Timeline for The space of positive definite orthogonal matrices

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Dec 20, 2015 at 16:06	comment	added	Robert Bryant		Of course, it's not true that each symmetric matrix $S$ has an anti-symmetric 'partner' $A$ such that $S+A$ is orthogonal. First of all, the eigenvalues of $S$ have to lie in the interval $[-1,1]$ or no such $A$ can exist. Even this is not enough, though. You also need that the eigenvalues of $S$ other than $\pm 1$ (if any) must have even multiplicity. If those conditions are satisfied, though, it is not hard to show that an $A$ exists such that $R = S+A$ is orthogonal.
Dec 20, 2015 at 6:18	comment	added	Pushpendre		For example, let $M = \begin{bmatrix}1 & 1\\1 & 2\end{bmatrix}$, then $M$ is positive definite but $A$ can be $M$ plus any anti symmetric matrix $M'$ divided by two. For example if $M' = \begin{bmatrix}0 & -1\\ -1 & 0\end{bmatrix}$ then A = $(M + M') / 2$ would not be orthogonal. Therefore, the above characterization is unsatisfactory.
Dec 20, 2015 at 6:05	comment	added	Pushpendre		If the symmetric matrix $M = A + A^T$ is positive definite, then it's not necessary that $A$ would be orthogonal, right? Or is there a unique antisymmetric matrix $M'$ that when added to $M$ would create an orthogonal matrix?
Dec 20, 2015 at 5:56	history	edited	Robert Israel	CC BY-SA 3.0	added 121 characters in body
Dec 20, 2015 at 5:50	history	answered	Robert Israel	CC BY-SA 3.0