# What is the canonical form of a matrix?

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!

• What you've done so far seems reasonable, given the paucity of information. If the person who asked has a different notion for "canonical form" s/he needs to tell you. You would be able to diagonalize this way even if the eigenvalues were not distinct provided there were enough - the term you want is "geometric multiplicity:/ Commented Feb 6, 2018 at 22:32
• Diagonalizability depends upon eigenvectors, notably you need a basis of eigenvectors to diagonilize. It is guaranteed if eigenvalues are distinct but you can also have diagonalizibility if eigenvalues are coincident.
– user
Commented Feb 6, 2018 at 22:33
• I would assume that you are meant to do Jordan Canonical Form (or perhaps another canonical form if you have learned about it in you class). Commented Feb 6, 2018 at 22:33
• The Jordan Canonical Form and the Rational Canonical Form are the only two that I can think of however if you've only been exposed to the Jordan Form I would assume that's the one they expect you to do produce. Commented Feb 7, 2018 at 2:16
• The Jordan form is probably the one emphasized in most undergraduate courses, so it seems reasonable to assume that is what is intended here, though as Ethan says, the person asking the question should really be more specific. If it is clear from context that you must work over a field which is not algebraically closed, so the Jordan form is not available, then (a suitable version of) the rational canonical form is probably required. Commented Apr 1, 2023 at 10:07

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:

$$AE=ED$$