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Similarity implies equality?

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Let A\in \mathbb{C}^{n\times n} be similar to A^2. Does A=A^2 hold?

Solution

No! Take A=\begin{pmatrix}1 &0 \\1&1 \end{pmatrix} then A^2=\begin{pmatrix}1 &0\\ 2&1 \end{pmatrix}. The matrices are similar but not equal.

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