On Eigenvalues of a Rayleigh Quotient Matrix

This note deals with the following problem: Let A be an n X n Hermitian matrix, and Q and 0 be two n X rn (n > m > 1) matrices both with orthonormal column vectors. How do the eigenvalues of the m X m Hermitian matrix Q”AQ differ from those of the m X m Hermitian matrix QHAQ? We give a positive answer to one of the unsolved problems raised recently by Sun. In what follows, we will consider the following interesting problem concerning the spectral variation of a Rayleigh quotient matrix: Let A be an n X n Hermitian matrix, and Q and Q be two n X m (n > m > 1) matrices both with orthonormal column vectors. How do the eigenvalues of the m X m Hermitian matrix Q”AQ differ from those of the m X m Hermitian matrix Q”AQ? Here the superscript H denotes the conjugate transpose of a matrix. This problem arises often in computation methods for symmetric matrix eigenvalues such as the power method, the QR method, the simultaneous method, and several other techniques now available. Thus it is of great importance from the point of view not only of theoretical analysis of some iterative algorithms but also of practical applications. Much work has been done for this problem in various aspects so far, e.g., [2], [S], [9], [ll], and [12]. In [S], Liu and Xu showed the following: Let A, ,..., A, and AL ,..., I, be the eigenvalues (arranged in ascending order) of Q”AQ and of Q”AQ, respectively. Zf AQ=QA, where h=diag(A, ,..., A,), (1) LZNEAR ALGEBRA AND ITS APPLlCATlONS 169:249-255 (1992) 0 Else&r Science Publishing Co., Inc., 1992 249 655 Avenue of the Americas, New York, NY 10010 0024-3795/92/$5.00