What is the canonical form of a matrix?

Question

I have been asked to find the canonical form of a matrix, but I am not exactly sure what is meant by this.

I found the eigenvalues and eigenvectors of the matrix, and the solution I was shown for the canonical form is a diagonal matrix with the eigenvalues on the diagonal (the explanation for this was that the eigenvalues were distinct)--I am not sure what I would do in the case that the eigenvalues were not distinct.

I cannot find a definition for "canonical form" other than Jordan Canonical Form, which is something different.

Any help would be appreciated!

Answer 1

The canonical or diagonal form of A is a diagonal matrix D with the eigenvalues of A on the main diagonal.

If the columns of E contain the eigenvectors of A, D is the matrix that satisfies that:

$$AE=ED$$