Is there a symmetric positive definite matrix $P$ such that $PA$ is diagonal?

Question

Let $A\in\mathbb{R}^{n\times n}$ be nonsingular. Can we show that there exists a symmetric positive definite matrix $P\in\mathbb{R}^{n\times n}$ such that $PA$ is a diagonal matrix? If not, can we come up with some conditions on $A$ that guarantees the existence of $P$?

Answer 1

Suppose such a matrix exists. Then $Q = P^{-1}$ is also positive definite symmetric, and $A = QD,$ so $A_{ij} = Q_{ij} d_j,$ and so $A_{ij}/A_{ji} = d_j/d_i.$ Since the RHS has ratios only, we can normalize by setting $d_1=1,$ and so we recover the other $d$s, namely $d_j = A_{1j}/A_{j1}.$ From here you can figure out the consistency conditions.