## 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
Full Text: