Schur complements and its applications to symmetric nonnegative and \(Z\)-matrices
The Schur complement \(H/A\) of a nonsingular principal submatrix \(A\) of a matrix \(H\) is a useful tool in matrix analysis. When \(H\) is irreducible nonnegative, the Perron complement is another interesting concept. For a Hermitian semidifinite matrix \(H\), the author obtains interlacing results for the eigenvalues of \(H\) and those of \(H/A\), refining the ones obtained by \textit{R. L. Smith} [Linear Algebra Appl. 177, 137-144 (1992; Zbl 0765.15007)]. For irreducible nonnegative matrices, analogous results involving Perron complements are also obtained. Perron complements are also used to get some bounds for the eigenvalues of irreducible symmetric nonnegative matrices and for irreducible symmetric \(Z\)-matrices, that are real matrices whose off-diagonal entries are nonpositive.
scientific article; zbMATH DE number 1800365