## 每週問題 October 19, 2015

Let $A$ and $B$ be $n\times n$ matrices such that $AB=BA$. If $A$ has $n$ distinct eigenvalues, show that $B$ can be expressed uniquely as a polynomial in $A$ with degree no more than $n-1$.

## 每週問題 October 12, 2015

(a) Let $A$ be an $m\times n$ matrix and $B$ be an $m\times p$ matrix. Show that

$C(A)+C(B)=C\left(\begin{bmatrix} A&B \end{bmatrix}\right)$,

where $C(X)$ denotes the column space of $X$.

(b) Let $P$ be an $m\times n$ matrix and $Q$ be a $p\times n$ matrix. Show that

$N(P)\cap N(Q)=N\left(\begin{bmatrix} P\\ Q \end{bmatrix}\right)$,

where $N(X)$ denotes the nullspace of $X$.

## 常係數線性遞迴關係式 (下)

$a_n=d_1a_{n-1}+d_2a_{n-2}+\cdots+d_ka_{n-k}+F(n),~~n\ge k$

$a_n=d_1a_{n-1}+d_2a_{n-2}+\cdots+d_ka_{n-k}$

$\displaystyle a^{(p)}_n=d_1a^{(p)}_{n-1}+d_2a^{(p)}_{n-2}+\cdots+d_ka^{(p)}_{n-k}+F(n)$

$\displaystyle b_n=d_1b_{n-1}+d_2b_{n-2}+\cdots+d_kb_{n-k}+F(n)$

$\displaystyle b_n-a^{(p)}_n=d_1(b_{n-1}-a^{(p)}_{n-1})+d_2(b_{n-2}-a^{(p)}_{n-2})+\cdots+d_k(b_{n-k}-a^{(p)}_{n-k})$

## 一個關於階乘的恆等式

$\displaystyle \sum_{i=0}^n(-1)^i\binom{n}{i}(x-i)^n=n!$

$\displaystyle \sum_{i=0}^n(-1)^i\binom{n}{i}p(i)=0$

## 每週問題 October 5, 2015

Let $A=\begin{bmatrix} B&C\\ D&E \end{bmatrix}$ be an $n\times n$ nonsingular matrix, where $B$ is $k\times k$. Denote the adjugate of $A$ by $\hbox{adj}\,A=\begin{bmatrix} P&Q\\ R&S \end{bmatrix}$, where $P$ is $k\times k$. Prove the Jacobi identity

$\det P=(\det A)^{k-1}(\det E)$.

## 每週問題 September 28, 2015

Let $A$ be an $n\times n$ matrix. If $\mathcal{X}\subseteq\mathbb{C}^n$ is an invariant subspace of $A$, i.e., $A\mathbf{x}\in\mathcal{X}$ for every $\mathbf{x}\in\mathcal{X}$, show that there exists a nonzero vector $\mathbf{x}$ in $\mathcal{X}$ such that $A\mathbf{x}=\lambda\mathbf{x}$.

## 常係數線性遞迴關係式 (上)

$k$ 階常係數線性遞迴關係式 (linear recurrence relation) 可表示如下：

$a_n=d_1a_{n-1}+d_2a_{n-2}+\cdots+d_ka_{n-k}+F(n),~~~n\ge k$

$0,1,1,2,3,5,8,13,21,34,55,89,\ldots$

$a_n=a_{n-1}+a_{n-2},~~n\ge 2$

## 每週問題 September 21, 2015

$Q$ 是正交矩陣，則 $\displaystyle\frac{dQ}{dt}Q^T$ 是反對稱矩陣。

Let $Q=[q_{ij}(t)]$ be an orthogonal matrix, where each entry $q_{ij}(t)$ is a differentiable function of $t$. Show that $\displaystyle\frac{dQ}{dt}Q^T$ is skew-symmetric.