MIT线性代数02_矩阵消元

1. Elimination

pivot 主元

$$\left[\begin{array}{ccc}1& 2& 1\\ 3& 8& 1\\ 0& 4& 1\end{array}\right]->\left[\begin{array}{ccc}1& 2& 1\\ 0& 2& -2\\ 0& 4& 1\end{array}\right]->\left[\begin{array}{ccc}1& 2& 1\\ 0& 2& -2\\ 0& 0& 5\end{array}\right]$$

2. Back-substitution

augumented matrix 增广矩阵

$$\left[\begin{array}{cccc}1& 2& 1& 2\\ 3& 8& 1& 12\\ 0& 4& 1& 2\end{array}\right]->\left[\begin{array}{cccc}1& 2& 1& 2\\ 0& 2& -2& 6\\ 0& 4& 1& 2\end{array}\right]->\left[\begin{array}{cccc}1& 2& 1& 2\\ 0& 2& -2& 6\\ 0& 0& 5& -10\end{array}\right]$$

3. Elimination matrices

• identity matrix 单位矩阵
• elementary matrix 初等矩阵
• associative law 结合律
• par·en·thesis n. /pəˈrenθəsɪs/= bracket
• permutation matrix 置换矩阵
• Inverse

Step 1: Matries: subtract 3 x row1 from row2

$$
\begin{bmatrix}
1 & 0 & 0 \
-3 & 1 & 0 \
0 & 0 & 1 \
\end{bmatrix}

\begin{bmatrix}
1 & 2 & 1 & 2 \
3 & 8 & 1 & 12 \
0 & 4 & 1 & 2 \
\end{bmatrix}

\begin{bmatrix}
1 & 2 & 1 & 2 \
0 & 2 & -2 & 6 \
0 & 4 & 1 & 2 \
\end{bmatrix}
$$

Step 2: Subtract 2 x row2 from row3

$$
\begin{bmatrix}
1 & 0 & 0 \
0 & 1 & 0 \
0 & -2& 1 \
\end{bmatrix}
\begin{bmatrix}
1 & 2 & 1 & 2 \
0 & 2 & -2 & 6 \
0 & 4 & 1 & 2 \
\end{bmatrix}

\begin{bmatrix}
1 & 2 & 1 & 2 \
0 & 2 & -2 & 6 \
0 & 0 & 5 & -10 \
\end{bmatrix}
$$

$$\begin{gathered} E_{32}(E_{21}A)=U \\ (E_{32}{E}_{21})A=U \end{gathered}$$

4. Matrix Multiplication

矩阵乘以一个列向量，相当于矩阵的列的线性组合。
一个行向量乘以一个矩阵，相当于矩阵的行的线性组合。

Permutation

Exchange row1 and row2

$$
\begin{bmatrix}
0 & 1 \
1 & 0 \
\end{bmatrix}

\begin{bmatrix}
a & b \
c & d \
\end{bmatrix}

\begin{bmatrix}
c & d \
a & b \
\end{bmatrix}
$$

$$
\begin{bmatrix}
a & b \
c & d \
\end{bmatrix}

\begin{bmatrix}
0 & 1 \
1 & 0 \
\end{bmatrix}

\begin{bmatrix}
b & a \
d & c \
\end{bmatrix}
$$

Inverses

$$
\begin{bmatrix}
1 & 0 & 0 \
3 & 1 & 0 \
0 & 0 & 1 \
\end{bmatrix}
\begin{bmatrix}
1 & 0 & 0 \
-3 & 1 & 0 \
0 & 0 & 1 \
\end{bmatrix}

\begin{bmatrix}
1 & 0 & 0 \
0 & 1 & 0 \
0 & 0 & 1 \
\end{bmatrix}
$$