【机器学习入门基础】Matrix

注：这里使用英文，感觉英文解释的更清楚

定义

Matrix: Pectangular array of numbers

dimension of matrix: the number of rows$\times$the number of column

$4\times 2$matrix($R^{4\times 2}$)
$$\left[\begin{array}{cc}1902& 191\\ 1371& 821\\ 949& 1437\\ 147& 1448\end{array}\right]$$
$2\times 3$matrix($R^{2\times 3}$)
$$\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$$

Matrix Element(entries of matrix)

$$A=\left[\begin{array}{cc}1902& 191\\ 1371& 821\\ 949& 1437\\ 147& 1448\end{array}\right]$$
$A_{ij}$:“i, j entry” in the $i^{th}$ row, $j^{th}$ column
$A_{11}$=1902 $A_{12}$=191 $A_{32}$=1437 $A_{41}$=147
$A_{43}$= undefined(error)(全部划掉)

Vector: An n$\times$ 1 matrix

4-dimensional vector ($R^4$)
$$y=\left[\begin{array}{c}460\\ 232\\ 315\\ 178\end{array}\right]$$
$y_i$=$i^{th}$ element
$y_1$=460 $y_2$=232 $y_3$=315 $y_4$=178

1-indexed vs 0-index

$$y=\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\\ y_4\end{array}\right]$$
$$y=\left[\begin{array}{c}{y}_0\\ y_1\\ y_2\\ y_3\end{array}\right]$$
Matices$\to$一般大写字母 Vector$\to$小写

Scalar$\to$ 对象是单个值，不是vector or matrix.(标量)

R$\to$the set of sccalar real numbers(标量实数集)
$R^n\to$ n-dimensional vectors of real numbers(实数n维向量)

Matrix Addition

$$\left[\begin{array}{cc}1& 0\\ 2& 5\\ 3& 1\end{array}\right]+\left[\begin{array}{cc}4& 0.5\\ 2& 5\\ 0& 2\end{array}\right]=\left[\begin{array}{cc}5& 0.5\\ 4& 10\\ 3& 2\end{array}\right]$$
$3\times 2matrix+3\times 2matrix=3\times 2matrix$
$$\left[\begin{array}{cc}1& 0\\ 2& 5\\ 3& 1\end{array}\right]+\left[\begin{array}{cc}4& 0.5\\ 2& 5\end{array}\right]=error(没有意义)$$

Scalar Multiplication

$$3\times \left[\begin{array}{cc}1& 0\\ 2& 5\\ 3& 1\end{array}\right]=\left[\begin{array}{cc}3& 0\\ 6& 15\\ 9& 3\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 2& 5\\ 3& 1\end{array}\right]\times 3$$
$$\left[\begin{array}{cc}4& 0\\ 6& 3\end{array}\right]/4=\frac{1}{4}\left[\begin{array}{cc}4& 0\\ 6& 3\end{array}\right]=\left[\begin{array}{cc}1& 0\\ \frac{3}{2}& \frac{3}{4}\end{array}\right]$$

Combination of Operands

$$3\times \left[\begin{array}{c}1\\ 4\\ 2\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 5\end{array}\right]-\left[\begin{array}{c}3\\ 0\\ 2\end{array}\right]/3=\left[\begin{array}{c}3\\ 12\\ 6\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 5\end{array}\right]-\left[\begin{array}{c}1\\ 3\\ \frac{2}{3}\end{array}\right]=\left[\begin{array}{c}2\\ 12\\ 10\frac{1}{3}\end{array}\right]$$

Matrix-vector multiplication

example

$$\left[\begin{array}{cc}1& 3\\ 4& 0\\ 2& 1\end{array}\right]\left[\begin{array}{c}1\\ 5\end{array}\right]=\left[\begin{array}{c}16\\ 4\\ 7\end{array}\right]$$
$1\times 1+3\times 5=16$
$4\times 1+0\times 5=4$
$2\times 1+1\times 5=7$
$(3\times 2matrix)(2\times 1matrix)=(3\times 1matrix)$

Detail

$$\left[\begin{array}{ccc}{a}_{11}& \dots & a_{1n}\\ ⋮& \ddots & ⋮\\ a_{m1}& \dots & a_{mn}\end{array}\right]\times \left[\begin{array}{c}{x}_1\\ ⋮\\ x_n\end{array}\right]=\left[\begin{array}{c}{y}_1\\ ⋮\\ y_m\end{array}\right]$$
To get $y_i$, multiply A’s $i^{th}$row with elements of vector x,and add them up.

example

$$\left[\begin{array}{cccc}1& 2& 1& 5\\ 0& 3& 4& 0\\ -1& -2& 0& 0\end{array}\right]\left[\begin{array}{c}1\\ 3\\ 2\\ 1\end{array}\right]=\left[\begin{array}{c}14\\ 13\\ -7\end{array}\right]$$
$1\times 1+2\times 3+1\times 2+5\times 1=14$
$0\times 1+3\times 3+0\times 2+4\times 1=13$
$-1\times 1+-2\times 3+0\times 2+0\times 1=-7$

house size:

$\{\begin{array}{l}2104\\ 1416\\ 1534\\ 852\end{array}$ $h_{\theta }(x)=-a0+0.25x$
$$\left[\begin{array}{cc}1& 2104\\ 1& 1416\\ 1& 1534\\ 1& 852\end{array}\right]\times \left[\begin{array}{c}-40\\ 0.25\end{array}\right]=\left[\begin{array}{c}-40\times 1+0.25\times 2104\\ -40\times 1+0.25\times 1416\\ -40\times 1+0.25\times 1534\\ -40\times 1+0.25\times 842\end{array}\right]\begin{array}{c}\to h_{\theta }(2104)\\ \to h_{\theta }(1416)\\ \to h_{\theta }(1534)\\ \to h_{\theta }(852)\end{array}$$
prediction=datamatrix$\ast$parameters

Matrix-matrix multiplication

$$\left[\begin{array}{ccc}1& 3& 2\\ 4& 0& 1\end{array}\right]\times \left[\begin{array}{cc}1& 3\\ 0& 1\\ 5& 2\end{array}\right]=\left[\begin{array}{cc}11& 10\\ 9& 14\end{array}\right]$$
$$\left[\begin{array}{ccc}1& 3& 2\\ 4& 0& 1\end{array}\right]\times \left[\begin{array}{c}1\\ 0\\ 5\end{array}\right]=\left[\begin{array}{c}11\\ 9\end{array}\right]$$
$$\left[\begin{array}{ccc}1& 3& 2\\ 4& 0& 1\end{array}\right]\times \left[\begin{array}{c}3\\ 1\\ 2\end{array}\right]=\left[\begin{array}{c}10\\ 14\end{array}\right]$$

Details:

$$\begin{gathered} m\times nmatrix+n\times omatrix=n\times omatrix \\ A\times B=C \end{gathered}$$
The $i^{th}$ column of the matrix C is obtained by mutiplying A withthe $i^{th}$ column of B(for i =1,2$\dots$, o)

example

$$\left[\begin{array}{cc}1& 3\\ 2& 5\end{array}\right]\times \left[\begin{array}{cc}0& 1\\ 3& 2\end{array}\right]=\left[\begin{array}{cc}9& 7\\ 15& 12\end{array}\right]$$

house size:

$\{\begin{array}{l}2104\\ 1416\\ 1534\\ 852\end{array}$
Have 3 competing hypothesis
$\{\begin{array}{l}1、h_{\theta }(x)=-40+0.25x\\ 1、h_{\theta }(x)=200+0.1x\\ 1、h_{\theta }(x)=-150+0.4x\end{array}$
$$\left[\begin{array}{cc}1& 2104\\ 1& 1416\\ 1& 1543\\ 1& 852\end{array}\right]\times \left[\begin{array}{ccc}-40& 200& -150\\ 0.25& 0.1& 0.4\end{array}\right]=\left[\begin{array}{ccc}482& 410& 692\\ 314& 342& 416\\ 344& 352& 464\\ 173& 285& 191\end{array}\right]$$

Matrix multiplication

$3\times 5=5\times 3$ “Commutative”
Let Aand B be matrices.Then in general.
$A\times B\ne B\times A$(not commutative)
$$\begin{gathered} E.g.\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]\times \left[\begin{array}{cc}0& 0\\ 2& 0\end{array}\right]=\left[\begin{array}{cc}2& 0\\ 0& 0\end{array}\right] \\ \ne \\ \left[\begin{array}{cc}0& 0\\ 2& 0\end{array}\right]\times \left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 2& 2\end{array}\right] \end{gathered}$$
$$\begin{array}{l}A\times B\\ m\times nn\times m\\ A\times B\sim m\times m\\ B\times A\sim n\times n\end{array}$$
Associative
$3\times 5\times 2=(3\times 5)\times 2=3\times (5\times 2)$
$$\begin{gathered} A\times B\times C\to (A\times B)\times C \\ \to A\times (B\times C)(sameanswer) \end{gathered}$$
Let D=B$\times$C. computeA$A\times D$
Let E=A$\times$B. computeA$E\times C$

Identity Matrix

Denoted $I$(or $I_{n\times n}$) \quad\quad\quad $1\sim identity$ \quad\quad\quad
$1\times z=z\times 1=z$ \quad\quad\quad for any z

Example of identity matrix:

$$\left[\begin{array}{c}1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& & & \\ & 1& & \\ & & \ddots & \\ & & & 1\end{array}\right](Informally)$$
For any matrix A
$$\begin{gathered} A\times I=I\times A=A \\ m\times nn\times nm\times mm\times nm\times n \end{gathered}$$
$$I_{n\times n}\begin{array}{l}Note:\\ A\times B\ne B\times Aingeneral\\ A\times I=I\times A✔\end{array}$$

Inverse and Transpose(逆运算以及转置矩阵)

1=“Identity” \quad $3(3^{-1})=1$ \quad $12(12^{-1})=1$
Mot all numbers have an inverse \quad $\to 0(0^{-1})$but $0^{-1}\to$undefined

Matrix invers

If A is an $m\times m$ matrix(square matrix{has the same row &column}),and if it has an inverse.
$\to A{A}^{-1}=A^{-1}A=I$ \quad $A逆矩阵A^{-1}$ \quad
$A=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]无逆矩阵$
$$E.g.\left[\begin{array}{cc}3& 4\\ 2& 16\end{array}\right]\left[\begin{array}{cc}0.4& -0.1\\ -0.05& 0.75\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=I_{2\times 2}$$
the way to caculate

caculate the inverse matrix

Octave

A=[3 4;2 16]
pinv(A)

Matlab

Matrices that don’t have an inverse are “singular” or “degenerate”

Matrix Transpose

example:

$A=\left[\begin{array}{ccc}1& 2& 0\\ 3& 5& 9\end{array}\right]$$A^T=\left[\begin{array}{cc}1& 3\\ 2& 5\\ 0& 9\end{array}\right]$$row\to column$
Let A be a $m\times n$ matrix,and let$B=A^T$
Then B is a $n\times m$ matrix, and
$B_{ij}=A_{ji}$
$B_{21}=A_{12}=2$
$B_{32}=A_{23}=9$

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