# Diagonalizing a matrix

How do you diagonalize a matrix without first determining its eigenvalues then eigenvectors? All of the examples I've seen first find the eigenvalues and eigenvectors, then diagonalize it. That seems to obscure why you'd want to diagonalize it in the first place - to easily compute the eigenvalues!

To diagonalize a matrix, you need to construct an eigenbasis--so you need to first know the eigenvalues. In particular, finding eigenvalues from a diagonalization is not practical in most cases. However, there are plenty of other good reasons for wanting to diagonalize a matrix--matrix exponentiation, which has applications in the solutions of linear ODEs, comes to mind, as does computing large powers of a given matrix easily (the two are related, of course). Aside from practical applications, diagonalization is also a very powerful theoretical tool.

There's a technique for diagonalizing the matrices of large systems in chemistry in order to simplify the matrix and be able to figure out the eigenvalues, which have physical meaning. I don't understand how it works. How can it work without having an eigenbasis?

http://en.wikipedia.org/wiki/GF_method

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