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A matrix $A$ is idempotent if:

$$AA = A$$

Is it true that all such matrices are symmetric?

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  • $\begingroup$ No. Consider an oblique, nonorthogonal, projection matrix. $\endgroup$
    – max_zorn
    Apr 24 '18 at 16:22
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No. Here's a simple $2 x 2$ counterexample:

Define:

$$A = \begin{pmatrix} 1 &0\\1 &0 \end{pmatrix}$$

Note that $A$ is not symmetric, i.e. $$A^T = \begin{pmatrix} 1 &1\\0 &0 \end{pmatrix} \neq A$$

However, $A$ is idempotent:

$$AA = \begin{pmatrix} 1 &0\\1 &0 \end{pmatrix} \begin{pmatrix} 1 &0\\1 &0 \end{pmatrix} = \begin{pmatrix} 1 &0\\1 &0 \end{pmatrix} = A$$

Therefore, it cannot be the case that an idempotent matrix has to be symmetric.

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