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Factorizable matrices

Miroslav FiedlerFrank J. Hall — 2013

Special Matrices

We study square matrices which are products of simpler factors with the property that any ordering of the factors yields a matrix cospectral with the given matrix. The results generalize those obtained previously by the authors.

G-matrices, J -orthogonal matrices, and their sign patterns

Frank J. HallMiroslav Rozložník — 2016

Czechoslovak Mathematical Journal

A real matrix A is a G-matrix if A is nonsingular and there exist nonsingular diagonal matrices D 1 and D 2 such that A - T = D 1 A D 2 , where A - T denotes the transpose of the inverse of A . Denote by J = diag ( ± 1 ) a diagonal (signature) matrix, each of whose diagonal entries is + 1 or - 1 . A nonsingular real matrix Q is called J -orthogonal if Q T J Q = J . Many connections are established between these matrices. In particular, a matrix A is a G-matrix if and only if A is diagonally (with positive diagonals) equivalent to a column permutation of...

Sign patterns of J-orthogonal matrices

This paper builds upon the results in the article “G-matrices, J-orthogonal matrices, and their sign patterns", Czechoslovak Math. J. 66 (2016), 653-670, by Hall and Rozloznik. A number of further general results on the sign patterns of the J-orthogonal matrices are proved. Properties of block diagonal matrices and their sign patterns are examined. It is shown that all 4 × 4 full sign patterns allow J-orthogonality. Important tools in this analysis are Theorem 2.2 on the exchange operator and Theorem...

Eigenvalue distribution of certain ray patterns

Carolyn A. EschenbachFrank J. HallZhongshan Li — 2000

Czechoslovak Mathematical Journal

In this paper, the eigenvalue distribution of complex matrices with certain ray patterns is investigated. Cyclically real ray patterns and ray patterns that are signature similar to real sign patterns are characterized, and their eigenvalue distribution is discussed. Among other results, the following classes of ray patterns are characterized: ray patterns that require eigenvalues along a fixed line in the complex plane, ray patterns that require eigenvalues symmetric about a fixed line, and ray...

Linear preservers of row-dense matrices

Sara M. MotlaghianAli ArmandnejadFrank J. Hall — 2016

Czechoslovak Mathematical Journal

Let 𝐌 m , n be the set of all m × n real matrices. A matrix A 𝐌 m , n is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions T : 𝐌 m , n 𝐌 m , n that preserve or strongly preserve row-dense matrices, i.e., T ( A ) is row-dense whenever A is row-dense or T ( A ) is row-dense if and only if A is row-dense, respectively. Similarly, a matrix A 𝐌 n , m is called a column-dense matrix if every column of A is a column-dense vector. At the end, the structure of linear...

When does the inverse have the same sign pattern as the transpose?

Carolyn A. EschenbachFrank J. HallDeborah L. HarrellZhongshan Li — 1999

Czechoslovak Mathematical Journal

By a sign pattern (matrix) we mean an array whose entries are from the set { + , - , 0 } . The sign patterns A for which every real matrix with sign pattern A has the property that its inverse has sign pattern A T are characterized. Sign patterns A for which some real matrix with sign pattern A has that property are investigated. Some fundamental results as well as constructions concerning such sign pattern matrices are provided. The relation between these sign patterns and the sign patterns of orthogonal matrices...

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