Relative eigenvalues of a definite matrix pair

AbstractLet A and B be Hermitian matrices, and let c(A,B)≡min‖x‖2=1‖xH(A+iB)x‖. The matrix pair {A, B} is called a definite pair, and the corresponding eigenvalue problem βAx=αBx is definite if c(A,B)>0. The relationship between the eigenvalues of {A,B} and those of a definite pair of the form {XH1AX1,XH1BX1} is studied in this paper. The eigenvalues of {XH1AX1,XH1BX1} are said to be the relative eigenvalues of {A,B} with respect to the subspace R(X1). From the main result of this paper one can deduce a corresponding conclusion on Hermitian matrices