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- Feb 5, 2012

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**saravananbs's question from Math Help Forum,**

Hi saravananbs,if A and B are similar matrices then the eigenvalues are same. is the converse is true? why?

thank u

No. The converse is not true in general. Take the two matrices, \(A=\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}\mbox{ and }B=\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}\).

\[\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}\begin{pmatrix}1 \\ 1\end{pmatrix}=1.\begin{pmatrix}1 \\ 1\end{pmatrix}\]

\[\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}\begin{pmatrix}1 \\ 1\end{pmatrix}=1.\begin{pmatrix}1 \\ 1\end{pmatrix}\]

Therefore both matrices have the same eigenvalue 1. However it can be easily shown that \(A\mbox{ and }B\) are not similar matrices.