Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly positive eigenvalues (note that $A$ is not required to be symmetric). 

> **My question.** Do there exist an orthogonal matrix $T\in\mathbb{R}^{n\times n}$ and a symmetric positive definite matrix $P\in\mathbb{R}^{n\times n}$ such that
$$
TAPT^\top = D+S,
$$
where $D\in\mathbb{R}^{n\times n}$ is a diagonal matrix with *positive* diagonal entries and $S\in\mathbb{R}^{n\times n}$ is a skew-symmetric matrix?


Of course, if the diagonal entries of $D$ are not required to be positive then the answer is in the affirmative (see, e.g., [this related question][1]).


  [1]: https://mathoverflow.net/questions/282694/is-every-real-matrix-conjugate-to-a-semi-antisymmetric-matrix