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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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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

slide4
[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

slide12
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

slide13
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

slide14
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