I wonder whether $\{0,1\}$ matrix always diagonalizable? I tried some examples, seems all can be diagonalized. But I seems can not figure out if its algebraic multiplicity is always equal to geometric multiplicity. Could someone help me with it?
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1A counterexample is shown in this question: https://math.stackexchange.com/q/472915/42969 – Martin R Feb 20 '23 at 14:24
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Thank you and I also find one: \begin{matrix} 1 & 1 & 1 & 1\ 0 & 1 & 1 & 1\ 1&1&1&0\ 1&1&1&1\ \end{matrix} – Duber Feb 20 '23 at 14:25
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See https://oeis.org/A000409 for a statistics for increading n. – R. J. Mathar Feb 20 '23 at 14:40
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And how about $\begin{bmatrix} 0 & 1 \\ 0 & 0\end{bmatrix}$? All its eigenvalues are $0$, but it is not similar to $\begin{bmatrix} 0 & 0 \\ 0 & 0\end{bmatrix}$.
GEdgar
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1Here is a simple problem: Take all the $2 \times 2$ matrices with entries from ${0,1}$. Of those $16$ matrices, how many of them are diagonalizable? – GEdgar Feb 20 '23 at 14:37