Hi Mohammed,
I tossed in some 'double' functions to make the comparison easier
A = [3 2 ; 7 -1];
syms x;
a2a = sym(A);
polya2a = charpoly(a2a,x);
eigenvalues = solve(polya2a);
eig1 = double(eigenvalues(1,1))
v1 = double(null(A-eig1*eye(2)))
eig2 = double(eigenvalues(2,1))
v2 = double(null(A-eig2*eye(2)))
[w, lambda] = eig(A)
eig1 = -3.2426
v1 = 0.3051
-0.9523
eig2 = 5.2426
v2 = 0.6656
0.7463
lambda = 5.2426 0
0 -3.2426
w = 0.6656 -0.3051
0.7463 0.9523
Compared to the first way of doing things, the second way has the order of the eigenvalues reversed. So the first column of w, which corresponds to eigenvalue lambda = 5.2426, should be compared to v2. And the second column of w, which corresponds to eigenvalue lambda = -3.2426, should be compared to v1. Both cases agree.
There is no requirement on the overall scaling of eigenvectors and there can be a constant of proportionality between w(:,1) and v2 for example. In that case the constant equals 1, and it's -1 for w(:,2) vs v1.
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