Are eigenvectors preserved by conjugating by a diagonal matrix?

Question

Let $A$ be a real symmetric matrix and $D$ a real diagonal matrix with non-zero entries along the diagonal.

Is it true that the eigenvectors of $D^{-1}AD$ are the same as those of $A$?

If not, can anything be said relating the eigenvectors/eigenvalues of these two matrices?

This conjugation amounts to a change of basis in which each of the standard basis vectors is scaled by some, possibly different, amount. — amd, May 11 '17 at 23:09

lhf · Accepted Answer · 2017-05-11T23:00:54.127

4

Let $A$ be an arbitrary matrix and $B=P^{-1}AP$, where $P$ is an arbitrary invertible matrix.

If $Av=\lambda v$, then $Bw=\lambda w$ for $w=P^{-1}v$.

In this sense, similar matrices have the same eigenvalues but not necessarily the same eigenvectors.

In particular, the eigenvectors are different when $P$ is a diagonal matrix that is not a scalar multiple of the identity matrix.

edited May 11 '17 at 23:00

answered May 11 '17 at 22:38

lhf

The eigenvalues are the same, but the eigenvectors most certainly aren't – eyeballfrog May 11 '17 at 22:41
See also https://math.stackexchange.com/a/87705/589. – lhf May 11 '17 at 22:51
More specific to OP's question, pretty much any non-diagonal $2 \times 2$ real symmetric matrix and any nonscalar diagonal $D$ should do the trick. – Dustan Levenstein May 11 '17 at 22:55
An further question is under what conditions is the OP's statement true. Clearly having $A$ be diagonal itself is sufficient. Is it also necessary? – eyeballfrog May 11 '17 at 23:01

1 Answers1