4
$\begingroup$

What are the conditions on two different matrices having the same eigenvalues and eigenvectors?

$\endgroup$
4
$\begingroup$

You would have to avoid having a full basis of eigenvectors. Thus some "generalized eigenvectors" would have to be present.

For example, the $2\times 2$ matrices:

$$ \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \; , \; \begin{pmatrix} 0 & 2 \\ 0 & 0 \end{pmatrix} $$

have the same eigenvalue (zero) and the same eigenvector $[1\; 0]^T$ (or a nonzero multiple thereof).

$\endgroup$
1
$\begingroup$

If you mean "What does it mean for $A$ and $B$ both to have the same set of eigenvalues, occurring in each care with the same multiplicity", then the answer is that $A$ and $B$ are similar: $B=P^{-1}AP$ for some non-singular $P$.

$\endgroup$

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.