Matrix

• 定义
• • Matrix: Pectangular array of numbers
  • • dimension of matrix: the number of rows$\times$the number of column
    • Matrix Element(entries of matrix)
  • Vector: An n$\times$ 1 matrix
  • 1-indexed vs 0-index
  • • Scalar$\to$ 对象是单个值，不是vector or matrix.(标量)
• Matrix Addition
• Scalar Multiplication
• Combination of Operands
• Matrix-vector multiplication
• • example
  • Detail
  • example
  • house size:
• Matrix-matrix multiplication
• • Details:
  • example
  • house size:
• Matrix multiplication
• Identity Matrix
• • Example of identity matrix:
• Inverse and Transpose(逆运算以及转置矩阵)
• • Matrix invers
  • • caculate the inverse matrix
    • • Octave
      • Matlab
  • Matrix Transpose
  • • example:
• the end

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( R4×2 R^{4\times2} R4×2)
[ 1902 191 1371 821 949 1437 147 1448 ] \begin{bmatrix} {1902}&{191}\\ {1371}&{821}\\ {949}&{1437}\\ {147}&{1448}\\ \end{bmatrix} ⎣⎢⎢⎡​19021371949147​19182114371448​⎦⎥⎥⎤​
$2\times 3$matrix( R2×3 R^{2\times3} R2×3)
[ 1 2 3 4 5 6 ] \begin{bmatrix} {1}&{2}&{3}\\ {4}&{5}&{6}\\ \end{bmatrix} [14​25​36​]

Matrix Element(entries of matrix)

A =[ 1902 191 1371 821 949 1437 147 1448 ] A=\left [ \begin{matrix} {1902}&{191}\\ {1371}&{821}\\ {949}&{1437}\\ {147}&{1448} \end{matrix} \right] A=⎣⎢⎢⎡​19021371949147​19182114371448​⎦⎥⎥⎤​
Aij A_{ij} Aij​:“i, j entry” in the ith i^{th} ith row, jth j^{th} jth column
A 11 A_{11} A11​=1902 A 12 A_{12} A12​=191 A 32 A_{32} A32​=1437 A 41 A_{41} A41​=147
A43 A_{43} A43​= undefined(error)(全部划掉)

Vector: An n$\times$ 1 matrix

4-dimensional vector ( R 4 R^4 R4)
y =[ 460 232 315 178 ] y=\left [ \begin{matrix} {460}\\ {232}\\ {315}\\ {178} \end{matrix} \right] y=⎣⎢⎢⎡​460232315178​⎦⎥⎥⎤​
y i y_i yi​= ith i^{th} ith element
y 1 y_1 y1​=460 y 2 y_2 y2​=232 y 3 y_3 y3​=315 y 4 y_4 y4​=178

1-indexed vs 0-index

y =[ y 1 y 2 y 3 y 4 ] y=\left [ \begin{matrix} {y_1}\\ {y_2}\\ {y_3}\\ {y_4} \end{matrix} \right] y=⎣⎢⎢⎡​y1​y2​y3​y4​​⎦⎥⎥⎤​
y =[ y 0 y 1 y 2 y 3 ] y=\left [ \begin{matrix} {y_0}\\ {y_1}\\ {y_2}\\ {y_3} \end{matrix} \right] y=⎣⎢⎢⎡​y0​y1​y2​y3​​⎦⎥⎥⎤​
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 → R^n\to Rn→ n-dimensional vectors of real numbers(实数n维向量)

Matrix Addition

[ 1 0 2 5 3 1 ] +[ 4 0.5 2 5 0 2 ] =[ 5 0.5 4 10 3 2 ] \begin{bmatrix} {1}&{0}\\ {2}&{5}\\ {3}&{1}\\ \end{bmatrix}+\begin{bmatrix} {4}&{0.5}\\ {2}&{5}\\ {0}&{2}\\ \end{bmatrix}=\begin{bmatrix} {5}&{0.5}\\ {4}&{10}\\ {3}&{2}\\ \end{bmatrix} ⎣⎡​123​051​⎦⎤​+⎣⎡​420​0.552​⎦⎤​=⎣⎡​543​0.5102​⎦⎤​
$3\times 2matrix+3\times 2matrix=3\times 2matrix$
[ 1 0 2 5 3 1 ] +[ 4 0.5 2 5 ] = e r r o r ( 没 有 意 义 ) \begin{bmatrix} {1}&{0}\\ {2}&{5}\\ {3}&{1}\\ \end{bmatrix}+\begin{bmatrix} {4}&{0.5}\\ {2}&{5}\\ \end{bmatrix}=error( 没有意义) ⎣⎡​123​051​⎦⎤​+[42​0.55​]=error(没有意义)

Scalar Multiplication

3 ×[ 1 0 2 5 3 1 ] =[ 3 0 6 15 9 3 ] =[ 1 0 2 5 3 1 ] × 3 3\times\begin{bmatrix} {1}&{0}\\ {2}&{5}\\ {3}&{1}\\ \end{bmatrix}=\begin{bmatrix} {3}&{0}\\ {6}&{15}\\ {9}&{3}\\ \end{bmatrix}=\begin{bmatrix} {1}&{0}\\ {2}&{5}\\ {3}&{1}\\ \end{bmatrix}\times3 3×⎣⎡​123​051​⎦⎤​=⎣⎡​369​0153​⎦⎤​=⎣⎡​123​051​⎦⎤​×3
[ 4 0 6 3 ] / 4 =14 [ 4 0 6 3 ] =[ 1 0 3 2 3 4 ] \begin{bmatrix} {4}&{0}\\ {6}&{3}\\ \end{bmatrix}/4=\frac{1}{4}\begin{bmatrix} {4}&{0}\\ {6}&{3}\\ \end{bmatrix}=\begin{bmatrix} {1}&{0}\\ {\frac{3}{2}}&{\frac{3}{4}}\\ \end{bmatrix} [46​03​]/4=41​[46​03​]=[123​​043​​]

Combination of Operands

3 ×[ 1 4 2 ] +[ 0 0 5 ] −[ 3 0 2 ] / 3 =[ 3 12 6 ] +[ 0 0 5 ] −[ 1 3 2 3 ] =[ 2 12 10 1 3 ] 3\times\begin{bmatrix} {1}\\ {4}\\ {2} \end{bmatrix}+\begin{bmatrix} {0}\\ {0}\\ {5} \end{bmatrix}-\begin{bmatrix} {3}\\ {0}\\ {2} \end{bmatrix}/3=\begin{bmatrix} {3}\\ {12}\\ {6} \end{bmatrix}+\begin{bmatrix} {0}\\ {0}\\ {5} \end{bmatrix}-\begin{bmatrix} {1}\\ {3}\\ {\frac{2}{3}} \end{bmatrix}=\begin{bmatrix} {2}\\ {12}\\ {10\frac{1}{3}} \end{bmatrix} 3×⎣⎡​142​⎦⎤​+⎣⎡​005​⎦⎤​−⎣⎡​302​⎦⎤​/3=⎣⎡​3126​⎦⎤​+⎣⎡​005​⎦⎤​−⎣⎡​1332​​⎦⎤​=⎣⎡​2121031​​⎦⎤​

Matrix-vector multiplication

example

[ 1 3 4 0 2 1 ] [ 1 5 ] =[ 16 4 7 ] \begin{bmatrix} {1}&{3}\\ {4}&{0}\\ {2}&{1} \end{bmatrix}\begin{bmatrix} {1}\\ {5}\\ \end{bmatrix}=\begin{bmatrix} {16}\\ {4}\\ {7} \end{bmatrix} ⎣⎡​142​301​⎦⎤​[15​]=⎣⎡​1647​⎦⎤​
$1\times 1+3\times 5=16$
$4\times 1+0\times 5=4$
$2\times 1+1\times 5=7$
$(3\times 2matrix)(2\times 1matrix)=(3\times 1matrix)$

Detail

[ a 11 … a 1 n ⋮⋱ ⋮a m 1 … a m n ] ×[ x 1 ⋮x n ] =[ y 1 ⋮y m ] \begin{bmatrix} {a_{11}}&\ldots&{a_{1n}} \\ {\vdots}&\ddots&{\vdots} \\ {a_{m1}}&\ldots&{a_{mn}} \\ \end{bmatrix}\times \begin{bmatrix} {x_{1}}\\ \vdots\\ {x_n} \end{bmatrix}=\begin{bmatrix} {y_1}\\ {\vdots}\\ {y_m} \end{bmatrix} ⎣⎢⎡​a11​⋮am1​​…⋱…​a1n​⋮amn​​⎦⎥⎤​×⎣⎢⎡​x1​⋮xn​​⎦⎥⎤​=⎣⎢⎡​y1​⋮ym​​⎦⎥⎤​
To get y i y_i yi​, multiply A’s ith i^{th} ithrow with elements of vector x,and add them up.

example

[ 1 2 1 5 0 3 4 0 − 1 − 2 0 0 ] [ 1 3 2 1 ] =[ 14 13 − 7 ] \begin{bmatrix} {1}&2&1&5\\ {0}&3&4&0\\ {-1}&{-2}&0&0 \end{bmatrix}\begin{bmatrix} 1\\3\\2\\1 \end{bmatrix}=\begin{bmatrix} 14\\13\\{-7} \end{bmatrix} ⎣⎡​10−1​23−2​140​500​⎦⎤​⎣⎢⎢⎡​1321​⎦⎥⎥⎤​=⎣⎡​1413−7​⎦⎤​
$1\times 1+2\times 3+1\times 2+5\times 1=14$
$0\times 1+3\times 3+0\times 2+4\times 1=13$
$-1\times 1+-2\times 3+0\times 2+0\times 1=-7$

house size:

{ 2104 1416 1534 852 \begin{cases} 2104\\ 1416\\ 1534\\ 852 \end{cases} ⎩⎪⎪⎪⎨⎪⎪⎪⎧​210414161534852​ h θ ( x ) = − a 0 + 0.25 x h_\theta(x)=-a0+0.25x hθ​(x)=−a0+0.25x
[ 1 2104 1 1416 1 1534 1 852 ] ×[ − 40 0.25 ] =[ − 40 × 1 + 0.25 × 2104 − 40 × 1 + 0.25 × 1416 − 40 × 1 + 0.25 × 1534 − 40 × 1 + 0.25 × 842 ] → h θ ( 2104 ) → h θ ( 1416 ) → h θ ( 1534 ) → h θ ( 852 ) \begin{bmatrix} {1}&{2104}\\ {1}&{1416}\\ {1}&{1534}\\ {1}&{852}\\ \end{bmatrix}\times\begin{bmatrix} {-40}\\ {0.25}\\ \end{bmatrix}=\begin{bmatrix} {{-40}\times1+0.25\times2104}\\ {{-40}\times1+0.25\times1416}\\ {{-40}\times1+0.25\times1534}\\ {{-40}\times1+0.25\times842}\\ \end{bmatrix} \begin{matrix} \to h_\theta(2104)\\ \to h_\theta(1416)\\ \to h_\theta(1534)\\ \to h_\theta(852) \end{matrix} ⎣⎢⎢⎡​1111​210414161534852​⎦⎥⎥⎤​×[−400.25​]=⎣⎢⎢⎡​−40×1+0.25×2104−40×1+0.25×1416−40×1+0.25×1534−40×1+0.25×842​⎦⎥⎥⎤​→hθ​(2104)→hθ​(1416)→hθ​(1534)→hθ​(852)​
prediction=datamatrix$\ast$parameters

Matrix-matrix multiplication

[ 1 3 2 4 0 1 ] ×[ 1 3 0 1 5 2 ] =[ 11 10 9 14 ] \begin{bmatrix} {1}&{3}&{2}\\ {4}&{0}&{1}\\ \end{bmatrix}\times\begin{bmatrix} {1}&{3}\\ 0&1 \\5&2 \end{bmatrix}=\begin{bmatrix} {11}&{10}\\ 9&14 \end{bmatrix} [14​30​21​]×⎣⎡​105​312​⎦⎤​=[119​1014​]
[ 1 3 2 4 0 1 ] ×[ 1 0 5 ] =[ 11 9 ] \begin{bmatrix} {1}&{3}&{2}\\ {4}&{0}&{1}\\ \end{bmatrix}\times\begin{bmatrix} {1}\\ 0 \\5 \end{bmatrix}=\begin{bmatrix} {11}\\ 9 \end{bmatrix} [14​30​21​]×⎣⎡​105​⎦⎤​=[119​]
[ 1 3 2 4 0 1 ] ×[ 3 1 2 ] =[ 10 14 ] \begin{bmatrix} {1}&{3}&{2}\\ {4}&{0}&{1}\\ \end{bmatrix}\times\begin{bmatrix} {3}\\ 1 \\2 \end{bmatrix}=\begin{bmatrix} {10}\\ 14 \end{bmatrix} [14​30​21​]×⎣⎡​312​⎦⎤​=[1014​]

Details:

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

example

[ 1 3 2 5 ] ×[ 0 1 3 2 ] =[ 9 7 15 12 ] \begin{bmatrix} {1}&3\\ 2&5 \end{bmatrix}\times \begin{bmatrix} {0}&1\\ 3&2 \end{bmatrix}=\begin{bmatrix} {9}&7\\ 15&12 \end{bmatrix} [12​35​]×[03​12​]=[915​712​]

house size:

{ 2104 1416 1534 852 \begin{cases} 2104\\ 1416\\ 1534\\ 852 \end{cases} ⎩⎪⎪⎪⎨⎪⎪⎪⎧​210414161534852​
Have 3 competing hypothesis
{1 、 h θ ( x ) = − 40 + 0.25 x 1 、 h θ ( x ) = 200 + 0.1 x 1 、 h θ ( x ) = − 150 + 0.4 x \begin{cases} 1、h_\theta (x)=-40+0.25x\\ 1、h_\theta (x)=200+0.1x\\ 1、h_\theta (x)=-150+0.4x\\ \end{cases} ⎩⎪⎨⎪⎧​1、hθ​(x)=−40+0.25x1、hθ​(x)=200+0.1x1、hθ​(x)=−150+0.4x​
[ 1 2104 1 1416 1 1543 1 852 ] ×[ − 40 200 − 150 0.25 0.1 0.4 ] =[ 482 410 692 314 342 416 344 352 464 173 285 191 ] \begin{bmatrix} {1}&2104\\ 1&1416\\ 1&1543\\ 1&852 \end{bmatrix}\times \begin{bmatrix} {-40}&200&{-150}\\ 0.25&0.1&0.4 \end{bmatrix}=\begin{bmatrix} 482&410&692\\ 314&342&416 \\344&352&464\\ 173&285&191 \end{bmatrix} ⎣⎢⎢⎡​1111​210414161543852​⎦⎥⎥⎤​×[−400.25​2000.1​−1500.4​]=⎣⎢⎢⎡​482314344173​410342352285​692416464191​⎦⎥⎥⎤​

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)
E . g .[ 1 1 0 0 ] ×[ 0 0 2 0 ] =[ 2 0 0 0 ]≠ [ 0 0 2 0 ] ×[ 1 1 0 0 ] =[ 0 0 2 2 ] E.g. \begin{bmatrix} {1}&1\\ 0&0\\ \end{bmatrix} \times\begin{bmatrix} {0}&0\\ 2&0\\ \end{bmatrix}=\begin{bmatrix} {2}&0\\ 0&0\\ \end{bmatrix}\\\ne\\\begin{bmatrix} {0}&0\\ 2&0\\ \end{bmatrix}\times \begin{bmatrix} {1}&1\\ 0&0\\ \end{bmatrix} =\begin{bmatrix} {0}&0\\ 2&2\\ \end{bmatrix} E.g.[10​10​]×[02​00​]=[20​00​]​=[02​00​]×[10​10​]=[02​02​]
A × B m × n n × m A × B ∼ m × m B × A ∼ n × n \begin{array}{|lll} A\times B\\ m\times n\quad n\times m\\ A\times B\sim m\times m \\ B\times A\sim n\times n \end{array} A×Bm×nn×mA×B∼m×mB×A∼n×n​
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 In×n I_{n\times n} In×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:

[ 1 ] [ 1 0 0 1 ] [ 1 0 0 0 1 0 0 0 1 ] [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] [ 1 1 ⋱ 1 ] ( I n f o r m a l l y ) \begin{bmatrix} 1 \end{bmatrix}\begin{bmatrix} {1}&0\\ 0&1\\ \end{bmatrix}\begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}\begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{bmatrix}\begin{bmatrix} 1&{}&{}&{} \\ {}&1&{}&{} \\ {}&{}& \ddots &{} \\ {}&{}&{}&1 \end{bmatrix}(Informally) [1​][10​01​]⎣⎡​100​010​001​⎦⎤​⎣⎢⎢⎡​1000​0100​0010​0001​⎦⎥⎥⎤​⎣⎢⎢⎡​1​1​⋱​1​⎦⎥⎥⎤​(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 × nN o t e : A × B ≠ B × A i n  g e n e r a l A × I = I × A ✔ I_{n\times n}\begin{array}{|lll} Note:\\ A\times B\ne B\times A \quad in \,general\\ A\times I= I\times A ✔ \end{array} In×n​Note:A×B​=B×AingeneralA×I=I×A✔​

Inverse and Transpose(逆运算以及转置矩阵)

1=“Identity” \quad 3 ( 3−1 ) = 1 3(3^{-1})=1 3(3−1)=1 \quad 12 ( 1 2−1 ) = 1 12(12^{-1})=1 12(12−1)=1
Mot all numbers have an inverse \quad → 0 ( 0−1 ) \to0(0^{-1}) →0(0−1)but 0−1 → 0^{-1}\to 0−1→undefined

Matrix invers

If A is an $m\times m$ matrix(square matrix{has the same row &column}),and if it has an inverse.
→ A A−1 = A−1 A = I \to AA^{-1}=A^{-1}A=I →AA−1=A−1A=I \quad A 逆 矩 阵 A−1 A逆矩阵A^{-1} A逆矩阵A−1 \quad
A = [ 0 0 0 0 ] 无 逆 矩 阵 A=\begin{bmatrix} {0}&0\\ 0&0\\ \end{bmatrix}无逆矩阵 A=[00​00​]无逆矩阵
E . g .[ 3 4 2 16 ] [ 0.4 − 0.1 − 0.05 0.75 ] =[ 1 0 0 1 ] =I 2 × 2E.g.\begin{bmatrix} 3&4\\ 2&16\\ \end{bmatrix}\begin{bmatrix} 0.4&{-0.1}\\ {-0.05}&0.75\\ \end{bmatrix}=\begin{bmatrix} 1&0\\ 0&1\\ \end{bmatrix}=I_{2\times 2} E.g.[32​416​][0.4−0.05​−0.10.75​]=[10​01​]=I2×2​
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 = [ 1 2 0 3 5 9 ] A=\begin{bmatrix} 1&2&0\\ 3&5&9\\ \end{bmatrix} A=[13​25​09​] A T = [ 1 3 2 5 0 9 ] \quad A^T=\begin{bmatrix} 1&3\\ 2&5\\ 0&9 \end{bmatrix} AT=⎣⎡​120​359​⎦⎤​$row\to column$
Let A be a $m\times n$ matrix,and let B = A T B=A^T B=AT
Then B is a $n\times m$ matrix, and
Bij = Aji B_{ij}=A_{ji} Bij​=Aji​
B 21 = A 12 = 2 B_{21}=A_{12}=2 B21​=A12​=2
B 32 = A 23 = 9 B_{32}=A_{23}=9 B32​=A23​=9

the end

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