3. 向量和矩阵

2020-08-15 本文已影响0人玄语梨落

Matrix and Vector

What is Matrix and Vector

Matrix: Rectangular array of numbers.

$R^{2\times3}$ ：一个2乘3的矩阵
Dimension of matrix: number of rows x number of clumns.
Matrix Elements(entries of matrix): $A_{ij}= "i,j\ entry"\ in\ the\ i^{th}\ row, j^{th}\ column.$

Vector: An n x 1 matrix.
n dimension vector

$R^4$
1-indexed vs 0-indexed
一般使用1-indexed 在机器学习中会使用0-indexed 。

Addition and Scalar Multiplication

Matrix Addition(or Subtration)

只有相同维度的矩阵可以相加。新的矩阵与原来维度相同，仅仅是对应部分的相加。

Scalar(real nmuber) Multiplication（or division）

实数和矩阵对应位置的数相乘得到新的矩阵。

Matrix-vector multiplication

m $\times$ n matrix multiply n $\times$ 1 matrix equals a m-dimensional vector

To get $y_i$ , multiply A's $i^{th}$ row with elemets of vector x, and add them up.

We can use matrix and vector to calculate prediction at the same time or with only one line code.

Matrix-Matrix multiplication

m $\times$ n matrix multiply n $\times$ o equals a m $\times$ o matrix.

The $i^{th}$ column of the matrixC is obtained by multiplying A with the $i^{th}$ column of B.

Matrix multiplication properties

not commutative（不满足交换律）
associative(结合律)

Identity Matrix:
Denoted I(or $I_{n\times n}$ ).
$I\times T$ = $T\times I$ = $T$
Informally:
对角线上都是1，其他位置都是0
For any matrix A: $A\times I=I\times A=A$

Inverse and Transpose

Not all number have an inverse.

Matrix inverse: If A is an m $\times$ m matrix, and if it has an inverse, $AA^{-1}=A^{-1}A=I$

Square matrix: m $\times$ m matrix.
A matirx which have inverse must be a square matrix.
Matrix that don't have an inverse are "singular"（奇异矩阵） or "degenerate"（退化矩阵）.

Matrix Transpose:Let A be an $m\times n$ matrix, and let $B=A^T$ . Then B is an $n\times m$ matrix, and $B_{ij}=A_{ji}$ .