5. Generalized Hermitian Eigenvalue Problems
M. Gu, A. Ruhe, G.L.G. Sleijpen, Henk A. van der Vorst, Z. Bai, Ruiyang Li · Society for Industrial and Applied Mathematics eBooks · 2000
5.1 Introduction A generalized Hermitian eigenvalue problem (GHEP) is given by Ax=λBx, 5.1 where A and B are Hermitian, , and . We call the pair {A, B} of matrices in (5.1) a matrix pencil. In this chapter we make the additional assumption that A or B or αA + βB for some scalars α and β is positive definite, in which case we talk about a Hermitian definite pencil. This assumption is true for a wide class of practically important cases, and the theory is very closely related to the standard Hermitian eigenproblem, as expounded in Chapter 4. If no positive definite combination exists, we could as well regard {A, B} as a general pencil and use the theory and algorithms described in Chapter 8. The nonstandard case, where αA + βB is positive definite, may be reduced to the standard case when B is positive definite, by noting that the pencil (A−θ (αA+βB) )x=0 5.2 has eigenvalues and the same eigenvectors as the original pencil (5.1). One may apply any algorithm applicable for positive definite B to this modified pencil and recover the from the . The GHEP (5.1) has n real eigenvalues , which we may order increasingly so that . Several eigenvalues may coincide, as in the standard case, except that some eigenvalues may be infinite. If the matrices A and B are positive definite, , but if A is positive semidefinite we can only say that .