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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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