0
$\begingroup$

I wanted to know if the following statement is true or false: are two invertible diagonalisable matrices with same eigenvalues similar?

My answer is no by considering the following diagonal matrices:

$$\begin{pmatrix} 1 && 0 && 0 \\ 0 && 2 && 0 \\ 0 && 0 && 2 \end{pmatrix} \text{ and } \begin{pmatrix} 1 && 0 && 0 \\ 0 && 1 && 0 \\ 0 && 0 && 2 \end{pmatrix}$$

Both are invertible ($\det \neq 0$) and obviously diagonalisable. They also have the same eigenvalues (with different multiplicities but nothing is imposed on the multiplicity). They are not similar because they do not have the same determinant and two similar matrices must have the same determinant.

Is this counterexample correct?

$\endgroup$
2
  • $\begingroup$ yes, it is correct $\endgroup$ Jun 30 at 1:23
  • $\begingroup$ @HagenvonEitzen thanks ! $\endgroup$
    – Kilkik
    Jun 30 at 1:24
3
$\begingroup$

I would say that your counter-example is correct under your understanding of the terminology, but under what I believe to be a more common interpretation, you're incorrect.

The issue is that if someone tells me two matrices have the same eigenvalues, I assume that they mean counting multiplicity, so that if one matrix has eigenvalues $1,2,2$ then the one with the same eigenvalues must have eigenvalues $1,2,2$, not $1,1,2$.

Under this understanding, it is indeed true that two diagonalizable matrices with the same eigenvalues are similar. Write them as:

$$A_1 = P_1 D_1 P_1^{-1},$$

$$A_2 = P_2 D_2 P_2^{-1}.$$

If the diagonal elements on the two diagonal matrices are in the same order, then we can see similarity as

$$A_1 = P_1 P_2^{-1} D_2 P_2 P_1^{-1} = P_1 P_2^{-1} D_2 (P_1 P_2^{-1})^{-1}$$.

If the diagonal elements happen to be in different orders, you can just use permutation matrices to swap the order. For example, applying a matrix permuting rows $i,j$ on both sides (this matrix is its own inverse) will swap the $(ii),(jj)$ elements of the diagonal matrix.

$\endgroup$
3
  • $\begingroup$ I would tend to side with the asker here. As he pointed out, nothing was mentioned about multiplicity. I would say, if your interpretation was intended, then the question is poorly-worded. That said, your interpretation is indeed very common in many circles of mathematics, so it's valuable to point out this potential pitfall. +1 $\endgroup$ Jun 30 at 2:08
  • $\begingroup$ I wrote the question in my own words, the original question was "are two invertible diagonalisable matrices with the same spectrum similar?" so it has nothing to do with multiplicity, sorry for the confusion $\endgroup$
    – Kilkik
    Jun 30 at 10:07
  • $\begingroup$ No problem. Good luck with your pursuits. $\endgroup$
    – David
    Jul 1 at 4:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.