A note on Stewart's theorem for definite matrix pairs☆

Abstract Let A and B be n × n Hermitian matrices. The matrix pair ( A , B ) is called definite pair and the corresponding eigenvalue problem β Ax = α Bx is definite if c ( A , B ) ≡ inf ‖ x ‖= 1 {| H ( A + iB ) x |} > 0. In this note we develop a uniform upper bound for differences of corresponding eigenvalues of two definite pairs and so improve a result which is obtained by G.W. Stewart [2]. Moreover, we prove that this upper bound is a projective metric in the set of n × n definite pairs.