97年 - 97 國立高雄大學_博士班招生考試試題_統計學博士班：線性代數#147983

a. If \( A^2 = A \) then the eigenvalues of \( A \) are either 0 or 1.

b. Let \( A, B \in M_{n \times n}(\mathbb{R}) \). If \( AB = 0 \) then \( \operatorname{rank}(A) + \operatorname{rank}(B) \leq n \).

c. If \( A, B \in M_{n \times n}(\mathbb{R}) \) then \( AB \) and \( BA \) have the same minimal polynomial.

d. If \( A, B \in M_{n \times n}(\mathbb{R}) \) then the determinant of \( AB - BA \) is zero.

a. (6) Find the projection of \( \mathbf{b} \) onto the column space of \( A \).

b. (6) Let \( \mathbf{b}_A \) be the projection of \( \mathbf{b} \) onto the column space of \( A \). Find solution \( \mathbf{s} \) to \( A\mathbf{x} = \mathbf{b}_A \) such that \( \| \mathbf{s} \| \leq \| \mathbf{u} \| \) for all other solutions \( \mathbf{u} \).

3 (10) Let \( A \in M_{n \times m}(\mathbb{R}) \), \( B \in M_{m \times n}(\mathbb{R}) \). Prove that if \( \lambda \) is an eigenvalue of \( AB \) and \( \lambda \neq 0 \) then \( \lambda \) is an eigenvalue of \( BA \).

4 (10) Let \( A \in M_{n \times n}(\mathbb{R}) \) and \( r \in \mathbb{R} \). Suppose \( A - rI_n = QR \) where \( Q \) is orthogonal and \( R \) is upper-triangular. Let \( B = RQ + rI_n \). Show that \( A \) and \( B \) have the same eigenvalues.

a. (5) Let  x_Ty ≠1. Show that
\[ A^{-1} = I_n - \frac{1}{\mathbf{x}^T\mathbf{y} - 1}\mathbf{x}\mathbf{y}^T \]

b. (8) Let x^Ty≠0  . Find the eigenvalues and corresponding to eigenspaces of \( A \)

c. (5) Show that \( \operatorname{det}(A) = 1 - \mathbf{x}^T\mathbf{y} \).

d. (8) Let \( \mathbf{x} \) and \( \mathbf{y} \) be nonzero vectors. Show that if \( \mathbf{x}^T\mathbf{y} = 0 \) then \( A \) is not diagonalizable.

a. (5) Show that \( T \) is a linear transformation.

b. (10) Find the eigenvalues and eigenvectors of \( T \).

c. (3) Find the minimal polynomial of \( T \).