For a project I am working on, I would like to find symmetric matrices depending on a parameter for which the eigenvalues can be written analytically and the corresponding eigenvectors vary as the elements of the matrix vary.
I did research on the internet and there are some classes of matrices for which the eigenvalues are known:
1) Circulant matrices (https://en.wikipedia.org/wiki/Circulant_matrix)
2) specific kind of tridiagonal matrices: https://epubs.siam.org/doi/pdf/10.1137/070695411
but the problem is that the eigenvectors stay constant so it is not appropriate for my case. Can anyone give me a parametrized matrix with analytic formula for the eigenvalues for which the eigenvector also depend on the parameter?
Thanks in advance, Koen