# An example of a square matrix with the same eigenvectors but different eigenvalues

Is there an example such that $$A$$ and $$B$$, three by three, that have the same eigenvectors, but different eigenvalues?

What would be the eigenvectors and eigenvalues if it exists because I'm stuck on this practice problem.

I know that if matrices $$A$$ and $$B$$ can be written such that $$AB=BA$$, they share the same eigenvectors, but what about their eigenvalues? precisely if they're squared matrices ($$3\times 3$$ case)

## 1 Answer

How about $$I$$ and $$-I$$? Then all $$x\neq0$$ are eigenvectors. But the eigenvalues are $$1$$ and $$-1$$ respectively.

For a less trivial example, how about $$\begin{pmatrix}1&0&0\\0&2&1\\0&0&2\end{pmatrix}$$ and $$\begin{pmatrix}4&0&0\\0&3&1\\0&0&3\end{pmatrix}$$?

They share eigenvectors $$(1,0,0)$$ and $$(0,1,0)$$, but for different eigenvalues.

As to your question about commuting matrices, you could take any two diagonal matrices, with different entries on the respective diagonals.

• It only works for the identity matrices? – xim Jan 29 at 3:04
• Hmm. I'll have to think about it. It looks like multiples of identity matrices will work. How about a permutation matrix? – Chris Custer Jan 29 at 3:33
• It would work if you put different values on the diagonal. Maybe using Jordan normal form we can make examples. – Chris Custer Jan 29 at 3:55