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On matrices with all minors negative. (English) Zbl 0962.15014

An \(m \times n\) matrix is called sign regular of order \(k\) if every minor of order \(i\) has the same sign \(\varepsilon_i\) for each \(i = 1,\ldots,k.\) \(A\) is totally positive (negative) if \(k = \min\{m,n\}\) and \(\varepsilon = +1\) (\(\varepsilon = -1\)). Whereas totally positive matrices have been studied extensively [cf., e.g., T. Ando, Linear Algebra Appl. 90, 165-219 (1987; Zbl 0613.15014)] and enjoy applications in economics, almost nothing was known about totally negative matrices [except results by M. Gasca and J. M. Peña, Linear Algebra Appl. 197/198, 133-142 (1994; Zbl 0796.65060)]. This paper presents a thorough investigation of totally negative matrices including the following main topics: spectral properties, generating algorithms, and triangular factorizations. Some open problems are mentioned.

MSC:

15B48 Positive matrices and their generalizations; cones of matrices
15A18 Eigenvalues, singular values, and eigenvectors
65F05 Direct numerical methods for linear systems and matrix inversion
15A09 Theory of matrix inversion and generalized inverses
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