## 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

### Citations:

Zbl 0613.15014; Zbl 0796.65060
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