Decompositions of strictly sign regular matrices
An \(n\times n\) matrix \(A\) is strictly sign regular (SSR) if all its minors have the same strict sign. A matrix is totally positive (TP) if all its minors are positive. A triangular matrix \(A\) is called \(\Delta\)TP if its nontrivial minors are all positive. A nonsingular matrix \(A\) is said to be lowerly signed with signature \(\varepsilon_1,\dots,\varepsilon_n\) if it can be decomposed in the form \(A = LDU\) and \(LD\Sigma\) is \(\Delta\)TP, where \(\Sigma\) is a diagonal matrix with diagonal entries \(\varepsilon_1, \varepsilon_1\varepsilon_2 , \dots , \varepsilon_{n-1}\varepsilon_n\). If, in addition, \(U^{-1}\) satisfies that \(\Sigma U^{-1}\Sigma\) is \(\Delta\)TP, then the matrix is called strictly signed \(\gamma\)-matrix with signature \(\varepsilon_1 ,\dots , \varepsilon_n\). The authors study several decompositions of strictly sign regular matrices. They give the following characterization of strictly sign regular matrices: Let \(A\) be an \(n \times n\) matrix. Then \(A\) is SSR with signature \(\varepsilon_1 ,\dots , \varepsilon_n\) if and only if for each \(i = 1, \dots , n\), \[ A[i, \dots , n] = Q_i R_i ,\quad (A[i,\dots , n])^T = \widetilde Q_i\widetilde R_i , \] where \(Q_i\), \(\widetilde Q_i\) are orthogonal strictly signed \(\gamma\)-matrices with signature \(\varepsilon_1 ,\dots , \varepsilon_{n-i+1}\), and \(R_i\), \(\widetilde R_i\) are nonsingular and upper triangular \(\Delta\)TP matrices. They get similar results related to Schur decomposition, singular value decomposition, and symmetric-triangular decomposition.
scientific article; zbMATH DE number 5307050