99年 - 99 國立高雄大學_博士班招生考試試題_應用數學學系：線性代數#147986

1. (10 pts) Let $A \in \mathsf{M}{m \times n}(\mathbb{R})$ and $B \in \mathsf{M}{n \times r}(\mathbb{R})$. If $AB = 0$, prove that

$$\operatorname{rank}(A) + \operatorname{rank}(B) \leq n.$$

2. (10 pts) Show that

$$\begin{pmatrix} 1 & 2 \\ 2 & i \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} i & 4 \\ 1 & 1 \end{pmatrix}$$

are not unitarily equivalent.

(a) (5 pts) Let \( P^{-1}AP = D \) be a diagonal matrix. Prove that \( e^A = Pe^D P^{-1} \).

(b) (10 pts) Find \( A, B \in \mathsf{M}_{2 \times 2}(\mathbb{R}) \) such that \( e^A e^B \neq e^{A + B} \).

(a) (6 pts) Show that \( \pm 1 \) are the only eigenvalues of \( T \).

(b) (6 pts) Describe the eigenvectors corresponding to each eigenvalue of \( T \).

(c) (8 pts) Find an ordered basis \( \beta \) for \( \mathsf{M}{n\times n}(\mathbb{R}) \) such that \( [T]{\beta} \) is a diagonal matrix.

5. (15 pts) Let

$$A = \begin{pmatrix} 3 & 1 & 2 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix}.$$

Find a Jordan canonical form $J$ of $A$ and an invertible matrix $Q$ such that $J = Q^{-1}AQ$.

6. (15 pts) State and prove the Cayley-Hamilton Theorem.

7. (15 pts) Let \( T \) be a projection on a finite-dimensional inner product space \( V \) such that \( \| T(x) \| \leq \| x \| \) for all \( x \in V \). Prove that \( T \) is an orthogonal projection.