Lab 3 matrix operation
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LAB 3 : Matrix Operation. Computing Matrix Functions. Norm :(measurement) For vectors... norm(V,P) = sum(abs(V).^P)^(1/P). norm(V) = norm(V,2). norm(V,inf) = max(abs(V)). Ex: x=[1 2 3]; , norm(x) ans = 3.7417. Eigenvalues and Eigenvectors.

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LAB 3 : Matrix Operation

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Lab 3 matrix operation

LAB 3 : Matrix Operation

NCU ME DCS-Lab


Computing matrix functions

Computing Matrix Functions

  • Norm:(measurement)

  • For vectors...

  • norm(V,P) = sum(abs(V).^P)^(1/P).

  • norm(V) = norm(V,2).

  • norm(V,inf) = max(abs(V)).

  • Ex: x=[1 2 3]; , norm(x)

  • ans =

  • 3.7417

NCU ME DCS-Lab


Eigenvalues and eigenvectors

Eigenvalues and Eigenvectors

  • 一個 n × n 方陣A, 存在λ, 滿足Ax= λx

  • 稱λ為 eigen value,

  • x 為eigen vector.

  • Ex: A=[1 2 3; 4 5 6; 7 8 9];

  • eig(A) 求 eigen value

  • ans = 16.1168

  • -1.1168

  • 0.0000

NCU ME DCS-Lab


Lab 3 matrix operation

  • [X,D]=eig(A)

  • (X:eigen vector;D:eigen value)

  • x = 0.2320 0.7858 0.4082

  • 0.5253 0.0868 -0.8165

  • 0.8187 -0.6123 0.4082

  • d = 16.1168 0 0

  • 0 -1.1168 0

  • 0 0 0.0000

NCU ME DCS-Lab


Characteristic equation

Characteristic equation

  • p = poly(A)

  • p = 1.0000 -15.0000 -18.0000 0.0000

  • characteristic equation : x3-15x2-18x+0=0

  • 其根為:r = roots(p)

  • r = 16.1168

  • -1.1168

  • 0.0000

  • p2 = poly(r)

  • p2 = 1.0000 -15.0000 -18.0000 0.0000

NCU ME DCS-Lab


Product of polynomials

Product of polynomials

  • a(s) = s2+2s+3

  • b(s) = 4s2+5s+6

  • a = [1 2 3];

  • b = [4 5 6];

  • c = conv(a,b)

  • c = 4 13 28 27 18

  • c(s) = 4s4+13s3+28s2+27s+18

NCU ME DCS-Lab


Division of polynomials

Division of polynomials

  • [q, r] = deconv(c, a)

  • q = 4 5 6

  • r = 0 0 0 0 0

  • [q2, r2] = deconv(c, b)

  • q2 = 1 2 3

  • r2 = 0 0 0 0 0

NCU ME DCS-Lab


Polynomial evaluation

Polynomial evaluation

  • evaluate polynomial a(s) = s2+2s+3 at s = 5

  • polyval(a,5)

  • ans = 38

  • 若 ploynomial 為矩陣運算:

  • =>A2+2A+3I

  • polyvalm(a,A)

  • ans = 35 40 48

  • 74 94 108

  • 116 142 171

NCU ME DCS-Lab


Absolute value and phase angle

Absolute value and phase angle

  • abs(A) = sqrt(real(A).^2 + image(A).^2)

  • angle(A) returns phase angles (π ~ - π)

  • A = [2+2*i 1+3*i; 4+5*i 6-i]

  • abs(A) (complex returns magnitude)

  • ans = 2.8284 3.1623

  • 6.4031 6.0828

  • angle(A)

  • ans = 0.7854 1.2490

  • 0.8961 -0.1651

NCU ME DCS-Lab


Magnitude and phase angle

Magnitude and phase angle

  • Complex number:

  • z = x + y*i = r*eiθ

  • r = abs (z)

  • theta= angle(z)

  • => z = r*exp(i*theta)

NCU ME DCS-Lab


Matrix function

Matrix function

  • expm:矩陣的指數函式運算

  • expm(A)=I+A+A2/2!+A3/3!+...

  • logm:矩陣的對數函式運算

  • sqrtm:矩陣的開方根運算

  • Note that a function is interpreted as a matrix function if an “m” is appended to the function name.

NCU ME DCS-Lab


Lab 3 matrix operation

實用矩陣函式

  • zeros:zeros(m,n) => m×n matrix of zeros

  • zeros(A) => size(A) matrix of zeros

  • ones:ones(m,n) => m×n matrix of ones

  • rand:平均分布的亂數矩陣

  • randn:高斯分布的亂數矩陣

  • eye:Identity matrix

  • diag:Diagonal matrix

NCU ME DCS-Lab


Lab 3 matrix operation

矩陣操作函式

  • rot90:矩陣旋轉90度

  • fliplr:矩陣左右反

  • flipup:矩陣上反置

  • tril:下三角矩陣

  • triu:上三角矩陣

  • reshape:改變矩陣的維數

NCU ME DCS-Lab


Lab 3 matrix operation

常用資料分析函式

  • max(A):矩陣A中每行向量的最大值

  • min(A):矩陣A中每行向量的最小值

  • mean(A):矩陣A中每行向量的平均值

  • median(A):矩陣A中每行向量的中間值

  • std(A):矩陣A中每行向量的標準差

  • sort(A):矩陣A中每行由小到大排序

  • sum(A):矩陣A中每行向量的總和

  • prod(A):矩陣A中每行元素的連乘積

NCU ME DCS-Lab


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