How to characterize matrices such that $Ax\cdot x=0$ for any $x$?

Question

As written in the title: what are the properties of matrices $A \in \mathbb{R}^{d\times d}$ such that $Ax\cdot x=0 $ for any $x\in \mathbb{R}^d$? In particular, as special case, assume that $A=M^{-1}P$, where $P$ is any symmetric matrix and $M$ is given. What can be said about the matrices $M$ satisfying that relation?

Answer 1

Let $e_i$ refer to the column vector with a $1$ in position $i$ and all other components equal to $0.$

First, let $x = e_i.$ Then $0 =x^T A x = A_{ii}.$ That is, all diagonal elements of $A$ are required to be $0.$

Second, with $i \neq j$ let $x = e_i + e_j.$ Since we already know that all $A_{kk} = 0,$ we calculate that $0 =x^T A x = A_{ij} + A_{ji}.$ That is, $A$ is skew symmetric.

It should help to do these calculations by hand with a sample 3 by 3 matrix, say $$ A = \left( \begin{array}{ccc} o&p&q \\ r&s&t \\ u&v&w \\ \end{array} \right) $$