## 每週問題 June 13, 2016

Let $\boldsymbol{\beta}=\{\mathbf{x}_1,\ldots,\mathbf{x}_k\}$ and $\boldsymbol{\gamma}=\{\mathbf{y}_1,\ldots,\mathbf{y}_k\}$ be bases for a subspace $\mathcal{V}$ in $\mathbb{R}^n$. Let $X=\begin{bmatrix} \mathbf{x}_1&\cdots&\mathbf{x}_k \end{bmatrix}$ and $Y=\begin{bmatrix} \mathbf{y}_1&\cdots&\mathbf{y}_k \end{bmatrix}$. Show that the change of coordinates matrix from $\boldsymbol{\beta}$ to $\boldsymbol{\gamma}$ is

$P=(Y^TY)^{-1}Y^TX$.

## 每週問題 June 6, 2016

Let $A$ be an $m\times n$ complex matrix. If $A\mathbf{x}=\mathbf{b}$ is consistent for some $\mathbf{b}$, prove that there exists a unique solution $\mathbf{x}$ in the column space of $A^\ast$.

## 每週問題 May 30, 2016

Let $A$ be an $m\times n$ matrix and $S$ be the solution set for a consistent system of linear equations $A\mathbf{x}=\mathbf{b}$ for some $\mathbf{b}\neq\mathbf{0}$.
(a) If $S_{\max}$ is a maximal independent subset of $S$ and $\mathbf{x}_p$ is any particular solution, show that

$\hbox{span}(S_{\max})=\hbox{span}\{\mathbf{x}_p\}+N(A)$,

where $N(A)$ denotes the nullspace of $A$.
(b) If $\hbox{rank}A=r$, show that $A\mathbf{x}=\mathbf{b}$ has at most $n-r+1$ independent solutions.

## 每週問題 May 23, 2016

Let $A$ and $B$ be $m\times n$ matrices. If $\hbox{rank}A=r$ and $\hbox{rank}B=k\le r$, show that

$r-k\le \hbox{rank}(A+B)\le r+k$.

In words, a perturbation of rank $k$ can change the rank by at most $k$.

## Cayley-Hamilton 定理的一個錯誤「證明」

\begin{aligned} p(\lambda)&=\det(A-\lambda I)=\begin{vmatrix} a-\lambda&b\\ c&d-\lambda \end{vmatrix}\\ &=\lambda^2-(a+d)\lambda+ad-bc,\end{aligned}

Cayley-Hamilton 定理宣稱

$p(A)=A^2-(a+d)A+(ad-bc)I=0$

Cayley-Hamilton 定理有很多種證法 (見“Cayley-Hamilton 定理”)，但其中幾乎挑不出一個簡單的證明。底下這個看似快捷實乃錯誤的「證明」曾經不斷地被初學者重複發現：

$p(A)=\det(A-AI)=\det(A-A)=\det 0=0$

## 「零」究竟是有還是沒有？

2000年3月27日，交通大學舉辦「人文與科技三賢鼎談」，三賢是法鼓山聖嚴法師，交通大學校長張俊彥與清華大學校長劉炯朗。會中有一段談話，抄錄於下[1]

## 反對角矩陣的特徵值

$A=[a_{ij}]$ 為一個 $n\times n$ 階反對角矩陣 (anti-diagonal matrix)。例如，若 $n=5$

$A=\begin{bmatrix} &&&&a_{15}\\ &&&a_{24}&\\ &&a_{33}&&\\ &a_{42}&&&\\ a_{51}&&&& \end{bmatrix}$

## 每週問題 May 16, 2016

Let $A$ and $B$ be $m\times n$ matrices. Show that

$|\hbox{rank}A-\hbox{rank}B|\le \hbox{rank}(A-B)$.

## 不說廢話──克拉瑪公式的證明

You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.
― Carl Friedrich Gauss

$A=\begin{bmatrix} \mathbf{a}_1&\cdots&\mathbf{a}_n \end{bmatrix}$ 為一個 $n\times n$ 階矩陣且 $\mathbf{b}$ 為一個 $n$ 維行向量 (column vector)。若 $A$ 是可逆的，克拉瑪公式 (Cramer’s rule) 給出線性方程 $A\mathbf{x}=\mathbf{b}$ 的解 $\mathbf{x}=(x_1,\ldots,x_n)^T$，如下 (見“克拉瑪公式的證明”)：

$\displaystyle x_i=\frac{\det A_i(\mathbf{b})}{\det A},~~i=1,\ldots,n$

$A_i(\mathbf{b})=\begin{bmatrix} ~&~&~&~&~&~&~\\ \mathbf{a}_1&\cdots&\mathbf{a}_{i-1}&\mathbf{b}&\mathbf{a}_{i+1}&\cdots&\mathbf{a}_n\\ ~&~&~&~&~&~&~ \end{bmatrix}$