試題詳解

16. Let A = be a nonzero matrix of size n×n with its real part and imaginary part being denoted by B and C, respectively. Moreover, let Ω be a symmetric matrix with A and C as its four blocks. Which of the following statements are true?
(A) Matrix C has at least one real eigenvalues.
(B) Let λ and μ be any two eigenvalues of B and C, respectively. Then λ + μ ≠ 0.
(C) Any eigenvalue of A is also an eigenvalue of Ω.
(D) Any eigenvalue of Ω is also an eigenvalue of A.
(E) Let be an eigenvalue-eigenvector-pair of Ω. Then (λ, x+iy) is an eigenvalue- eigenvector-pair of A.