Can I reduce a matrix inequality such as $\mathbf x^\prime\mathbf A\mathbf x > \mathbf x^\prime\mathbf x$?

Question

I'm new to Mathematica. When I do linear algebra, I wonder if I can have an inequality such as $\mathbf x^\prime\mathbf A\mathbf x > \mathbf x^\prime\mathbf x$, where $\mathbf x$ is a column vector and $\mathbf A$ is an $n\times n$ matrix, reduced to something like "$\mathbf I-(\mathbf A^\ast+\mathbf A)/2$ is positive definite".

Answer 1

Here is something that might at least go in the right direction. It's not at the fully symbolic level you're looking for, because most matrix calculations require specifying an actual number for the dimension. So here I'll look at $2\times 2$ matrices only:

Resolve[
 ForAll[
  {x1, x2}, 
  {x1, x2}.{{a11, a12}, {a21, a22}}.{x1, x2} >= {x1, x2}.{x1, x2}
 ]
]

(*
==> (a12 | a21 | a22) \[Element] 
  Reals && ((a11 == 1 && a12 + a21 == 0 && a22 >= 1) || (a11 >= 1 && 
     a12 + a21 == 0 && 
     a22 >= 1 && -4 a11 - a12^2 - 2 a12 a21 - a21^2 - 4 a22 + 
       4 a11 a22 >= -4) || (a11 > 
      1 && -4 a11 - a12^2 - 2 a12 a21 - a21^2 - 4 a22 + 
       4 a11 a22 >= -4))
*)

The answer gives the combination of conditions under which the expression holds, in terms of the four matrix elements individually.