2.):

((CN)^{∗} adj(C(CN)^{∗}))_{ij} = ∑_{1≤q_{1}<...<q_{r}≤n} \overline{(CN)} ( \begin{matrix}1 & ... & ... & ... & r \\ q_{1} & ... & i & ... & q_{r} \end{matrix} ) C_{ji} ( \begin{matrix}1 & ... & ... & ... & r \\ q_{1} & ... & i & ... & q_{r} \end{matrix} ) .

Now (x_{M•,•N}^{†})_{i} is equal to

\frac{∑_{k=1}^{r} ∑_{q1<...<qr} \overline{(CN)} ( \begin{matrix}1 & ... & ... & ... & r \\ q1 & ... & i & ... & qr \end{matrix} ) C_{ki} ( \begin{matrix}1 & ... & ... & ... & r \\ q1 & ... & i & ... & qr \end{matrix} ) ∑_{p1<...<pr} \overline{(MB)} ( \begin{matrix}p1 & ... & pr \\ 1 & ... & r \end{matrix} ) B ( \begin{matrix}p1 & ... & pr \\ 1 & ... & r \end{matrix} ) (k → _{p}z)}{| C(CN)^{∗} | | (MB)^{∗}B |} ; = \frac{∑_{\begin{matrix}1≤q1<...<qr ≤n \\ 1≤p1<...<pr ≤m \end{matrix}} \overline{(MAN)} ( \begin{matrix}p1 & ... & ... & ... & pr \\ q1 & ... & i & ... & qr \end{matrix} ) [ ∑_{k=1}^{r} C_{ki} ( \begin{matrix}1 & ... & ... & ... & r \\ q1 & ... & i & ... & qr \end{matrix} ) B ( \begin{matrix}p1 & ... & pr \\ 1 & ... & r \end{matrix} ) (k → _{p}z)]}{∥A∥_{M•,•N}^{r}} .

By using Laplace’s development on the kth column of the square matrix

B ( \begin{matrix}p_{1} & ... & p_{r} \\ 1 & ... & r \end{matrix} ) (k → _{p}z), we get

(x_{M•,•N}^{†})_{i} =

\frac{∑_{\begin{matrix}1≤q_{1}<...<q_{r}≤n \\ 1≤p_{1}<...<p_{r}≤m \end{matrix}} \overline{(MAN)} ( \begin{matrix}p_{1} & ... & ... & ... & p_{r} \\ q_{1} & ... & i & ... & q_{r} \end{matrix} ) [∑_{l=1}^{r} z_{p_{l}} ∑_{k=1}^{r} C_{ki} ( \begin{matrix}1 & ... & ... & ... & r \\ q_{1} & ... & i & ... & q_{r} \end{matrix} ) B_{p_{l} k} ( \begin{matrix}p_{1} & ... & p_{r} \\ 1 & ... & r \end{matrix})]}{∥A∥_{M•,•N}^{r}}

= \frac{∑_{\begin{matrix}1≤q_{1}<...<q_{r}≤n \\ 1≤p_{1}<...<p_{r}≤m \end{matrix}} \overline{(MAN)} ( \begin{matrix}p_{1} & ... & ... & ... & p_{r} \\ q_{1} & ... & i & ... & q_{r} \end{matrix} ) ∑_{l=1}^{r} z_{p_{l}} A_{p_{l} i} ( \begin{matrix}p_{1} & ... & ... & ... & p_{r} \\ q_{1} & ... & i & ... & q_{r} \end{matrix} )}{∥A∥_{M•,•N}^{r}}

= \frac{∑_{\begin{matrix}1≤q_{1}<...<q_{r}≤n \\ 1≤p_{1}<...<p_{r}≤m \end{matrix}} \overline{(MAN)} ( \begin{matrix}p_{1} & ... & ... & ... & p_{r} \\ q_{1} & ... & i & ... & q_{r} \end{matrix} ) A ( \begin{matrix}p_{1} & ... & p_{r} \\ q_{1} & ... & q_{r} \end{matrix} ) (i → _{p}z)}{∥A∥_{M•,•N}^{r}}.

As we mentioned above, in [4] it is showed that the weighted least-squares solution lies in the convex hull of the solutions to the square subsystems of the original system. But, this result includes positive definite diagonal weighted matrices. In the following theorem we generalize this result and prove that arbitrary weighted Moore-Penrose solution of a linear system lies in the convex hull of the solutions to the square subsystems of the original system.

Theorem 4.2 The weighted Moore-Penrose solution x_{M•,•N}^{†} of system of linear equations Ax = z is the convex combination

x_{M•,•N}^{†} = ∑_{\begin{matrix}1≤q_{1}<...<q_{r}≤n \\ 1≤p_{1}<...<p_{r}≤m \end{matrix}} β_{p}γ_{q}x^{(p,q)}

of the solutions of all uniquely solvable r-dimensional subsytems canonically imbedded into C^{m}, where

\begin{aligned}β & = ∑_{1≤α_{1}<...<α_{r}≤m} \overline{(MB)} ( \begin{matrix}α_{1} & ... & α_{r} \\ 1 & ... & r \end{matrix}) B ( \begin{matrix}α_{1} & ... & α_{r} \\ 1 & ... & r \end{matrix}) ; \\ γ & = ∑_{1≤β_{1}<...<β_{r}≤n} \overline{(CN)} ( \begin{matrix}1 & ... & r \\ β_{1} & ... & β_{r} \end{matrix} ) B ( \begin{matrix}1 & ... & r \\ β_{1} & ... & β_{r} \end{matrix} ); \\ β_{p} & = \frac{1}{β} \overline{(MB)} ( \begin{matrix}p_{1} & ... & p_{r} \\ 1 & ... & r \end{matrix}) B ( \begin{matrix}p_{1} & ... & p_{r} \\ 1 & ... & r \end{matrix}) ; \\ γ_{q} & = \frac{1}{γ} \overline{(CN)} ( \begin{matrix}1 & ... & r \\ q_{1} & ... & q_{r} \end{matrix} ) C ( \begin{matrix}1 & ... & r \\ q_{1} & ... & q_{r} \end{matrix} ) . \end{aligned}

Proof. According to Theorem 4.1. (x_{M•,•N}^{†})_{i} has the following determinantal representation

\frac{∑_{\begin{matrix}1≤q1<...<qr ≤n \\ 1≤p1<...<pr ≤m \end{matrix}} \overline{(MB)} ( \begin{matrix}p1 & ... & pr \\ 1 & ... & r \end{matrix} ) \overline{(CN)} ( \begin{matrix}1 & ... & ... & ... & r \\ q1 & ... & i & ... & qr \end{matrix} ) A ( \begin{matrix}p1 & ... & pr \\ q1 & ... & qr \end{matrix} ) (i → _{p}z)}{[  ∑_{1≤α1<...<αr ≤m} P ( \begin{matrix}α1 & ... & αr \\ 1 & ... & r \end{matrix} ) \overline{(MP)} ( \begin{matrix}α1 & ... & αr \\ 1 & ... & r \end{matrix} )] [  ∑_{1≤β1<...<βr ≤n} Q ( \begin{matrix}1 & ... & r \\ β1 & ... & βr \end{matrix} ) \overline{(QN)} ( \begin{matrix}1 & ... & r \\ β1 & ... & βr \end{matrix} )]}

= ∑_{\begin{matrix}q1<...<qr \\ p1<...<pr \end{matrix}} \frac{1}{β} \overline{(MB)} ( \begin{matrix}p1 & ... & pr \\ 1 & ... & r \end{matrix} ) B ( \begin{matrix}p1 & ... & pr \\ 1 & ... & r \end{matrix} ) \frac{1}{γ} \overline{(CN)} ( \begin{matrix}1 & ... & ... & ... & r \\ q1 & ... & i & ... & qr \end{matrix} ) C ( \begin{matrix}1 & ... & ... & ... & r \\ q1 & ... & i & ... & qr \end{matrix} ) ×

× \frac{A ( \begin{matrix}p1 & ... & pr \\ q1 & ... & qr \end{matrix} ) (i → _{p}z)}{A ( \begin{matrix}p1 & ... & pr \\ q1 & ... & qr \end{matrix} )} .

In the case A ( \begin{matrix}p_{1} & ... & p_{r} \\ q_{1} & ... & q_{r} \end{matrix} ) ≠ 0 let x^{(p,q)} be the canonical imbedding of the solution

of A [ \begin{matrix}p_{1} & ... & p_{r} \\ q_{1} & ... & q_{r} \end{matrix} ] x = _{p}z into the m-dimensional space. This means that, according to

Cramer’s rule, x^{(p,q)} possesses the components

x_{i}^{(p,q)} = \frac{A( \begin{matrix}p_{1} & ... & p_{r} \\ q_{1} & ... & q_{r} \end{matrix} )(i→_{p}z)}{A( \begin{matrix}p_{1} & ... & p_{r} \\ q_{1} & ... & q_{r} \end{matrix} )}

for i contained in combination 1 ≤ q_{1} < … < q_{r} ≤ n, and x_{i}^{(p,q)} = 0 otherwise. In the singular case A ( \begin{matrix}p_{1} & ... & p_{r} \\ q_{1} & ... & q_{r} \end{matrix} ) = 0 we define x^{(p,q)} to be the zero vector. Now it is evident that

x_{M•,•N}^{†} = ∑_{\begin{matrix}1≤q_{1}<...<q_{r}≤n \\ 1≤p_{1}<...<p_{r}≤m \end{matrix}} β_{p}γ_{q}x^{(p,q)}.

Since

∑_{1≤p_{1}<...<p_{r}≤m} β_{p} = 1,  ∑_{1≤q_{1}<...<q_{r}≤n} γ_{q} = 1

the proof is completed.

REFERENCES

[1] Arghiriade, E. et Dragomir, A., Une nouvelle definition de l’inverse generalisee d’une matrice, Lincei - Rend. Sc. fis. mat. e nat. XXXV (1963), 158–165

[2] Ben-Israel, A. and Grevile, T.N.E., Generalized inverses: Theory and applications, Wiley-Interscience, New York 1974

[3] Ben-Israel, A., Generalized inverses of matrices: a perspective of the work of Penrose, Math. Proc. Camb. Phil. Soc. 100 (1986), 407–425

[4] Ben-Tal, A., A Geometric Property of the Least Squares Solution of Linear Equations, Linear Algebra and Applications 139 (1990), 165–170

[5] Berg, L., Three results in connection with inverse matrices, Linear algebra and applications 84 (1986), 63–77

[6] Duane Pyle, L., The Weighted Generalized Inverse in Nonlinear Programming-Active Set Selection Using a Variable-metric Generalization of the Simplex Algorithm, International simposium on extremal methods and systems analysis, Lecture Notes in Economics and Mathematical Systems 174 (1977), 197–231

[7] Gabriel, R., Extinderea Complementilor Algebrici Generalizati la Matrici Oarecare, Studii si cercetari matematice 17, 10 (1965), 1566–1581

[8] Gabriel, R., Das verallgemeinerte Inverse einer Matrix, deren Elemente einem beliebigen Körper angehören, J. Reine angew Math. 234 (1967), 107–122

[9] Gabriel, R., Das verallgemeinerte Inverse einer Matrix, über einem beliebigen Körper - analytish betrachtet, J. Reine angew Math. 244(V) (1970), 83–93

[10] ���������� ����, ������ ������, ������� ����� ����

[11] Horn, R.A. and Johnson, C.R., Matrix Analysis, Cambridge University press, Cambridge, New York, Melbourne, Sydney 1985

[12] S. Kumar Mitra and C. Radhakrishna Rao, Extensions of a Duality Theorem Concerning g-inverses of Matrices, The Indian Journal of Statistics 37 (1975), 439–445

[13] Lancaster, P and Tismenetsy, M., The Theory of Matrices, Academic Press 1985

[14] Moore, E.H., General Analysis, Part I. The Algebra of Matrices, (compiled and edited by R.W. Barnard), The Amer. Philos. Soc. 1935

[15] Moore, E.H., On the reciprocal of the general algebraic matrix (Abstract), Bull. Amer. Math. Soc. 26 (1920), 394–395

[16] Penrose, R., A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51 (1955), 406–413

[17] Radhakrishna Rao, C. and Kumar Mitra, S., Generalized inverse of matrices and its applications, John Wiley & Sons, Inc, New York, London, Sydney, Toronto 1971

[18] Radić, M., Some contributions to the inversion of rectangular matrices, Glasnik matematički 1(21), 1 (1966), 23–37

(received 14.06.1994.)

University of Niš, Faculty of Philosophy, Departement of Mathematics, Ćirila i Metodija 2, 18000 Niš, Yugoslavia