the eigenvaluas and the eigenvctors of the complex matrix?

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dongdong 2012-11-15

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https://ww2.mathworks.cn/matlabcentral/answers/53738-the-eigenvaluas-and-the-eigenvctors-of-the-complex-matrix

As we know, the eigenvctors of the complex matrix has the property of the orthogonality,but in fact, the results from the matlab obey the conlusion? why? the code is list as following:

mass matrix,stiffness matrix, and damping matrix

M = [1  0; 0  1];
C = [0.7   -0.1; -0.1  0.2];
K = [20000  -1;  -1  20000];
f = [1 0];
%%system matrix
A = [C  M; M zeros(2)];   
B = [K  zeros(2); zeros(2)  -M];  
F =[1 0 0 0]';
%%check the orthogonality
As = -inv(A)*B;
Bs = inv(A)*F;
Cs = [1, 0, 0, 0];
Ds = 0;
sys = ss(As,Bs,Cs,Ds);
t = 0:0.001:1;
impulse(sys,t);
[v,d] = eig(As,eye(4));
[m,n] = size(v);
vr = v(:,1:2:n);
vrr = v(:,2:2:n);
v = [vr vrr];
A = v'*v;
B = v'*As*v
A = inv(A)*A;