## On nonsingular sign regular matrices.(English)Zbl 1025.15038

An $$m\times n$$ real matrix $$A$$ is called sign regular of order $$k$$, $$1\leq k\leq \min\{m,n\}$$ if for each $$l$$, $$l=1,\ldots,k$$, all $$l\times l$$-minors of $$A$$ have the same sign $$\varepsilon_l$$. The matrix $$A$$ is called sign regular if $$k = \min\{m,n\}$$. The list $$(\varepsilon_1, \ldots,\varepsilon_k)$$ of signs is called the signature of $$A$$. A matrix is totally nonpositive (nonnegative) if all its minors are nonpositive (nonnegative).
The paper deals with the zero pattern of nonsingular matrices which are sign regular of order $$k$$. The author shows that the amount of zero entries that can appear in a nonsingular sign regular matrix depends on its signature.
A characterization of nonsingular totally nonpositive matrices is obtained and a test of $$O(n^3)$$ operations to check, if a square matrix of size $$n$$ is nonsingular totally nonpositive, is provided. A key tool for the test is a Neville elimination procedure.
Nonsingular matrices with all principle minors being nonpositive (called weak $$N$$-matrices) are considered. It is known that all principal minors of a nonsingular totally nonnegative matrix are positive. However, a similar result does not hold for a nonsingular totally nonpositive matrix. It is proved that if $$A=(a_{ij})$$ is a nonsingular totally nonpositive matrix with $$a_{11}<0$$ and $$a_{nn}<0$$ then $$A$$ is a weak $$N$$-matrix.

### MSC:

 15B57 Hermitian, skew-Hermitian, and related matrices 15B48 Positive matrices and their generalizations; cones of matrices
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