# Eigenvalues of a tridiagonal matrix

## Main Question or Discussion Point

Some algorithms of finding the eigenvalues of symmetric matrices first transform the matrix to a tridiagonal matrix which is similar to the original matrix and then find the eigenvalues of the tridiagonal matrix. . Are there special algorithms for a tridiagonal matrix, or do the same algorithms for general matrixes converge faster for a tridiagonal one?

Related Linear and Abstract Algebra News on Phys.org
HallsofIvy
Homework Helper
A general n by n matrix has $n^2$ entries. An n by n "tri-diagonal" matrix has only 3n- 2 entries (not counting the ones that you know are 0 and don't have to keep track of). There are numerical algorithms for working with tri-diagonal matrices that do NOT "fill in the zeros" so, yes, there is an enormous saving in both time and storage working with tri-diagonal matrices.

Thanks HalsofIvy,

Do you know any of such algorithms?
The general algorithms that I know, like QR eigenvalue algorithm ,ruin the sparsity of the matrix in the very first iteration. Or maybe I misunderstood it.

Added: There was a mistake in my code. The QR transformation does preserve the tridiagonality.

Thanks again.

Last edited: