I have a large (and sparse) matrix with size 1000x1000 -- 10000x10000. I believe i know all eigenvalues for the matrices. All entries are integers and so are the eigenvalues.
I want to check this by calculating the algebraic multiplicity of the eigenvalues and see if they sum up to the dimension my matrix implying I have all the eigenvalues. I know the matrices to be non-diagonalizable, which makes it non-sufficient to calculate the dimension of each corresponding nullspace. I also need to do the calculations symbolically (numerically I seem to have found them all at least within error margin). I have tried calculation the characteristic polynomial but this seems to be very slow. Once I have the characteristic polynomial checking multiplicity would have been a simple task.
Anyone have any ideas?
I guess the question simplifies to: How do i find the algebraic multiplicity of a known eigenvalue when the matrix is large and sparse?