This Gist describes simultaneous iteration, an eigenvalue algorithm for a symmetric NxN matrix A, which can be seen as a continuation of my previous Gist on a conceptually straightforward (albeit practically sub-optimal) eigenvalue algorithm for a symmetric NxN matrix. The algorithm presented here is more practically useful than the one presented in the previous Gist, although still not as practically useful as the implicit QR algorithm.
We want to find the eigenvalue decomposition for the symmetric matrix A; that is, we want to find matrices U and D which satisfy A == U * D * U^T , where U is orthogonal (IE U * U^T equals the identity matrix) and D is diagonal.
Using the power method, we can take an initial guess of an eigenvector u, and repeatedly replace u with the matrix-vector product `A *