The matrix $\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$ is orthogonal and indefinite. $\begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}$ is positive definite and not orthonormal. and the Identity matrix $I$ is of course both orthogonal and positive definite. Let $S$ be the intersection of orthogonal matrices and positive definite matrices. We have seen that $S$ is non-empty.

By a positive definite matrix, I mean either a symmetric or asymmetric matrix $M \in \mathbb{R}^{p^2}$ whose quadratic form satisfies $\forall x \in \mathbb{R}^p \setminus \{0\}: x^TMx > 0$. Let $P$ denote the set of all positive definite matrices. Then $P$ is a convex cone.

$S$ is not a convex cone, unlike $P$. ~~Also unlike $P$, $S$ is closed under multiplication~~ The product of any two matrices in $S$ is guaranteed to be at least PSD.[1]

I am interested in complete characterizations of this space $S$, which globally behaves more like the space of orthogonal matrices. *My real motivation is that* I want to know whether there are efficient procedures for testing whether a matrix $M $ is in $S$ that are faster than testing for positive definiteness which requires the calculation of eigenvalues? E.g. such procedures might take only $O(p^2)$ computations. I tried to google for resources but nothing relevant came up.

[1] Orthogonal matrices are of course closed under multiplication. The product of two ~~PD matrices is PD~~ PSD matrices is PSD iff their product itself is normal, which is true in $S$.

Reference: On a product of positive semidefinite matrices, A.R. Meenakshi, C. Rajian, Linear Algebra and its Applications, Volume 295, Issues 1–3, 1 July 1999, Pages 3–6.

*In case someone is wondering, the real real reason, why* I am interested in this question is because I want to "efficiently" find a "reasonably good" positive definite approximation of a matrix whose $QR$ decomposition is known to me. The details of this part are best left for future.