# Calculate the algebraic multiplicity of known eigenvalues of a large, sparse matrix

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?

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So you want something like JordanDecomposition[] ... but faster? –  belisarius May 24 '14 at 16:42
@belisarius Yes! –  user14574 May 24 '14 at 16:54