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. Does there exists 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).

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).

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. Does there exists 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?

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. Does there exists 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).

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 questions:

1. Does there exists 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?
2. In case the answer to my previous question is in the affirmative, is it possible to find a matrix pair $(T,P)$ such that $\mathrm{trace}(D)=\frac{1}{2\,\mathrm{trace}(A)}$?

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. Does there exists 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?