無年度 - [非官方正解]主題課程_對角化：特徵值、特徵向量、特徵空間#107848

科目：主題課程專用 | 選擇題數：6 | 申論題數：6

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所屬科目：主題課程專用

選擇題 (6)

110台大電信丙組
(10%) Consider a matrix

Which of the following is an eigenvalue of this matrix M? (A)1 (B)-3 (C)9 (D) -1+2j (E) -1+4j

(10%) Consider the matrix M in. It is known that M has four distinct eigernvalues, and let us denote the four cotresponding eigenvectors as Which of the following matrices also has as its eigenvectors? (A) (B) (C)(D) (E)
109台大電信乙組、資料科學_工數D
(5%) Which of the following is an eigenvector of A =? (A) x=；(B) x= ；(C)x=；(D) x=； (E) None of the above.
複選題
109中山_電機工數乙
 Let A . Which of the following statements are true? (A) If A is singular, then O is an eigenvaluc of A. (B) A and AT share the same eigenvalues and eigenvectors. (C) If A is diagonalizable, then A has n distinct eigenvalues. (D) Suppose that A is nonsingular. The condition Ax = λλx implies (E) Suppose that all the eigenvalues of A are real and positive. Then we have det( A ) ＞ 0.
複選題
108台大電機資訊學院_工數C
 .(10%) For the matrix below, which of the following are correct?(A)-1 is its eigenvalue. (B) 1 is its eigenvalue. (C)-3 is its eigenvalue. (D)Zero vector belongs to its eigenvector. (E)[1 0 2]T is a basis of its eigenspace.
複選題
108台大電機資訊學院_工數C 
(5%) Which of the following are correct? (A) An eigenvalue of a matrix has infinite number of eigenvectors. (B) If det(A -λln) = 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.

申論題 (6)

(a) (6%) Plcase find the eigenvalues of627dc9e66c5ed.jpg.
(b) (6%) Please find the eigenvalues of 2A.
(c) (6%) Please find the eigenvalues of A2. 
109中興電機 (15 pts)
 The eigenvalues of A and AT are the same, because det(A- λ l) -det(A-  λ l)T= det(AT-  λ l). By coming up with a 2 X 2 counter-example, show an example that the cigenvectors of A and ATneed not be the same.
106中興電機 
(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.
105中興電機 
(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.