Timeline for Condition for two matrices to share at least one eigenvector?
Current License: CC BY-SA 3.0
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| Oct 9, 2013 at 22:02 | comment | added | SashaKolpakov | @sasquires: all right, I see what you mean. I just picked up the above two matrices virtually at random (I mean I did several tries, but without too much consideration). Your example is much more clever in this regard and finally dots the "i". | |
| Oct 9, 2013 at 21:17 | comment | added | sasquires | The only thing which struck me about this example is that $A$ has the eigenvalue $0$ with multiplicity 2, and I was hoping that the "converse" discussed above might be true when all eigenvalues have multiplicity 1 (the typical case I'm interested in). But I've just constructed a simple example where it doesn't: $A=\begin{pmatrix} 1 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 3 \end{pmatrix}$, $B=A^T$. In this case, $\textrm{det}([A,B])=0$ but none of the eigenvectors are equal. | |
| Oct 9, 2013 at 20:49 | history | answered | SashaKolpakov | CC BY-SA 3.0 |