Criteria for sufficient matrices
The author proposes new criteria for identifying column sufficient, row sufficient and sufficient matrices. A matrix \(A\in\mathbb{R}^{n\times n}\) is column sufficient if for all \(x \in\mathbb{R}^n\), \(x^*(Ax) \leq 0 \Rightarrow x^*(Ax)=0\) where \(*\) denotes the Hadamard product. If \(A^T\) is column sufficient, \(A\) is row sufficient and \(A\) is sufficient if it is both column and row sufficients. It is proved that if a column (row) sufficient matrix \(A\) is of rank \(r\) and some collection of \(r\) columns (rows) are linearly independent, then the corresponding principal submatrix is nonsingular. As a consequence, if some columns (rows) of \(A\) are linearly independent, then so are the corresponding rows (columns). Based on principal pivoting, the paper gives some new criteria for (column, row) sufficient matrices. A matrix \(A\in\mathbb{R}^{n \times n}\) of rank \(r<n\) is (column, row) sufficient if it is (column, row) sufficient of order \(r+1\). A matrix \(A \in\mathbb{R}^{n \times n}\) with positive determinant is (column, row) sufficient if it is (column, row) sufficient of order \(k<n\) and \(A^{-1}\) is (column, row) sufficient of order \(n-k\). Necessary and sufficient conditions for a (column, row) sufficient matrix \(A\in\mathbb{R}^{n \times n}\) of order \(n-1\) to be (column, row) sufficient are given. Using these conditions, inductive criteria are proposed for (column, row) sufficiency, that is, sequentially check for \(k=2,3, \dots\) whether (column, row) sufficient \(A\) (or \(A^{-1})\) of order \(k-1\) is (column, row) sufficient of order \(k\).
scientific article; zbMATH DE number 841741
scientific article