• Journal of applied mathematics & informatics
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• v.9 no.2
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• pp.519-529
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• 2002

We investigate implementation of the determinantal representation of generalized inverses for complex and rational matrices in the symbolic package MATHEMATICA. We also introduce an implementation which is applicable to sparse matrices.

• Communications of the Korean Mathematical Society
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• v.29 no.2
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• pp.227-237
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• 2014

For an $m{\times}n$ nonnegative integral matrix A, a generalized inverse of A is an $n{\times}m$ nonnegative integral matrix G satisfying AGA = A. In this paper, we characterize nonnegative integral matrices having generalized inverses using the structure of nonnegative integral idempotent matrices. We also define a space decomposition of a nonnegative integral matrix, and prove that a nonnegative integral matrix has a generalized inverse if and only if it has a space decomposition. Using this decomposition, we characterize nonnegative integral matrices having reflexive generalized inverses. And we obtain conditions to have various types of generalized inverses.

• Journal of applied mathematics & informatics
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• v.18 no.1_2
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• pp.321-328
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• 2005

We investigate a matrix inequality on Schur complements defined by {1}-generalized inverses, and obtain simultaneously a necessary and sufficient condition under which the inequality turns into an equality. This extends two existing matrix inequalities on Schur complements defined respectively by inverses and Moore-Penrose generalized inverses (see Wang et al. [Lin. Alg. Appl., 302-303(1999)163-172] and Liu and Wang [Lin. Alg. Appl., 293(1999)233-241]). Moreover, the non-uniqueness of $\{1\}$-generalized inverses yields the complicatedness of the extension.

• Journal of applied mathematics & informatics
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• v.25 no.1_2
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• pp.415-424
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• 2007

The generalized inverses have many important applications in the aspects of theoretic research and numerical computations and therefore they were studied by many authors. In this paper we get some necessary and sufficient conditions of the forward order law for {1}-inverse of multiple matrices products $A\;=\;A_1A_2{\cdots}A_n$ by using the maximal rank of generalized Schur complement.

• Communications of the Korean Mathematical Society
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• v.13 no.4
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• pp.875-888
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• 1998

The use of the zeros of Chebyshev polynomial of the first kind $T_{4n＋4(x}$ ) and second kind $U_{2n＋1}$ (x) for Gauss-Chebyshev quad-rature and collocation of singular integral equations of Cauchy type yields computationally accurate solutions over other combinations of $T_{n}$ /(x) and $U_{m}$(x) as in [8]. We show that the coefficient matrix of the overdetermined system has the generalized inverse. We estimate the residual error using the norm of the generalized inverse.e.

• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.541-547
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• 2008

Suppose R is an idempotence-diagonalizable ring. Let n and m be two arbitrary positive integers with $n\;{\geq}\;3$. We denote by $M_n(R)$ the ring of all $n{\times}n$ matrices over R. Let ($J_n(R)$) be the additive subgroup of $M_n(R)$ generated additively by all idempotent matrices. Let ($D=J_n(R)$) or $M_n(R)$. In this paper, by using an additive idem potence-preserving result obtained by Coo (see [4]), I characterize (i) the additive preservers of tripotence from D to $M_m(R)$ when 2 and 3 are units of R; (ii) the additive preservers of inverses (respectively, Drazin inverses, group inverses, {1}-inverses, {2}-inverses, {1, 2}-inverses) from $M_n(R)$ to $M_n(R)$ when 2 and 3 are units of R.

• Journal of applied mathematics & informatics
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• v.12 no.1_2
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• pp.211-218
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• 2003

Let H be a Hilbert space, X be a real Banach space, A : H \longrightarrow X be an operator with D(A) dense in H, G： H \longrightarrow H be positive definite, $\chi$ $\in$ D(A) and b $\in$ H. Consider the quadratic programming problem: QP： Minimize $\frac{1}{2}$〈p, $\chi$〉 + 〈$\chi$, G$\chi$〉 subject to A$\chi$= b In this paper, we obtain an explicit solution to the above problem using generalized inverses.

• The Mathematical Education
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• v.22 no.2
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• pp.13-15
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• 1984

• Kyungpook Mathematical Journal
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• v.43 no.4
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• pp.605-616
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• 2003

In this paper, some rank equalities related to generalized inverses $A^{(2)}_{T,S}$ of a matrix are presented. As applications, a variety of rank equalities related to the M-P inverse, the Drazin inverse, the group inverse, the weighted M-P inverse, the Bott-Duffin inverse and the generalized Bott-Duffin inverse are established.

• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.61-74
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• 2014

This paper describes an iterative method for orthogonal projections $AA^+$ and $A^+A$ of an arbitrary matrix A, where $A^+$ represents the Moore-Penrose inverse. Convergence analysis along with the first and second order error estimates of the method are investigated. Three numerical examples are worked out to show the efficacy of our work. The first example is on a full rank matrix, whereas the other two are on full rank and rank deficient randomly generated matrices. The results obtained by the method are compared with those obtained by another iterative method. The performance measures in terms of mean CPU time (MCT) and the error bounds for computing orthogonal projections are listed in tables. If $Z_k$, k = 0,1,2,... represents the k-th iterate obtained by our method then the sequence of the traces {trace($Z_k$)} is a monotonically increasing sequence converging to the rank of (A). Also, the sequence of traces {trace($I-Z_k$)} is a monotonically decreasing sequence converging to the nullity of $A^*$.