On complex Benzi-Golub matrices
A class of complex \(N\times N\) matrices is studied which are called Benzi-Golub (BG) matrices by the author. A complex square matrix \(A=B+iC\) with \(B=B^*\), \(C=C^*\) is said to be strictly accretive if \(B\) is a positive definite matrix. A matrix of the form \[ A=\begin{pmatrix} P&Q \\ Q^*&R\end{pmatrix}\tag{1} \] with \(P\) and \(-R\) strictly accretive matrices (respectively \(n\times n\) and \(m\times m,\;n+m=N)\) is called a BG-matrix. The aim of the paper is to show that the BG-matrices retain almost all the properties of the (real) symmetric quasidefinite matrices. The main results show that if \(A\) is a BG-matrix as in (1) then: {\parindent=5mm \begin{itemize}\item[1.] the Schur complements \(-A/P\) and \(A/R\) are strictly accretive; \item[2.] \(A\) is nonsingular and if \[ A^{-1}=\begin{pmatrix} S&T \\ U&V\end{pmatrix} \] is a partition similar to that in (1), then the blocks \(S\) and \(-V\) are strictly accretive; \item[3.] \(A\) has not purely imaginary eigenvalues; \item[4.] \(A\) has \(n\) eigenvalues with positive real parts and \(m\) eigenvalues with negative real parts; \item[5.] if \(\lambda\) and \(\mu\) are eigenvalues of \(A\) with respectively positive and negative real parts and the associated eigenvectors \(x\) and \(y\) have the partitioned form \[ x=\binom{c x_1}{x_2},\quad y=\binom{cy_1}{y_2}, \quad x_1,y_1\in \mathbb C^n,\;x_2,y_2\in \mathbb C^m, \] then \(\| x_1\|_2>\| x_2\|_2\) and \(\| y_2\|_2>\| y_1\|_2\). \end{itemize}}
scientific article; zbMATH DE number 5621453