How to numerically calculate the complex roots (eigenvalues) of a determinant?

Please consider the following determinant:

"aij" and "bij" take on values lager than zero and smaller than one, and "q" has a range between pi and -pi. I want to calculate the complex eigenvalues (lambda) of the determinant using some sort of a numerical scheme.

I'm well aware that Computer Algebra Systems like MATLAB have no problem calculating the roots of a determinant of this size, but I am working with matrices that are ridiculously larger than this one and without an efficient numerical solution, they would take forever to get solved.

I would immensely appreciate any help or insight.

Thank you very much

[Sam Chak]

Hi @Moein M

Can you at least type out all 16 expressions in the matrix entries?

lambda
aij     % from a11 to a44
bij     % from b11 to b44
q

[Walter Roberson]

Given numeric inputs, eig() is fairly efficient.

[Matt J]

Your expression doesn't look like a characteristic equation in lambda, so I'm not sure why the solutions can be called "eigenvalues".

One thing that is fairly certain is that you will not be able to do it by writing down the determinant as a function of lambda. For NXN matrices with N>10, that will give you a a polynomial of order greater than 20, which cannot be solved stably in double float precision.

[Moein M]

Hello @Sam Chak

I aplogize if the question was a bit vague. I have substituted the parameters of the matrix with numercial values, assuming q=pi/3. Please see the attached file.

The goal is to calculate lambda.

Thanks again.

[Moein M]

Thank you very much for your feedbacks @Walter Roberson @Matt J @Sergio Yanez-Pagans

[Walter Roberson]

To be honest, I do not have enough enthusiasm to bother typing in those equations by hand starting from an image.

If you had used syms lambda and posted the code to construct the matrix, then I would have been more likely to take a look.

Answer 1

[Sergio Yanez-Pagans]

Maybe this helps:

https://www.mathworks.com/matlabcentral/fileexchange/94730-matrix-methods-of-linear-algebra?s_tid=srchtitle