I am implementing an algorithm to calculate eigenvalues and eigenvectors of a symmetric matrix in a GPU. In order to calculate the eigenvalues I first reduced the matrix to the tridiagonal form using Householder transformations. Now I want to use those eigenvalues to calculate the corresponding eigenvectors through inverse iteration.
I was reading about inverse iteration here and I noticed that in step 4 I must invert a matrix or, alternatively, solve a set of linear equations:
As noted here this is much easier to do with tridiagonal matrices. I already have the reduction, but its eigenvectors are not the same of the original matrix. So my question is, what do I have to do to get the eigenvectors of the original matrix?