Matrix Similarity and Eigenvalues

[MikeDietrich]

Homework Statement

If two 3 x 3 matrices A and B have the eigenvalues 1, 2, and 3, then A must be similar to B. True or False and why.

Homework Equations

A is similar to B iff B = S^-1AS

The Attempt at a Solution

I know that if A and B are similar then they have the same eigenvalues but the same does not always hold true the other way. For example, [1 0 $0 1] and [1 1$ 0 1] both have eigenvalues of 1 and 1 but the first is diagonalizable and the second is not so they are not similar. However, I cannot find a counterexample for a 3 x 3 matrix. Any thoughts? Thank you!

 

[vela]

Think about why you can't diagonalize the matrix

$$\begin{pmatrix}1 & 1 \\ 0 & 1\end{pmatrix}$$

What problem do you run into when you try to diagonalize it? Will that same problem crop up with A and B?

 

[MikeDietrich]

Since the eigenvalues are given and they are 3 unique eigenvalues then there will be an eigenbasis and the matrices will both have to be diagonalizable therefore, the statement is true (am I on the right track?).

 

[vela]

Yup, you got it.