Matrix

矩阵基本概念 (Basic Concepts of Matrices)

矩阵定义 (Matrix Definition)

一个矩阵是一个矩形数组，包含 $m$ 行和 $n$ 列的元素，记作 $\mathbf{A}$：

A matrix is a rectangular array of elements with $m$ rows and $n$ columns, denoted as $\mathbf{A}$:

$$\mathbf{A}=\left(\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right)$$

矩阵类型 (Types of Matrices)

1. 零矩阵 (Zero Matrix): 所有元素均为零的矩阵，记作 $0$。 A matrix with all elements being zero, denoted as $0$.

2. 单位矩阵 (Identity Matrix): 对角线元素为 1，非对角线元素为 0 的方阵，记作 $\mathbf{I}$。 A square matrix with ones on the diagonal and zeros elsewhere, denoted as $\mathbf{I}$.

3. 对角矩阵 (Diagonal Matrix): 仅对角线元素可能为非零的方阵。 A square matrix where only the diagonal elements may be non-zero.

4. 对称矩阵 (Symmetric Matrix): 转置后等于自身的矩阵，即 $\mathbf{A}={\mathbf{A}}^T$。 A matrix that is equal to its transpose, i.e., $\mathbf{A}={\mathbf{A}}^T$.

5. 反对称矩阵 (Skew-Symmetric Matrix): 转置后等于其负的矩阵，即 $\mathbf{A}=-{\mathbf{A}}^T$。 A matrix that is equal to the negative of its transpose, i.e., $\mathbf{A}=-{\mathbf{A}}^T$.

6. 矩形矩阵 (Rectangular Matrix): 行数和列数不相等的矩阵。 A matrix with unequal number of rows and columns.

7. 方阵 (Square Matrix): 行数和列数相等的矩阵。 A matrix with an equal number of rows and columns.

矩阵运算 (Matrix Operations)

1. 矩阵加法 (Matrix Addition)

   两个同维数矩阵的对应元素相加：

   Adding corresponding elements of two matrices of the same dimension:

   $$\mathbf{A}+\mathbf{B}=\left(\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right)+\left(\begin{array}{cccc}{b}_{11}& b_{12}& \cdots & b_{1n}\\ b_{21}& b_{22}& \cdots & b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ b_{m1}& b_{m2}& \cdots & b_{mn}\end{array}\right)=\left(\begin{array}{cccc}{a}_{11}+b_{11}& a_{12}+b_{12}& \cdots & a_{1n}+b_{1n}\\ a_{21}+b_{21}& a_{22}+b_{22}& \cdots & a_{2n}+b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}+b_{m1}& a_{m2}+b_{m2}& \cdots & a_{mn}+b_{mn}\end{array}\right)$$

$$2.\ast \ast 矩阵减法(MatrixSubtraction)\ast \ast 两个同维数矩阵的对应元素相减：Subtractingcorrespondingelementsoftwomatricesofthesamedimension:$$

\mathbf{A} - \mathbf{B} = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}

\begin{pmatrix} b_{11} & b_{12} & \cdots & b_{1n} \ b_{21} & b_{22} & \cdots & b_{2n} \ \vdots & \vdots & \ddots & \vdots \ b_{m1} & b_{m2} & \cdots & b_{mn} \end{pmatrix}

\begin{pmatrix}

a_{11} - b_{11} & a_{12} - b_{12} & \cdots & a_{1n} - b_{1n} \

a_{21} - b_{21} & a_{22} - b_{22} & \cdots & a_{2n} - b_{2n} \

\vdots & \vdots & \ddots & \vdots \

a_{m1} - b_{m1} & a_{m2} - b_{m2} & \cdots & a_{mn} - b_{mn}

\end{pmatrix}

3. **数乘矩阵 (Scalar Multiplication)** 矩阵的每个元素乘以一个标量$c$： Multiplying each element of a matrix by a scalar $c$:

c\mathbf{A} = c \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}

\begin{pmatrix}

ca_{11} & ca_{12} & \cdots & ca_{1n} \

ca_{21} & ca_{22} & \cdots & ca_{2n} \

\vdots & \vdots & \ddots & \vdots \

ca_{m1} & ca_{m2} & \cdots & ca_{mn}

\end{pmatrix}

4. **矩阵乘法 (Matrix Multiplication)** 矩阵$\mathbf{A}$与矩阵$\mathbf{B}$的乘积定义为： The product of matrix $\mathbf{A}$ and matrix $\mathbf{B}$ is defined as:

\mathbf{C} = \mathbf{A} \mathbf{B}, \quad \text{其中} \ \mathbf{C} = \begin{pmatrix}

c_{11} & c_{12} & \cdots & c_{1p} \

c_{21} & c_{22} & \cdots & c_{2p} \

\vdots & \vdots & \ddots & \vdots \

c_{m1} & c_{m2} & \cdots & c_{mp}

\end{pmatrix}

其中 $c_{ij}$ 为第 $i$ 行与第 $j$ 列的元素的乘积和： where $c_{ij}$ is the sum of the products of the elements from the $i$-th row and the $j$-th column:

c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}

## 矩阵的性质 (Properties of Matrices) 1. **交换律 (Commutative Property)**: 矩阵加法满足交换律，即$\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}$。 Matrix addition is commutative, i.e., $\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}$. 2. **结合律 (Associative Property)**: 矩阵加法和数乘满足结合律，即$(\mathbf{A} + \mathbf{B}) + \mathbf{C} = \mathbf{A} + (\mathbf{B} + \mathbf{C})$和$c(\mathbf{A} \mathbf{B}) = (c\mathbf{A}) \mathbf{B}$。 Matrix addition and scalar multiplication are associative, i.e., $(\mathbf{A} + \mathbf{B}) + \mathbf{C} = \mathbf{A} + (\mathbf{B} + \mathbf{C})$ and $c(\mathbf{A} \mathbf{B}) = (c\mathbf{A}) \mathbf{B}$. 3. **分配律 (Distributive Property)**: 矩阵乘法对矩阵加法满足分配律，即$\mathbf{A}(\mathbf{B} + \mathbf{C}) = \mathbf{A}\mathbf{B} + \mathbf{A}\mathbf{C}$和$(\mathbf{A} + \mathbf{B})\mathbf{C} = \mathbf{A}\mathbf{C} + \mathbf{B}\mathbf{C}$。 Matrix multiplication is distributive over matrix addition, i.e., $\mathbf{A}(\mathbf{B} + \mathbf{C}) = \mathbf{A}\mathbf{B} + \mathbf{A}\mathbf{C}$ and $(\mathbf{A} + \mathbf{B})\mathbf{C} = \mathbf{A}\mathbf{C} + \mathbf{B}\mathbf{C}$. ## 逆矩阵 (Inverse Matrix) 一个方阵$\mathbf{A}$的逆矩阵$\mathbf{A}^{-1}$满足以下条件： The inverse of a square matrix $\mathbf{A}$, denoted as $\mathbf{A}^{-1}$, satisfies:

\mathbf{A} \mathbf{A}^{-1} = \mathbf{A}^{-1} \mathbf{A} = \mathbf{I}

- 只有当$\mathbf{A}$是非奇异矩阵（即$\det(\mathbf{A}) \neq 0$）时，$\mathbf{A}$才存在逆矩阵。 - The inverse matrix $\mathbf{A}^{-1}$ exists if and only if $\mathbf{A}$ is a non-singular matrix (i.e., $\det(\mathbf{A}) \neq 0$). ## 伪逆矩阵 (Pseudo-Inverse Matrix) 矩阵$\mathbf{A}$的伪逆矩阵$\mathbf{A}^+$是最小二乘解的一种扩展，常用于求解无法求逆的矩阵。 The pseudo-inverse matrix $\mathbf{A}^+$ of a matrix $\mathbf{A}$ is an extension for finding least-squares solutions, often used when $\mathbf{A}$ is not invertible. - $\mathbf{A}^+$ 是唯一的，满足以下四个条件： - $\mathbf{A} \mathbf{A}^+ \mathbf{A} = \mathbf{A}$ - $\mathbf{A}^+ \mathbf{A} \mathbf{A}^+ = \mathbf{A}^+$ - $(\mathbf{A} \mathbf{A}^+)^T = \mathbf{A} \mathbf{A}^+$ - $(\mathbf{A}^+ \mathbf{A})^T = \mathbf{A}^+ \mathbf{A}$ - $\mathbf{A}^+$ is unique and satisfies the following four conditions: - $\mathbf{A} \mathbf{A}^+ \mathbf{A} = \mathbf{A}$ - $\mathbf{A}^+ \mathbf{A} \mathbf{A}^+ = \mathbf{A}^+$ - $(\mathbf{A} \mathbf{A}^+)^T = \mathbf{A} \mathbf{A}^+$ - $(\mathbf{A}^+ \mathbf{A})^T = \mathbf{A}^+ \mathbf{A}$ ## 矩阵的迹 (Trace of a Matrix) 矩阵$\mathbf{A}$的迹是其对角线元素之和，记作$\mathrm{tr}(\mathbf{A})$： The trace of a matrix $\mathbf{A}$, denoted as $\mathrm{tr}(\mathbf{A})$, is the sum of its diagonal elements:

\mathrm{tr}(\mathbf{A}) = \sum_{i=1}^{n} a_{ii}

- 迹的性质： - $\mathrm{tr}(\mathbf{A} + \mathbf{B}) = \mathrm{tr}(\mathbf{A}) + \mathrm{tr}(\mathbf{B})$ - $\mathrm{tr}(c\mathbf{A}) = c \cdot \mathrm{tr}(\mathbf{A})$ - $\mathrm{tr}(\mathbf{A} \mathbf{B}) = \mathrm{tr}(\mathbf{B} \mathbf{A})$ - Properties of the trace: - $\mathrm{tr}(\mathbf{A} + \mathbf{B}) = \mathrm{tr}(\mathbf{A}) + \mathrm{tr}(\mathbf{B})$ - $\mathrm{tr}(c\mathbf{A}) = c \cdot \mathrm{tr}(\mathbf{A})$ - $\mathrm{tr}(\mathbf{A} \mathbf{B}) = \mathrm{tr}(\mathbf{B} \mathbf{A})$ ## 矩阵的秩 (Rank of a Matrix) 矩阵$\mathbf{A}$的秩是其行向量（或列向量）线性无关的最大数目： The rank of a matrix $\mathbf{A}$ is the maximum number of linearly independent row vectors (or column vectors): - 秩的性质： - $\mathrm{rank}(\mathbf{A}) = \mathrm{rank}(\mathbf{A}^T)$ - 若$\mathbf{A}$是$m \times n$矩阵，则$\mathrm{rank}(\mathbf{A}) \leq \min(m, n)$ - $\mathrm{rank}(\mathbf{A} \mathbf{B}) \leq \min(\mathrm{rank}(\mathbf{A}), \mathrm{rank}(\mathbf{B}))$ - Properties of the rank: - $\mathrm{rank}(\mathbf{A}) = \mathrm{rank}(\mathbf{A}^T)$ - If $\mathbf{A}$ is an $m \times n$ matrix, then $\mathrm{rank}(\mathbf{A}) \leq \min(m, n)$ - $\mathrm{rank}(\mathbf{A} \mathbf{B}) \leq \min(\mathrm{rank}(\mathbf{A}), \mathrm{rank}(\mathbf{B}))$ ## 矩阵的行列式 (Determinant of a Matrix) 一个方阵$\mathbf{A}$的行列式，记作$\det(\mathbf{A})$，是一个标量，反映了矩阵是否可逆： The determinant of a square matrix $\mathbf{A}$, denoted as $\det(\mathbf{A})$, is a scalar that indicates whether the matrix is invertible: - $\det(\mathbf{A}) = 0$表示矩阵$\mathbf{A}$是奇异矩阵，不可逆。 - $\det(\mathbf{A}) \neq 0$表示矩阵$\mathbf{A}$是非奇异矩阵，可逆。 - $\det(\mathbf{A}) = 0$ indicates that the matrix $\mathbf{A}$ is singular and not invertible. - $\det(\mathbf{A}) \neq 0$ indicates that the matrix $\mathbf{A}$ is non-singular and invertible. - 行列式的性质： - $\det(\mathbf{A} \mathbf{B}) = \det(\mathbf{A}) \det(\mathbf{B})$ - $\det(\mathbf{A}^T) = \det(\mathbf{A})$ - $\det(c\mathbf{A}) = c^n \det(\mathbf{A})$，其中$c$是一个标量，$\mathbf{A}$是$n \times n$矩阵 - Properties of the determinant: - $\det(\mathbf{A} \mathbf{B}) = \det(\mathbf{A}) \det(\mathbf{B})$ - $\det(\mathbf{A}^T) = \det(\mathbf{A})$ - $\det(c\mathbf{A}) = c^n \det(\mathbf{A})$, where $c$ is a scalar and $\mathbf{A}$ is an $n \times n$ matrix