Fallat, Shaun M.; van den Driessche, P. On matrices with all minors negative. (English) Zbl 0962.15014 Electron. J. Linear Algebra 7, 92-99 (2000). 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. Reviewer: Arnold Richard Kräuter (Leoben) Cited in 1 ReviewCited in 18 Documents 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 Keywords:minors; sign regular matrices; totally positive matrices; totally negative matrices; triangular facotrizations; spectral properties; generating algorithms Citations:Zbl 0613.15014; Zbl 0796.65060 PDF BibTeX XML Cite \textit{S. M. Fallat} and \textit{P. van den Driessche}, Electron. J. Linear Algebra 7, 92--99 (2000; Zbl 0962.15014) Full Text: EuDML EMIS OpenURL