Lanczos.md

Different from Arnoldi, here we suppose matrix A is an n x n Hermitian matrix and we plan to find the m most valueable eigenvalues. In the following, we will introduce Lanczos iterative algorithm.

Let matrix V be an n x m matrix, which can be written as , where are orthonormal Lanczos vectors. Let matrix T be a m x m tridiagonal real symmetric matrix with , which can be also written as

.

Note that can be expressed as AV = VT. Comparing the j-th column of both sides, we have

.

Furthermore, we have

since .

In addition, we have

by multiplying at both sides of since , , and are orthogonal vectors.

However, in order to guarantee numerical stability, and should add additional terms. Therefore, they are expressed as follows:

1. ,

since and are orthogonal vectors.

2. 

since has unit norm.

The algorithm is shown as follow, which is from wiki: