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Disc separation of the Schur complement of diagonally dominant matrices and determinantal bounds. (English) Zbl 1107.15022

Let \(A\) be an \(n\times n\) complex matrix. For subsets \(\alpha,\beta\subset\{1,2,\ldots,n\}\), define the matrix \(A(\alpha,\beta)\) to be the submatrix of \(A\) consisting of the rows \(\alpha\) and the columns \(\beta\) of \(A\). Denote \(A(\alpha,\alpha)\) by \(A(\alpha)\). The Schur complement of \(A\) with respect to a nonsingular submatrix \(A(\alpha)\) is the matrix \[ A/\alpha=A(\alpha^c)-A(\alpha^c,\alpha)\,[A(\alpha)]^{-1}\,A(\alpha,\alpha^c). \]
We say that \(A\) is strictly diagonally (row) dominant (SD) if \[ | a_{ii}| > P_i(A)=\sum_{j=1,i\neq j}^n | a_{ij}| \] The authors study the differences \(| a_{ii}| - P_i(A)\) and prove that they increase when passing to the Schur complement of an SD matrix. As a consequence, several bounds for determinants and localization of eigenvalues are presented.

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
15A45 Miscellaneous inequalities involving matrices
15A42 Inequalities involving eigenvalues and eigenvectors
15A15 Determinants, permanents, traces, other special matrix functions
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