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LAB 3 : Matrix Operation

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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  1. LAB 3 : Matrix Operation NCU ME DCS-Lab

  2. 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

  3. 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

  4. [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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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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