How to obtain eigenvalues as functions of a variable?

I want to solve an eigenvalue problem as such:

In which the matrix B is constant, and the matrix A varies with the term u. My objective is to analyze the variation of each eigenvalue with u (for example):

u=0:100;
B=[1 2 3 4;5 6 7 8;9 10 11 12;13 14 15 16];

lambda=zeros(size(B,1),size(u,2));

for i=1:size(u,2)
    A=[u(i)*5 7 u(i)*10 9;10 15 17 20;11 13 12 20;3 7 1 9];
    lambda(:,i)=eig(A,B);
end

If I am not wrong, I believe that each time MATLAB solves the eigenvalue problem, the order in which it places each eigenvalue in the output column vector is completely random (please correct me if I am wrong). For me, this is a problem because I want to determine how each eigenvalue varies, so I would like the '"same" eigenvalue to be placed (or sorted) in the same position (row) every time. Because I am working with big matrices, with complex eigenvalues, in which the variations are very unpredictable, I have no criteria to use the function sort().

I have also tried to use symbolic computation, defining u as a symbolic variable, and trying to obtain each eigenvalue as symbolic expressions, functions of u (for example):

syms u
B=[1 2 3 4;5 6 7 8;9 10 11 12;13 14 15 16];
B=sym(B);

A=[u*5 7 u*10 9;10 15 17 20;11 13 12 20;3 7 1 9];

S=B\A;
lambda=eig(S);

But this solution proved to be unsustainable, due to my computer's poor memory (I don't know if there is any way to improve this approach).

Is there any way to keep track of each eigenvalue and know to which eigenvalue it "corresponds"? (sorry if I am not very clear)

[Ameer Hamza]

Since you mentioned that there are no criteria for sorting the eigenvalues, how do you determine which order is correct? For example, for one value of u, the eigenvalues were

diag([0.1 0.5 -0.2 -0.3])

and then after changing the value of u, you get

diag([1 2 -0.9 -0.2])

How do you determine if the 2nd order is consistent or not? If you can describe the criteria in words, then most probably it can be implemented in the code.

[Pedro Camacho]

What I meant is that the order in which they appear in the output vector is not important, but it has to be consistent. So, there is no appearent reason to use sort().

Because each eigenvalue is a root of the characteristic polynomial, what is important is that all the eigenvalues in the same row corresponds to the "same" root. That is why I tried the symbolic approach.

[David Goodmanson]

Hi Ameer,

You can sort the eigenvalues with, say, sort(eigvalues,'abs') and get a dependable order. But if you want to follow the eigenvalues by continuity in the variable u, you might have to use small steps for u. That works, but it is of course a time comsuming process .

Answer 1

[the cyclist]

Take a look at John D'Errico's eigenshuffle function in the File Exchange. I believe it does what you want (and there is some theoretical discussion there as well).

[Pedro Camacho]

Thank you very much.