# 「統計解析のための線形代数」復習筆記21

## 行的基本變形

Theorem 1 (行的基本變形) 對矩陣進行下列操作的過程，被稱爲是行的基本變形（行的基本操作, elementary row operations）。

1. 給任意一行乘以/除以一個非零的數。
2. 給任意一行加上/減去另外任意行的倍數。
3. 將任意兩行的對應元素互換。

### 練習基本變形：

\left(\begin{array}{c} 1& 2& 1 & \vdots & 1 & 0 & 0\\ 2& 1& 1 & \vdots & 0 & 1 & 0\\ 1& 1& 2 & \vdots & 0 & 0 & 1\\ \end{array}\right) \begin{align} \left\{ \begin{array}{rr} (1)\\ (2)\\ (3) \end{array} \right. \end{align}

\left(\begin{array}{c} 1& 2& 1 & \vdots & 1 & 0 & 0\\ 0& -3& -1 & \vdots & -2 & 1 & 0\\ 0& -1& 1 & \vdots & -1 & 0 & 1\\ \end{array}\right) \begin{align} \left\{ \begin{array}{l} (1)\\ (2)=(2)-2\times(1)\\ (3)=(3)-(1) \end{array} \right. \end{align}

\left(\begin{array}{c} 1& 0& 3 & \vdots & -1 & 0 & 2\\ 0& -4& 0 & \vdots & -3 & 1 & 1\\ 0& 1& -1 & \vdots & 1 & 0 & -1\\ \end{array}\right) \begin{align} \left\{ \begin{array}{l} (1)=(1)+2\times(3)\\ (2)=(2)+(3)\\ (3)=-1\times(3) \end{array} \right. \end{align}

Next:

\left(\begin{array}{c} 1& 0& 3 & \vdots & -1 & 0 & 2\\ 0& 1& 0 & \vdots & 3/4 & -1/4 & -1/4\\ 0& 1& -1 & \vdots & 1 & 0 & -1\\ \end{array}\right) \begin{align} \left\{ \begin{array}{l} (1)=(1)\\ (2)=(2)\div(-4)\\ (3)=(3) \end{array} \right. \end{align}

Next:

\left(\begin{array}{c} 1& 0& 3 & \vdots & -1 & 0 & 2\\ 0& 1& 0 & \vdots & 3/4 & -1/4 & -1/4\\ 0& 0& -1 & \vdots & 1/4 & 1/4 & -3/4\\ \end{array}\right) \begin{align} \left\{ \begin{array}{l} (1)=(1)\\ (2)=(2)\\ (3)=(3)-(2) \end{array} \right. \end{align}

Next:

\left(\begin{array}{c} 1& 0& 0 & \vdots & -1/4 & 3/4 & -1/4\\ 0& 1& 0 & \vdots & 3/4 & -1/4 & -1/4\\ 0& 0& 1 & \vdots & -1/4 & -1/4 & -3/4\\ \end{array}\right) \begin{align} \left\{ \begin{array}{l} (1)=(1)+3\times(3)\\ (2)=(2)\\ (3)=-1\times(3) \end{array} \right. \end{align}

$X^{-1}=\left(\begin{array}{c} -1/4 & 3/4 & -1/4\\ 3/4 & -1/4 & -1/4\\ -1/4 & -1/4 & 3/4\\ \end{array}\right)$

### A: 有。把行的基本變形中的定義 (1) 的行改成列，既是列的基本變形的定義。

##### 王　超辰 - Chaochen Wang
###### Real World Evidence Scientist

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