Given a primitive matrix $A$. Is it true that it is only similar to other primitive matrices?
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all entries positive doesn't mean eigenvalues are positive. See matrix with all entries as $1$. – Yimin Mar 22 '13 at 02:06
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O ok. I see, sorry. Let me rephrase my question though. – Steven-Owen Mar 22 '13 at 02:07
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still, we can try to make skew symmetric matrix $[0,1;1,0]$ to make the eigenvalues as $1,-1$. – Yimin Mar 22 '13 at 02:13
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what does it mean for a matrix to be primitive? – Ittay Weiss Mar 22 '13 at 03:00
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@IttayWeiss See Perron-Frobenius. – Julien Mar 22 '13 at 12:27
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No. A primitive matrix may be similar to a non-primitive matrix. For instance, $\begin{pmatrix}1&1\\1&1\end{pmatrix}\sim\begin{pmatrix}2&0\\0&0\end{pmatrix}$.
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