矩阵操作函式.PPT

1998/10/15 NCU ME DCS-Lab 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 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 [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 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 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 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 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 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 Magnitude and phase angle Complex number： z = x + y*i = r*eiθ r = abs (z) theta