Matrix (矩阵)

Definition

In general, a $m \times n$ matrix is represented as

$$A = \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{bmatrix}$$

where $m$ is the number of rows and $n$ is the number of columns.

We can also write it as a function:

$$A: \set{(i, j) | 1 \leq i \leq m, 1 \leq j \leq n} \to R$$

We write the output by $A(i, j) = a_{i, j}$, where the input $(i, j)$ represents the $i$th row and $j$th column, and the output $a_{i,j}$ is the number at the $i$th row and $j$th column.

Every $a_{i, j}$ is called an entry of the matrix $A$.

Special Matrices

Square Matrix (方阵)

A matrix is called a square matrix if the number of rows equals the number of columns, i.e., $m = n$.

Zero Matrix (零矩阵)

A matrix is called a zero matrix if all of its entries are zero.

A zero matrix is a diagonal matrix.

Diagonal Matrix (对角矩阵)

A square matrix is called a diagonal matrix if all of its entries off the main diagonal are zero.

$$A = \begin{bmatrix} a_{1,1} & 0 & \cdots & 0 \\ 0 & a_{2,2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_{n,n} \end{bmatrix}$$

where the entries $a_{1,1}, a_{2,2}, \cdots, a_{n,n}$ are on the main diagonal of the matrix $A$.

Triangular Matrix (三角矩阵)

A square matrix is called an upper triangular matrix if all of its entries below the main diagonal are zero, and is called a lower triangular matrix if all of its entries above the main diagonal are zero.

For example, the following is an upper triangular matrix:

$$A = \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ 0 & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_{n,n} \end{bmatrix}$$

Identity Matrix (单位矩阵)

The identity matrix is a square matrix in which all the entries on the main diagonal are $1$ and all other entries are $0$:

$$I_n = \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{bmatrix}_{n \times n}$$

An identity matrix is a diagonal matrix.

Matrix Operations

Matrix Addition (矩阵加法)

If $A$ and $B$ are two $m \times n$ matrices, then their sum is the $m \times n$ matrix obtained by adding corresponding entries:

$$A + B = \begin{bmatrix} a_{1,1} + b_{1,1} & a_{1,2} + b_{1,2} & \cdots & a_{1,n} + b_{1,n} \\ a_{2,1} + b_{2,1} & a_{2,2} + b_{2,2} & \cdots & a_{2,n} + b_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m,1} + b_{m,1} & a_{m,2} + b_{m,2} & \cdots & a_{m,n} + b_{m,n} \end{bmatrix}$$

Adding two matrices is only defined when they are of the same size ($m \times n$).

Scalar Multiplication (数乘)

If $A$ is an $m \times n$ matrix and $c$ is a scalar, then the scalar multiple of $A$ by $c$ is the $m \times n$ matrix obtained by multiplying each entry of $A$ by $c$:

$$cA = \begin{bmatrix} ca_{1,1} & ca_{1,2} & \cdots & ca_{1,n} \\ ca_{2,1} & ca_{2,2} & \cdots & ca_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ ca_{m,1} & ca_{m,2} & \cdots & ca_{m,n} \end{bmatrix}$$

Matrix Multiplication (矩阵乘法) and Power of a Matrix (矩阵的幂)

See 4. Matrix Multiplication.

Transpose (转置)

The transpose of an $m \times n$ matrix $A$ is the $n \times m$ matrix obtained by interchanging the rows and columns of $A$:

$$A^T = \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{bmatrix}^T = \begin{bmatrix} a_{1,1} & a_{2,1} & \cdots & a_{m,1} \\ a_{1,2} & a_{2,2} & \cdots & a_{m,2} \\ \vdots & \vdots & \ddots & \vdots \\ a_{1,n} & a_{2,n} & \cdots & a_{m,n} \end{bmatrix}$$

An Example

$$\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}$$

Copyright

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