Spectral decomposition

The spectrum of a square matrix is the set of its eigenvalues. There is a cool theorem in linear algebra that states that all matrices representing a linear operator have the same spectrum. Before we use the spectrum though, we need to talk about diagonal matrices.

Diagonal matrices

The main diagonal of a matrix is every entry where the row index equals the column index. Examples make this very easy to see. All the following matrices have the letter d on their main diagonal:

Now, a diagonal matrix has zero on all entries outside the main diagonal. Here are examples of diagonal matrices:

Here are two cool features of diagonal matrices that make them all the rage at linear algebra parties. One, all their eigenvalues are on their main diagonal. Two, they are very easy to exponentiate. Let's see the latter in action real quick:

Try to exponentiate any regular old random matrix...