LAB 3 : Matrix Operation

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# LAB 3 : Matrix Operation - PowerPoint PPT Presentation

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
• 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
• 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

[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
• 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
• 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
• [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
• 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
• 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
• 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
• 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

• 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

• 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

• 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