• 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.

• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.171-184
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• 2014

A higher order iterative method to compute the Moore-Penrose inverses of arbitrary matrices using only the Penrose equation (ii) is developed by extending the iterative method described in [1]. Convergence properties as well as the error estimates of the method are studied. The efficacy of the method is demonstrated by working out four numerical examples, two involving a full rank matrix and an ill-conditioned Hilbert matrix, whereas, the other two involving randomly generated full rank and rank deficient matrices. The performance measures are the number of iterations and CPU time in seconds used by the method. It is observed that the number of iterations always decreases as expected and the CPU time first decreases gradually and then increases with the increase of the order of the method for all examples considered.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.251-263
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• 2004

In this paper, algorithms for computing the minimal polynomial and the common minimal polynomial of resultant matrices over any field are presented by means of the approach for the Grobner basis of the ideal in the polynomial ring, respectively, and two algorithms for finding the inverses of such matrices are also presented. Finally, an algorithm for the inverse of partitioned matrix with resultant blocks over any field is given, which can be realized by CoCoA 4.0, an algebraic system over the field of rational numbers or the field of residue classes of modulo prime number. We get examples showing the effectiveness of the algorithms.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.81-94
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• 2007

In this paper we investigate symbolic implementation of two modifications of the Leverrier-Faddeev algorithm, which are applicable in computation of the Moore-Penrose and the Drazin inverse of rational matrices. We introduce an algorithm for computation of the Drazin inverse of rational matrices. This algorithm represents an extension of the papers [11] and [14]. and a continuation of the papers [15, 16]. The symbolic implementation of these algorithms in the package MATHEMATICA is developed. A few matrix equations are solved by means of the Drazin inverse and the Moore-Penrose inverse of rational matrices.

• Bulletin of the Korean Mathematical Society
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• v.40 no.3
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• pp.425-435
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• 2003

In this paper, a new kind of matrices, i.e., $level-{\kappa}$ II-circulant matrices is considered. Algorithms for computing minimal polynomial of this kind of matrices are presented by means of the algorithm for the Grobner basis of the ideal in the polynomial ring. Two algorithms for finding the inverses of such matrices are also presented based on the Buchberger's algorithm.

• 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.

• Kyungpook Mathematical Journal
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• v.28 no.1
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• pp.45-50
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• 1988

Circulant and multiblock circulant matrices have many important applications, and therefore their inverses are of considerable interest. A simple recursive algorithm is presented to compute the inverse of a multiblock circulant matrix. The algorithm only uses complex variables, roots of unity and normal matrix/vector operations.

• 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 $A^*$.