A note on Stewart's theorem for definite matrix pairs

Let A and B be n×n Hermitian matrices. The matrix pair (A, B) is called definite pair and the corresponding eigenvalue problem βAx = αBx is definite if c(A, B) ≡ inf_{6x6= 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.