����������� ������ 48 (1996), 1–9

UDK 512.643

���������� ������ ��� research paper

GENERAL DETERMINANTAL REPRESENTATION OF PSEUDOINVERSES OF MATRICES

Predrag Stanimirović

Abstract. In this paper we establish general determinantal representation of generalized inverses and general form of different definitions of rectangular determinants and induced general inverses, in terms of minors of a matrix, satisfying certain conditions. Using this representation we obtain a general algorithm for exact computation of different classes of pseudoinverses: Moore-Penrose and weighted Moore-Penrose inverse, group inverse, {1, 2, 3}, {1, 2, 4}, {1, 2} inverses, left/right inverses, Radić’s and Stojaković’s generalized inverses.

We also give some examples which illustrate our results.

1. Introduction

Let C_{r}^{m×n} be the set of m × n complex/rational matrices whose rank is r. Conjugate, transpose and conjugate-transpose matrix of A will be denoted by \overline{A}, A^{T} and A^{∗} respectively. Minor of A containing rows α_{1}, …, α_{t} and colums β_{1}, …, β_{t} will be denoted by A ( \begin{matrix}α_{1} & … & α_{t} \\ β_{1} & … & β_{t} \end{matrix} ), while its algebraic complement

corresponding to the element a_{ij} is defined by 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)^{p+q}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 matrix A ∈ C^{m×n} consider the following equations in X:

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

and if m = n also

(5) AX = XA.

For a subset S od {1, 2, 3, 4, 5} the set of matrices G obeying the conditions represented in S will be denoted by A(S). A matrix G ∈ A(S) is called an S-inverse of A and 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^{†} [13]. In the case m = n, the group inverse, denoted as A^{♯}, of A is the unique {1, 2, 5} inverse, and exists if and only if ind(A) = min{k : k > 0 and rank(A^{k+1}) = rank(A^{k})} = 1.

AMS Subject Classification: 15A09, 65F05

The starting point of the investigations of this paper is the determinantal representation of Moore-Penrose inverse, studied in [1], [4], [6], [7], [8], [11]. The main result of these papers is:

Theorem 1.1 Element a_{ij}^{†} in the i-row and j-column of the Moore-Penrose pseudoinverse of a given matrix A ∈ C_{r}^{m×n} is given by

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

For the sake of completeness in the following definition we unify definitions of generalized inverses introduced by M. Radić [16], [17], M. Stojaković [20] and V.N.Joshi [9].

Definition 1.1 Let i, j be integers, 1 ⩽ i ⩽ n, 1 ⩽ j ⩽ m. Then the (i, j)-th entry of Radić’s, Stojaković’s and Joshi’s generalized inverse A ∈ C_{r}^{m×n} is defined by

\frac{∑_{\begin{matrix}1⩽j_{1}<···<j<···<j_{r}⩽n \\ 1⩽i_{1}<···<i<···<i_{r}⩽m \end{matrix}} ε^{(i_{1}+···+i_{r})+(j_{1}+···+j_{r})}A_{ji} ( \begin{matrix}i_{1} & ... & j & ... & i_{r} \\ j_{1} & ... & i & ... & j_{r} \end{matrix} )}{∑_{\begin{matrix}1⩽α_{1}<···<α_{r}⩽m \\ 1⩽β_{1}<···<β_{r}⩽n \end{matrix}} ε^{(α_{1}+···+α_{r})+(β_{1}+···+β_{r})}A ( \begin{matrix}α_{1} & ... & α_{r} \\ β_{1} & ... & β_{r} \end{matrix} )} , where ε ∈ \{−1, 1\}.

For ε = 1 we get Stojaković’s definition, and for ε = −1, we get Radić’s definition.

The following theorem [15] has described an useful representation of {i, j, k} generalized inverses:

Theorem 1.2 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:

A^{†} = Q^{†}P^{†} = Q^{∗}(QQ^{∗})^{−1}(P^{∗}P)^{−1}P^{∗};

the generalized solution of the equations (1), (2) is given by W_{1}(QW_{1})^{−1}(W_{2}P)^{−1}W_{2};

the generalized solution of the equations (1), (2), (3) is given by W_{1}(QW_{1})^{−1}(P^{∗}P)^{−1}P^{∗};

the generalized solution of the equations (1), (2), (4) is given by Q^{∗}(QQ^{∗})^{−1}(W_{2}P)^{−1}W_{2}.

In [19] we developed determinantal representation of class of {1, 2} inverses and the weighted Moore-Penrose inverse, as follows in the following two theorems:

Theorem 1.3 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(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} )}.

Theorem 1.4 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 i-th row and j-th 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} )}

Now we describe the main results of the paper. First we obtain determinantal representation of {i, j, k} generalized inverses and the group inverse. Then we show that determinantal representation of the Moore-Penrose and the weighted Moore-Penrose inverse, {1, 2}, {i, j, k} inverses are only partial cases of the common general determinantal inverses, generalization of the Arghiriade-Dragomir representation of the Moore-Penrose inverse [18], possesses the same general form. We also show that the class of left/right generalized inverses can be obtained from this general determinantal representation.

2. General determinantal representation

Applying method used in [19, Theorem 2.1] and using general form of {i, j, k} inverses (Theorem 1.2), the following lemma can be proved.

Lemma 2.1. If A ∈ C_{r}^{m×n} and A = P Q is a full rank factorization of A, and if W_{1} ∈ C^{m×r} and W_{2} ∈ C^{r×n} are some matrices such that rank(QW_{1}) = rank(W_{2}P) = rank(A), then a_{ij}^{(1,2,3)} ∈ A^{(1,2,3)} and a_{ij}^{(1,2,4)} ∈ A^{(1,2,4)} can be represented in this way:

a_{ij}^{(1,2,3)} = \frac{∑_{\begin{matrix}1⩽β_{1}<···<β_{r}⩽n \\ 1⩽α_{1}<···<α_{r}⩽m \end{matrix}}(W_{1}P^{∗})^{T} ( \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} ) (W_{1}P^{∗})^{T} ( \begin{matrix}γ_{1} & ... & γ_{r} \\ δ_{1} & ... & δ_{r} \end{matrix} )}

a_{ij}^{(1,2,4)} = \frac{∑_{\begin{matrix}1⩽β_{1}<···<β_{r}⩽n \\ 1⩽α_{1}<···<α_{r}⩽m \end{matrix}}(Q^{∗}W_{2})^{T} ( \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} ) (Q^{∗}W_{2})^{T} ( \begin{matrix}γ_{1} & ... & γ_{r} \\ δ_{1} & ... & δ_{r} \end{matrix} )}.

Note that conditions rank(QW_{1}) = rank(W_{2}P) = rank(A) are satisfied if and only if rank(W_{1}) = rank(W_{2}) = rank(A).

Determinantal representation of the group inverse of a singular n by n matrix can be obtained from [2] and [10]:

Lemma 2.2. The group inverse A^{♯} = (a_{ij}^{♯}) of A ∈ C_{r}^{n×n} exists if

u = ∑_{1⩽μ_{1}<···<μ_{r}⩽n} A ( \begin{matrix}μ_{1} & ... & μ_{r} \\ μ_{1} & ... & μ_{r} \end{matrix} ) ≠ 0.

Then A^{♯} has the following determinantal representation:

a_{ij}^{♯} = \frac{∑_{\begin{matrix}1⩽α_{1}<···<α_{r}⩽n \\ 1⩽β_{1}<···<β_{r}⩽n \end{matrix}} A^{T} ( \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⩽γ_{1}<···<γ_{r}⩽n \\ 1⩽δ_{1}<···<δ_{r}⩽n \end{matrix}} A^{T} ( \begin{matrix}γ_{1} & ... & γ_{r} \\ δ_{1} & ... & δ_{r} \end{matrix} ) A ( \begin{matrix}γ_{1} & ... & γ_{r} \\ δ_{1} & ... & δ_{r} \end{matrix} )}.

Proof. In view of the supposition u ≠ 0 from [10, Theorem 8] we get the following determinantal representation for the (i, j)-th entry of A^{♯}:

a_{ij}^{♯} = \frac{∑_{\begin{matrix}1⩽α_{1}<···<α_{r}⩽n \\ 1⩽β_{1}<···<β_{r}⩽n \end{matrix}} A ( \begin{matrix}β_{1} & ... & i & ... & β_{r} \\ α_{1} & ... & j & ... & α_{r} \end{matrix} ) A_{ji} ( \begin{matrix}α_{1} & ... & j & ... & α_{r} \\ β_{1} & ... & i & ... & β_{r} \end{matrix} )}{u^{2}}.

Now the proof of the lemma can be completed using the following relation [10, Theorem 8]

u^{−2} · A ( \begin{matrix}α_{1} & ... & α_{r} \\ β_{1} & ... & β_{r} \end{matrix} ) A ( \begin{matrix}β_{1} & ... & β_{r} \\ α_{1} & ... & α_{r} \end{matrix} ) = 1

and the following, evident relation: A ( \begin{matrix}α_{1} & ... & α_{r} \\ β_{1} & ... & β_{r} \end{matrix} ) = A^{T} ( \begin{matrix}β_{1} & ... & β_{r} \\ α_{1} & ... & α_{r} \end{matrix} ).

Now we introduce a general determinantal representation which include all of presented determinantal representations, generlaized inverses introduced by M. Stojaković, M. Radić and V.N. Joshi and the well known concept of the inversion of square regular matrices.

Theorem 2.1. Let A ∈ C_{r}^{m×n} and matrix R satisfies condition

(U_{1}) \{ \begin{matrix}rank(AR^{∗}) = rank(A), & if m ⩽ n \\ rank(R^{∗}A) = rank(A), & if n ⩽ m. \end{matrix}

An arbitrary (i, j)-th entry of all the above mentioned generalized inverses has the following determinantal representation:

g_{ij} = \frac{∑_{\begin{matrix}1⩽β_{1}<···<β_{t}⩽n \\ 1⩽α_{1}<···<α_{t}⩽m \end{matrix}} \overline{R} ( \begin{matrix}α_{1} & ... & j & ... & α_{t} \\ β_{1} & ... & i & ... & β_{t} \end{matrix} ) A_{ji} ( \begin{matrix}α_{1} & ... & j & ... & α_{t} \\ β_{1} & ... & i & ... & β_{t} \end{matrix} )}{∑_{\begin{matrix}1⩽δ_{1}<···<δ_{t}⩽n \\ 1⩽γ_{1}<···<γ_{t}⩽m \end{matrix}} A ( \begin{matrix}γ_{1} & ... & γ_{t} \\ δ_{1} & ... & δ_{t} \end{matrix} ) \overline{R} ( \begin{matrix}γ_{1} & ... & γ_{t} \\ δ_{1} & ... & δ_{t} \end{matrix} )}.  (2.1)

The numerator in (2.1) we simly denote by A_{ij}^{(R,t)} and denominator by DET_{(R,T)}(A). The expression A_{ij}^{(R,t)} will be called generalized albebraic complement corresponding to element a_{ji}, and DET_{(R,T)}(A) will be called generalized determinant.

Notation r_{c}(A) = t ⩽ r ⩽ min{m, n} denotes the greates integer which ensures DET_{(R,t)}(A) ≠ 0.

Proof. Consider the following cases:

1. Suppose that t = m ⩽ n. Using Laplace’s development for the square minors of A, we get

DET_{(R,m)}(A) = det(AR^{∗}) = ∑_{j_{1}<···<j_{m}} \overline{R} ( \begin{matrix}1 & ... & m \\ j_{1} & ... & j_{m} \end{matrix} ) [∑_{k=1}^{r} a_{ij_{k}}A_{ij_{k}} ( \begin{matrix}1 & ... & m \\ j_{1} & ... & j_{m} \end{matrix} )]

= ∑_{l=1}^{n} a_{il}[ ∑_{j_{1}<···<j_{m}} \overline{R} ( \begin{matrix}1 & ... & ... & ... & m \\ j_{1} & ... & l & ... & j_{m} \end{matrix} ) A_{il} ( \begin{matrix}1 & ... & ... & ... & m \\ j_{1} & ... & l & ... & j_{m} \end{matrix} )] = ∑_{l=1}^{n} a_{il}A_{li}^{(R,m)}.

For two integers p ≠ q, 1 ⩽ p, q ⩽ m, substituting in the minors of A in the expression

DET_{(R,m)}(A) = ∑_{j_{1}<···<j_{m}} \overline{R} ( \begin{matrix}1 & ... & m \\ j_{1} & ... & j_{m} \end{matrix} ) A ( \begin{matrix}1 & ... & m \\ j_{1} & ... & j_{m} \end{matrix} )

the q-row by the p-row, starting from DET_{(R,m)}(A) = 0, in a similar way it can be

proved ∑_{l=1}^{n} a_{pl}A_{lq}^{(R,m)} = 0.

In this way, g_{ij} = δ_{ij} DET_{(R,m)}(A), and consequently A · A_{(R,m)}^{−1} = I_{m}, which means that A_{(R,m)}^{−1} represents a right inverse of a full-rank matrix A.

In a similar way it can be proved that A_{(R,n)}^{−1} represents a left inverse in the case t = n ⩽ m. Now it is evident that (2.1) represents general determinantal representation of right/left inverses of a full rank matrix A.

2. For R = A we obtain determinantal representation of A^{†}, presented in Theorem 1.3. Then it is trivial to conclude r_{c}(A) = r, which represents a known result in [8].

3. If m = n, ind(A) = 1 and R = A^{∗}, the determinantal representation of the group inverse is obtained (Lemma 2.2).

4. If r = r_{c}(A) and the matrix R satisfies condition

(U_{2})  \overline{R} ( \begin{matrix}i_{1} & ... & i_{r} \\ j_{1} & ... & j_{r} \end{matrix} ) = K · ε^{(i_{1}+···+i_{r})+(j_{1}+···+j_{r})}, where K ∈ C and ε ∈ \{−1, 1\}, for all combinations 1 ⩽ i_{1} < ⋯ < i_{r} ⩽ m; 1 ⩽ j_{1} < ⋯ < j_{r} ⩽ n,

then, in the case ε = 1, A_{(R,r)}^{−1} is equal to Stojaković’s inverse and the Radić’s inverse in the case ε = −1 (Definition 1.1)

5. If A = P Q is a full-rank factorization of A ∈ C_{r}^{m×n} and if 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) = r, then:

– for R = P W_{1}^{∗} we obtain determinantal representation of A{1, 2, 3} (Lemma 2.1);

– for R = W_{2}^{∗}Q we obtain determinantal representation of A{1, 2, 4} (Lemma 2.1);

– for R = (W_{1}W_{2})^{∗} we obtain determinantal representation of A{1, 2} (Lemma 1.3).

6. For R = MAN, where M and N are positive definite matrices of the appropriate sizes, A_{(R,r)}^{−1} reduces to the weighted Moore-Penrose inverse (Theorem 1.4).

7. For a regular matrix A formula (2.1) is transformed into the well known inversion of regular square matrices, for an arbitrary n by n matrix R.

8. If A is a full-rank matrix and each minor of R is the degree of the corresponding minor of A with the exponent \frac{1}{2k−1}, for some positive integer k, we obtain Radić’s generalization of the Arghiriade-Dragomir representation of the Moore-Penrose inverse [18].

3. Examples

If a matrix R runs over the set of m by n matrices, in (2.1) we get various definitions of generalized inverses.

1. If r = r_{c}(A) and a matrix R satisfies condiiton (U_{1}), then A_{(R,r)}^{−1} is equal to Stojaković’s inverse, i.e. equivalent Joshi’s inverse, in the case ε = 1 and the Radić’s inverse, in the case ε = −1.

For example, consider matrix A = ( \frac{11}{2} \frac{22}{15} 1_{\frac{3}{20} \frac{−2}{7} \frac{234}{233}} ). Using R = ( \begin{matrix}2 & 0 & −2 \\ 1 & 1 & 0 \end{matrix} )

we get the following Stojaković’s inverse of A: A_{(R,2)}^{−1} = ( \begin{matrix}\frac{58600}{440191} & \frac{619780}{1320573} \\ \frac{139335}{880382} & \frac{366975}{440191} \\ \frac{22135}{880382} & \frac{1720705}{1320573} \end{matrix} ). Now,

using fixed point representation for the elements in A and the same matrix R we get

A = ( \begin{matrix}5.50000000000000000000 & 1.53333333333333344000 & 1.00000000000000000000 \\ 0.14999999999999999400 & −0.28571428571428569800 & 1.00429184549356232000 \end{matrix} ) ,

General determinantal representation of pseudoinverses of matrices 7

and the following Stojaković’s inverse of A:

A_{(R,2)}^{−1} = ( \begin{matrix}0.13312403025050492700 & −0.46932657263172883300 \\ 0.15826652521291895100 & 0.83367220138530784300 \\ 0.02514249496241404220 & 1.30299877401703657000 \end{matrix} ) .

2. Furthermore, if R = A satisfies (U_{1}), then A_{(R,r)}^{−1} = A^{†}, and both generalized inverses are identical to the Stojaković’s or Radić’s generalized inverse.

Concretely for R = A = ( \begin{matrix}\frac{5729}{327} & \frac{5729}{327} & 0 \\ 0 & \frac{5729}{327} & \frac{5729}{327} \\ −\frac{5729}{327} & 0 & −\frac{5729}{327} \end{matrix} ) we get the following Moore-

Penrose inverse of A:

A_{(R,2)}^{−1} = A^{†} = ( \begin{matrix}\frac{2008044837}{256295929} & 0 & −\frac{2008044837}{256295929} \\ \frac{2008044837}{256295929} & \frac{2008044837}{256295929} & 0 \\ 0 & \frac{2008044837}{256295929} & \frac{2008044837}{256295929} \end{matrix} ) ,

which is identical to the Stojaković’s generalized inverse.

3. If A ∈ C_{r}^{m×n} and R = A we get A_{(R,r)}^{−1} = A^{†}. For example, if we use

R = A = ( \begin{matrix}\frac{175}{23} & 0 & \frac{175}{23} \\ 0 & \frac{1}{13} & \frac{175}{23} \\ \frac{175}{23} & \frac{1}{13} & \frac{525}{46} \\ 0 & \frac{1}{13} & \frac{175}{23} \end{matrix} ), then we obtain

A_{(R,2)}^{−1} = A^{†} = ( \begin{matrix}\frac{192878339}{497627891} & \frac{−201395239}{995255782} & \frac{−4258450}{497627891} & \frac{−201395239}{995255782} \\ \frac{1684865000}{497627891} & \frac{−1263648750}{497627891} & \frac{−421216250}{497627891} & \frac{−2263648750}{497627891} \\ \frac{−655721205}{497627891} & \frac{−1075979571}{995255782} & \frac{1281633260}{995255782} & \frac{−1075979571}{995255782} \end{matrix} ) .

4. For a square matrix A, such that ind(A) ⩽ 1 and R = A^{∗} we get A_{(R,m)}^{−1} =

A^{♯}. For example, let A = ( \begin{matrix}21.93−3i & 4 & \frac{275}{35917} & 9.13570+2950.844725i \\ 11.35 & 35.75−2i & 0 & 1257420 \\ 257384 & \frac{91584}{23} & 12+15i & \frac{213574}{5762403} \\ 159384−135i & 109825.23 & \frac{183294}{7359} & 0.000579 \end{matrix} ), where

i = \sqrt{−1}. Using R = A^{∗}, we get A^{♯} = A^{†} = A^{−1} =

( \begin{matrix}−0.005992−0.010995i & −0.000026+0.000014i & 0.000004+0.000001i & 0 \\ −0.029221+0.010681i & 0.000025+0.000068i & −0.000004+0.000001i & 0.000010 \\ 167.244553−23.230652i & 0.053301−0.392650i & −0.0109+0.000438i & −0.005703−0.000733i \\ 0.000001 & 0.000001 & 0 & 0 \end{matrix} ) .

5. Consider the following matrix of rank 2: A = ( \begin{matrix}2 & 0 & −5 & 4 \\ 7 & −4 & −9 & \frac{3}{2} \\ 3 & −4 & 1 & − \frac{13}{2} \end{matrix} ). Matrices P = ( \begin{matrix}2 & 0 \\ 7 & −4 \\ 3 & −4 \end{matrix} ) and Q = ( \begin{matrix}1 & 0 & − \frac{5}{2} & 2 \\ 0 & 1 & − \frac{17}{8} & \frac{25}{8} \end{matrix} ) form a full-rank factorization of A. If we use

matrix W_{1} = ( \begin{matrix}0 & 11 \\ −2 & 2 \\ 4 & 3 \\ 2 & 7 \end{matrix} ) satisfying rank(QW_{1}) = rank(A) and R = P W_{1}^{T}, then the

8 P. Stanimirović

following {1, 2, 3} inverse of A can be obtained:

A_{(R,2)}^{−1} = ( \begin{matrix}\frac{143}{735} & −\frac{11}{147} & −\frac{341}{735} \\ −\frac{3}{245} & \frac{20}{245} & \frac{26}{245} \\ \frac{109}{735} & −\frac{31}{147} & −\frac{373}{735} \\ \frac{6}{35} & −\frac{1}{7} & −\frac{17}{35} \end{matrix} ) .

6. In this example we show necessity of the condition rank(QW_{1}) = rank(A). Given matrix A = ( \begin{matrix}2 & 0 & −5 & 4 \\ 7 & −4 & −9 & \frac{3}{2} \\ 3 & −4 & 7 & − \frac{13}{2} \end{matrix} ) of rank 3, full-rank factorization of A can be

obtained from matrices P = ( \begin{matrix}2 & 0 & −5 \\ 7 & −4 & −9 \\ 3 & −4 & 7 \end{matrix} ) and Q = ( \begin{matrix}1 & 0 & 0 & \frac{21}{8} \\ 0 & 1 & 0 & \frac{117}{32} \\ 0 & 0 & 1 & \frac{1}{4} \end{matrix} ). Now we use the

matrix W_{1} = ( \begin{matrix}0 & 11 & 0 \\ −2 & 2 & 4 \\ 4 & 3 & −8 \\ 2 & 7 & −4 \end{matrix} ) such that rank(QW_{1}) = 2. For R = P W_{1}^{T} we get:

X = A_{(R,2)}^{−1} = ( \begin{matrix}\frac{17}{5717} & \frac{990}{5717} & \frac{957}{5717} \\ \frac{129}{5717} & \frac{176}{5717} & −\frac{211}{5717} \\ −\frac{251}{5717} & \frac{278}{5717} & \frac{1031}{5717} \\ −\frac{120}{5717} & \frac{634}{5717} & \frac{994}{5717} \end{matrix} ) ,

and it is trivial to verify that the equations (1), (2) and (3) are not satisfied.

7. For A ∈ C^{m×n} using R = ( \begin{matrix}1 & ... & 0 & 0 & ... & 0 \\ ... & ... & ... & ... & ... & ... \\ 0 & ... & 1 & 0 & ... & 0 \\ 0 & ... & 0 & 0 & ... & 0 \\ ... & ... & ... & ... & ... & ... \\ 0 & ... & 0 & 0 & ... & 0 \end{matrix} ) = ( \begin{matrix}I_{r} & 0 \\ 0 & 0 \end{matrix} ), R ∈ C^{m×n}, we

obtain DET_{(R,r)}(A) = A ( \begin{matrix}1 & ... & r \\ 1 & ... & r \end{matrix} ) and the following algebraic complement of the element a_{ij}

A_{ij}^{(R,r)} = \{ \begin{matrix}0, & for j > r or i > r \\ A_{ji} ( \begin{matrix}1 & ... & i & ... & r \\ 1 & ... & j & ... & r \end{matrix} ) , & for j, i ⩽ r. \end{matrix}

Generalized inverse of A ∈ C^{m×n}, is equal to

A_{(R,r)}^{−1} = \frac{1}{A ( \begin{matrix}1 & ... & r \\ 1 & ... & r \end{matrix} )} ( \begin{matrix}A_{11} ( \begin{matrix}1 & ... & r \\ 1 & ... & r \end{matrix} ) & … & A_{r1} ( \begin{matrix}1 & ... & r \\ 1 & ... & r \end{matrix} ) & 0 & … & 0 \\ … & … & … & … & … & … \\ A_{1r} ( \begin{matrix}1 & ... & r \\ 1 & ... & r \end{matrix} ) & … & A_{rr} ( \begin{matrix}1 & ... & r \\ 1 & ... & r \end{matrix} ) & 0 & … & 0 \\ 0 & … & 0 & 0 & … & 0 \\ … & … & … & … & … & … \\ 0 & … & 0 & 0 & … & 0 \end{matrix} ) .

Concretely, for A = ( \begin{matrix}\frac{13}{56} & 115 & \frac{476}{13} \\ \frac{1}{3} & −372 & \frac{23}{26} \\ −3 & \frac{14}{3} & \frac{21}{17} \\ \frac{12}{13} & 1 & 0 \end{matrix} ) and R = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) the following right

generalized inverse of A can be obtained:

A_{(R,2)}^{−1} = ( \begin{matrix}\frac{10652600}{6188751} & \frac{−8558144}{80453763} & \frac{−1364947612}{26817921} & 0 \\ \frac{35980}{2062917} & \frac{−3615752}{8939307} & \frac{−7448669}{160907526} & 0 \\ \frac{76388480}{18566253} & \frac{7857808}{6188751} & \frac{−1097248}{6188751} & 0 \end{matrix} ) .

Problem. Find determinantal representation of the Drazin inverse, i.e. determine the value of the matrix R corresponding to the Drazin inverse.

REFERENCES

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[18] Radić, M., On a generalization of the Arghiriade-Dragomir representation of the Moore-Penrose inverse, Lincei-Rend. Sc. fis. mat. e nat. 44 (1968), 333–336

[19] Stanimirović, P. and Stanković, M., Determinantal representation of weighted Moore-Penrose inverse, Matematički Vesnik 46 (1994), 41–50

[20] Stojaković, M., Determinante nekvadratnih matrica, Vesnik D.M.N.R.S. 1–2 (1952), 9–21

(received 07.03.1995.)

University of Niš, Faculty of Philosophy, Department of Mathematics

Ćirila i Metodija 2, 18000 Niš, Yugoslavia