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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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NCU ME DCS-Lab

• 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

• 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

• 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

• 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

• [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

• 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

• 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

• Complex number：

• z = x + y*i = r*eiθ

• r = abs (z)

• theta= angle(z)

• => z = r*exp(i*theta)

NCU ME DCS-Lab

• 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