Linear Algebra
Linear Algebra notes.
Matrix multiplication
The origin of matrix multiplication
参考：数学家最初发明行列式和矩阵是为了解决什么问题？ - 马同学的回答 - 知乎
最初目的：解线性方程组
举例：\(YC_rC_b \to RGB\)
- 黑白电视到彩色电视
- 兼容问题
- \(Y\): 灰度图
$$
\begin{cases}
0.299R + 0.587G + 0.114B = Y \\
0.500R - 0.419G - 0.081B + 128 = C_r \\
-0.169R - 0.331G + 0.500B + 128 = C_b
\end{cases}
$$
演变过程：
- 解方程方法：
- 高斯消元法
- 初中？小学？解方程的方法，逐个元素消除
- 凯莱的高斯消元法
- 用
数块表示线性方程组 - 变换写在横线上很不数学
数块乘法- 数块被命名为
矩阵
- 用
- 高斯消元法
高斯消元法到数块乘法的对应：
$$
\begin{pmatrix}
1 & 2 & 3 \\
3 & 4 & 5
\end{pmatrix}
\xrightarrow{r^{'}_2 = r_2 - 3r_1}
\begin{pmatrix}
1 & 2 & 3 \\
0 & -2 & -4
\end{pmatrix}
$$
对应
$$
\begin{pmatrix}
1 & 0 \\
-3 & 1
\end{pmatrix}
\begin{pmatrix}
1 & 2 & 3 \\
3 & 4 & 5
\end{pmatrix} =
\begin{pmatrix}
1 & 2 & 3 \\
0 & -2 & -4
\end{pmatrix}
$$
高斯消元法完全用数块乘法表示：
$$
\begin{pmatrix}
1 & -2 \\
0 & 1
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
0 & -\frac{1}{2}
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
-3 & 1
\end{pmatrix}
\begin{pmatrix}
1 & 2 & 3 \\
3 & 4 & 5
\end{pmatrix} =
\begin{pmatrix}
1 & 0 & -1 \\
0 & 1 & 2
\end{pmatrix}
$$
PCA
Compute with Python
Properties of complex arithmetic
- commutativity (交换性):
\(\alpha + \beta = \beta = \alpha \ \text{and} \ \alpha\beta = \beta\alpha \ \text{for all} \ \alpha,\beta \in \mathbf{C}\) - associativity (结合性):
\((\alpha + \beta)+\lambda = \alpha + (\beta + \lambda) \ \text{and} \ (\alpha\beta)\lambda = \alpha(\beta\lambda) \ \text{for all} \ \alpha, \beta, \lambda \in \mathbf{C} \) - identities (单位元):
\(\lambda + 0 = \lambda \ \text{and} \ \lambda 1 = \lambda \ \text{for all} \ \lambda \in \mathbf{C} \) - additive inverse (加法逆元):
\(\text{for every } \alpha \in \mathbf{C}, \ \text{there exists a unique } \beta \in \mathbf{C} \ \text{such that} \alpha + \beta = 0\) - multiplicative inverse (乘法逆元):
\(\text{for every } \alpha \in \mathbf{C}, \ \text{with } \alpha \ne 0, \ \text{there exists a unique } \beta \in \textbf{C} \text{ such that } \alpha \beta = 1 \) - distributive property (分配性):
\(\lambda (\alpha + \beta) = \lambda \alpha + \lambda \beta \ \text{for all} \ \alpha,\beta \in \mathbf{C}\)