A Hoffman–Wielandt‐type residual bound for generalized eigenvalues of a definite pair

Let $\{A,B\}$ be a definite matrix pair of order $n$ , and let the columns of ${\tilde{X}}_1\in {‐{C}}^{n\times l}$ span an $l$ ‐dimensional approximate invariant eigenspace of $\{A,B\}$ . In this paper, we present a Hoffman–Wielandt‐type residual bound for eigenvalues between Rayleigh quotient matrix pair $\{{\tilde{X}}_1^{H}A{\tilde{X}}_1,{\tilde{X}}_1^{H}B{\tilde{X}}_1\}$ and $\{A,B\}$ , which extends a result of Kahan [W. Kahan, Numerical Linear Algebra, Canadian Mathematical Bulletin, 9 (1966) 757–801].