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What can be said about the properties of an $n\times n$ invertible matrix $A$ with real entries such that $A+A^T=\alpha I_n$, where $\alpha\in \mathbb{R}$?

Particularly, I am looking for properties regarding its determinant, if there are any interesting ones.

Also, for the case $n=2$, if it has determinant 1, then I know it must be orthogonal. But is this true in general?

Thanks for your help.

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  • $\begingroup$ Its skew symmetric, except at diagonals, so A can be always written as sum of skew symmetric matrix and diagonal matrix - infact not just diagonal, but constant multiple of identity matrix. $\endgroup$ Commented Feb 28 at 22:23
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    $\begingroup$ Suppose $A$ is any such matrix. Then $B := A - \frac{1}{2} \alpha I$ must be skew-symmetric. $\endgroup$ Commented Feb 28 at 22:26
  • $\begingroup$ Note that these matrices commute. Commuting matrices have some nice properties, like if one is diagnosable then the other is as well (in the same basis). $\endgroup$ Commented Feb 28 at 23:08

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In general, that statement is not true, and counterexamples are already known in the case $n=3$. Consider

$$ A = \begin{bmatrix} 1 & 1 & -1 \\ -1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix} $$

You can easily verify that $A + A^T = 2 I_3$, and that $\det A = 1$. However, $A$ is not orthogonal, as $(A A^T)_{11} = 3$.

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