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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 2:06
O ok. I see, sorry. Let me rephrase my question though. – Steven-Owen Mar 22 '13 at 2:07
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 2:13
what does it mean for a matrix to be primitive? – Ittay Weiss Mar 22 '13 at 3:00
@IttayWeiss See Perron-Frobenius. – 1015 Mar 22 '13 at 12:27
up vote 3 down vote accepted

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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As usual,you gave the best possible example, +1. – 1015 Mar 22 '13 at 12:28

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