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Report on test matrices for generalized inverses. (English) Zbl 0566.65026
This paper is a comprehensive report on test matrices for the generalized inversion of matrices. Two principles are described how to construct singular square or arbitrary rectangular test matrices and their Moore- Penrose inverses. By prescribing the singular values of the matrices or by suitably choosing the free parameters test matrices with condition numbers of any size can be obtained. We also deal with test matrices which are equal to their Moore-Penrose inverse. In addition to many advices how to construct test matrices the paper presents many test matrices explicitly, in particular singular square matrices of order n, sets of \(7\times 6\) and \(7\times 5\) matrices of different rank, a set of \(5\times 5\) matrices which are equal to their Moore-Penrose inverse and some special test matrices known from literature. For the set of \(7\times 6\) parameter matrices also the singular values corresponding to six values of the parameter are listed. For three simple parameter matrices of order \(5\times 4\) and \(6\times 5\) even test results obtained by eight different algorithms are quoted.
As ”by-products” the paper contains inequalities between condition numbers of different norms, representations for unitary, orthogonal, column-orthogonal and row-orthogonal matrices, a generalization of Hadamard matrices and representations of matrices which are equal to their Moore-Penrose inverse (or their inverse). All test matrices given in this paper may also be used for testing algorithms solving linear least squares problems.

65F20 Numerical solutions to overdetermined systems, pseudoinverses
15A09 Theory of matrix inversion and generalized inverses
65F25 Orthogonalization in numerical linear algebra
65F35 Numerical computation of matrix norms, conditioning, scaling
15B57 Hermitian, skew-Hermitian, and related matrices
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