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

Let $\left\{A,B\right\}$ be a definite matrix pair of order $n$ , and let the columns of ${\stackrel{˜}{X}}_{1}\in {‐{C}}^{n×l}$ span an $l$ ‐dimensional approximate invariant eigenspace of $\left\{A,B\right\}$ . In this paper, we present a Hoffman–Wielandt‐type residual bound for eigenvalues between Rayleigh quotient matrix pair $\left\{{\stackrel{˜}{X}}_{1}^{H}A{\stackrel{˜}{X}}_{1},{\stackrel{˜}{X}}_{1}^{H}B{\stackrel{˜}{X}}_{1}\right\}$ and $\left\{A,B\right\}$ , which extends a result of Kahan [W. Kahan, Numerical Linear Algebra, Canadian Mathematical Bulletin, 9 (1966) 757–801].