[無官方正解]主題課程_對角化：特徵值、特徵向量、特徵空間#107848　

(5%) Which of the following are correct?
(A) An eigenvalue of a matrix has infinite number of eigenvectors.
(B) If det(A -λl_n) = O, λ is the eigenvalue of n x n matrix A.
(C) IfA is the cigenvalue of matrix A, the columns of (A - λIn) are independent.
(D) Ifa is the eigenvalue of matrix A, there exist a non-zero vector V such that Av = av.
(E) The number of the eigenvalue of n x n matrix A may larger than n.

 The eigenvalues of A and AT are the same, because det(A- λ l) -det(A-  λ l)^T= det(A^T-  λ l). By coming up with a 2 X 2 counter-example, show an example that the cigenvectors of A and A^Tneed not be the same.

(10pts) Suppose A is a real and symmetric matrix with order n, and has a repeated eigenvalue. Then for every i in {1,2, ... n}, there exists an eigenvector whose ith component is 0.

(10pts) Suppose A and B are two matrices with size m✕n and n✕m resbectively. Show that the nonzero eigenvalues of AB and BA are the same.