Why are my eigenvalues complex? (eig)

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I Ahmed
I Ahmed 2017-9-1
编辑： Bruno Luong 2021-2-2
I wanted to find and plot the eigenvalues of large matrices (around1000x1000). But discovered when using the eig function, it gives complex eigenvalues when it shouldn't. In the code below I have a Tridiagonal Toeplitz matrix which should have all real eigenvalues. Tridiagonal Toeplitz
But it seems eig is unstable for n=90 and returns a small complex error in a few of the eigenvalues. Is there a way I can get the eigenvalues more accurately?
clear parameters
close all
clc
n=90;
dd=-2.*ones(n,1);
ud=1.8*ones(n,1);
ld=.1*ones(n,1);
A = spdiags([ld dd ud],-1:1,n,n);
C=full(A);
g=eig(C);
g=sort(g);
cond(C)
plot(g,'.')
Any help would be appreciated.

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回答（1 个）

Christine Tobler
Christine Tobler 2017-9-1
The eigenvalues of a real matrix are only real if the matrix is symmetric. The matrix C is not symmetric, therefore the eigenvalues are either real or complex conjugate pairs.
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Bruno Luong
Bruno Luong 2021-2-2
Geert Awater comment deserves to be on a separate answer (and accepted)
Bruno Luong
Bruno Luong 2021-2-2
编辑：Bruno Luong 2021-2-2
The condition number of A is not relevant in eigenvalue computation, what is more relevant is the condition number of the eigen-vectors matrix.
When they are large; the eigen spaces are almost parallel and it causes numerical algoritm to fail. The similarity transformation is the excellent trick to improve the conditioning without changing the eigen-values.
n=90;
dd=-2.*ones(n,1);
ud=1.8*ones(n,1);
ld=.1*ones(n,1);
A = spdiags([ld dd ud],-1:1,n,n);
[V,g]=eig(full(A));
cond(V) % 1.4975e+53
D=spdiags([sqrt(ld.*ud) dd sqrt(ld.*ud)],-1:1,n,n);
[W, g]=eig(full(D));
cond(W) % 1.0000

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