Signed-off-by: sairate <sairate@sina.cn>
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@ -22,5 +22,6 @@ $ mkdocs serve # 在本地启动服务器预览
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我们欢迎和鼓励所有人贡献自己的知识和经验！
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- 在工单中提交需求
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- 联系我:sairate@sina.cn
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@ -5,78 +5,64 @@
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**例如：**
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设有两个矩阵 \( A \) 和 \( B \)，如下：
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\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \]
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$$
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A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}
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$$
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**加法：**
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\[ A + B = \begin{pmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix} \]
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$$
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A + B = \begin{pmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix}
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$$
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**减法：**
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\[ A - B = \begin{pmatrix} 1-5 & 2-6 \\ 3-7 & 4-8 \end{pmatrix} = \begin{pmatrix} -4 & -4 \\ -4 & -4 \end{pmatrix} \]
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$$
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A - B = \begin{pmatrix} 1-5 & 2-6 \\ 3-7 & 4-8 \end{pmatrix} = \begin{pmatrix} -4 & -4 \\ -4 & -4 \end{pmatrix}
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$$
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### 2. 矩阵乘法
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矩阵乘法是矩阵运算中最常见的一种操作。矩阵 \( A \) 的列数必须等于矩阵 \( B \) 的行数。结果矩阵的维度为 \( A \) 的行数和 \( B \) 的列数。
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**例如：**
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设有矩阵 \( A \) 和 \( B \)，如下：
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\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \]
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$$
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A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}
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$$
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**乘法：**
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\[ A \times B = \begin{pmatrix} 1\times5 + 2\times7 & 1\times6 + 2\times8 \\ 3\times5 + 4\times7 & 3\times6 + 4\times8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix} \]
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$$
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A \times B = \begin{pmatrix} 1\times5 + 2\times7 & 1\times6 + 2\times8 \\ 3\times5 + 4\times7 & 3\times6 + 4\times8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}
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$$
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### 3. 矩阵转置
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矩阵的转置是将矩阵的行和列互换。对于矩阵 \( A \) 的转置矩阵记作 \( A^T \)。
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**例如：**
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设矩阵 \( A \) 如下：
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\[ A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \]
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$$
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A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}
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$$
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**转置：**
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\[ A^T = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix} \]
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$$
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A^T = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}
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$$
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### 4. 单位矩阵和逆矩阵
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- **单位矩阵** \( I \) 是一个对角线上全为1，其余元素为0的方阵。它在矩阵乘法中起着类似于数字1在数乘中的作用。
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**例如：**
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\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]
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$$
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I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
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$$
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- **逆矩阵** \( A^{-1} \) 是一个矩阵，使得 \( A \times A^{-1} = I \)。只有方阵（行数等于列数）并且行列式不为零的矩阵才有逆矩阵。
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### 5. 矩阵行列式
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行列式是一个标量值，可以用于判断矩阵是否可逆。对于 \( 2 \times 2 \) 的矩阵 \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \)，行列式 \( \det(A) \) 计算如下：
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\[ \det(A) = ad - bc \]
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$$
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\det(A) = ad - bc
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$$
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如果行列式不为零，矩阵 \( A \) 可逆。
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### 6. Python 中的矩阵运算
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在 Python 中，NumPy 是一个常用的库来进行矩阵运算。
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```python
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import numpy as np
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# 定义矩阵
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A = np.array([[1, 2], [3, 4]])
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B = np.array([[5, 6], [7, 8]])
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# 矩阵加法
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C = A + B
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# 矩阵乘法
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D = np.dot(A, B)
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# 矩阵转置
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E = A.T
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# 矩阵的行列式
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det_A = np.linalg.det(A)
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# 矩阵的逆
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inv_A = np.linalg.inv(A)
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print("矩阵加法结果：\n", C)
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print("矩阵乘法结果：\n", D)
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print("矩阵转置结果：\n", E)
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print("矩阵的行列式：", det_A)
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print("矩阵的逆：\n", inv_A)
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```
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这些基本运算为许多复杂的线性代数问题打下了基础。如果你对更高级的矩阵运算或应用有兴趣，可以继续深入学习特征值分解、奇异值分解等内容。
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这些基本运算为许多复杂的线性代数问题打下了基础。如果你对更高级的矩阵运算或应用有兴趣，可以继续深入学习特征值分解、奇异值分解等内容。
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@ -1,11 +1,10 @@
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document$.subscribe(({ body }) => {
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renderMathInElement(body, {
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delimiters: [
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{ left: "$$", right: "$$", display: true },
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{ left: "$", right: "$", display: false },
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{ left: "\\(", right: "\\)", display: false },
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{ left: "\\[", right: "\\]", display: true },
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{ left: "\[", right: "\]", display: true }
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{ left: "$$", right: "$$", display: true }, // 用于块级显示的数学公式
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{ left: "$", right: "$", display: false }, // 用于行内显示的数学公式
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{ left: "\\(", right: "\\)", display: false }, // 行内数学公式的另一种表示法
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{ left: "\\[", right: "\\]", display: true } // 块级数学公式的另一种表示法
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],
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})
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})
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})
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@ -1,7 +1,16 @@
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// markdown_extensions:
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// - pymdownx.arithmatex:
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// generic: true
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//
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// extra_javascript:
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// - javascripts/mathjax.js
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// - https://unpkg.com/mathjax@3/es5/tex-mml-chtml.js
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window.MathJax = {
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tex: {
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inlineMath: [["\\(", "\\)"]],
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displayMath: [["\\[", "\\]"]],
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processEscapes: true,
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processEnvironments: true
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},
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options: {
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ignoreHtmlClass: ".*|",
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processHtmlClass: "arithmatex"
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}
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};
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document$.subscribe(() => {
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MathJax.typesetPromise()
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})
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@ -33,18 +33,30 @@ markdown_extensions:
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- pymdownx.arithmatex:
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generic: true
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extra_javascript:
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- javascripts/katex.js
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- https://unpkg.com/katex@0/dist/katex.min.js
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- https://unpkg.com/katex@0/dist/contrib/auto-render.min.js
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extra_css:
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- https://unpkg.com/katex@0/dist/katex.min.css
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- themes/css/custom.css
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- themes/css/simpleLightbox.min.css
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- themes/css/pied_piper.css
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- https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css
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#markdown_extensions:
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# - pymdownx.arithmatex:
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# generic: true
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#
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#extra_javascript:
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# - javascripts/mathjax.js
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# - https://unpkg.com/mathjax@3/es5/tex-mml-chtml.js
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extra_javascript:
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- themes/js/custom.js
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- themes/js/simpleLightbox.min.js
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- themes/js/optionalConfig.js
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- themes/js/mermaidloader.js
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- themes/js/umlconvert.js
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- themes/js/mathjax.js
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- themes/js/katex.js
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- https://cdn.jsdelivr.net/npm/mermaid@10.6.1/dist/mermaid.min.js
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- https://cdnjs.cloudflare.com/ajax/libs/flowchart/1.17.1/flowchart.min.js
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- https://cdnjs.cloudflare.com/ajax/libs/raphael/2.3.0/raphael.min.js
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- https://cdnjs.cloudflare.com/ajax/libs/underscore.js/1.13.6/underscore-min.js
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- https://cdn.jsdelivr.net/npm/@mermaid-js/mermaid-mindmap@9.3.0/dist/diagram-definition.0faef4c2.min.js
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- https://cdn.jsdelivr.net/npm/markdown-it-plantuml@1.4.1/index.min.js
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- https://cdnjs.cloudflare.com/ajax/libs/webfont/1.6.28/webfontloader.js
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- https://cdnjs.cloudflare.com/ajax/libs/mathjax/3.2.0/es5/tex-mml-chtml.js
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- https://cdnjs.cloudflare.com/ajax/libs/mathjax/3.2.0/es5/tex-chtml.js
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- https://cdnjs.cloudflare.com/ajax/libs/mathjax/3.2.0/es5/tex-chtml-full.js
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- https://cdnjs.cloudflare.com/ajax/libs/mathjax/3.2.0/es5/tex-svg-full.js
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- https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.js
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- https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/contrib/auto-render.min.js
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