Let M > 0 be a fixed positive definite n × n matrix. A nonzero vector v ≠ 0 is called a generalized eigenvector of the n × n matrix K if
Kv = λMv, v ≠ 0, (8.31)
where the scalar λ is the corresponding generalized eigenvalue.
(a) Prove that λ is a generalized eigenvalue of the matrix K if and only if it is an ordinary eigenvalue of the matrix M-1 K. How are the eigenvectors related?
(b) Now suppose K is a symmetric matrix. Prove that its generalized eigenvalues are all real.
(c) Show that if K > 0, then its generalized eigenvalues are all positive: λ > 0.
(d) Prove that the eigenvectors corresponding to different generalized eigenvalues are orthogonal under the weighted inner product (v, w) = vT M w.
(e) Show that, if the matrix pair K, M has n distinct generalized eigenvalues, then the eigenvectors form an orthogonal basis for Rn.