Gaussian Elimination and diagonalization

Question

The standard procedure to diagonalize a square matrix $A$:

1) Find eigenvalues, eigenvectors and possibly generalized eigenvectors by solving $(A-\lambda I)\xi=0$.

2) Form a Linear transformation $T=[\xi_1\:\xi_2\:\cdots\xi_n]$ s.t. $\Lambda=T^{-1}AT$ is in Jordan form.

Suppose I wanted to perform a series of row and column operations on the matrix $A$ and sequentially transform it into Jordan form. Is there a general scheme available to do this? Gaussian elimination down and across in a pair originating from the same element would preserve the eigenvalues?

The motivation for this question is that I have a block matrix of the following form:

\begin{align} \left[\begin{array}{cccccc}A&B&0&0&C&0\\ -B&D&0&0&0&0\\ 0&0&E&0&0&0\\ 0&0&0&I&0&0\\ C&0&0&0&F&0\\ 0&0&0&0&0&I\end{array}\right] \end{align}

each block is $n\times n$ and I am trying to study the characteristic polynomial as a function of $n$. I've already done quite a bit to get it this nice. Im currently using symmetry adapted basis vectors but still have coupling through $C$ and $B$. If I could come up with some way to treat the $C$ and $B$ blocks before calculating the characteristic polynomial it would be awesome.

Answer 1

We have $J=T^{-1}AT$ for some invertible $T$. Now any invertible matrix is a product of elementary matrices, so we can write $$J=E_k^{-1}\cdots E_1^{-1}\,A\,E_1\cdots E_k\ ,$$ and this is the result of performing certain row and column operations on $A$.

If you are looking for a systematic way of finding which elementary matrices to use, I imagine that this could well be harder.

Answer 2

No, Gaussian elimination does not preserve eigenvalues. You need similarity transformations $A \to T^{-1} A T$. You can use Householder transformations to put the matrix into tridiagonal form, and this can be the first step in the QR algorithm.