Find a subspace of $\mathbb{R}^4$ for which $x^T*A*x$ = 0

Question

Given a matrix $A$ find a two dimensional subspace $V \subset\mathbb{R}^4$ for which $\forall x \in V : x^TAx=0$

$$A = \begin{pmatrix}1&2&0&1\\ 2&3&1&1\\ 0&1&0&1\\ 1&1&1&-1 \end{pmatrix}$$

I understand that I have to find a subspace whose image in relation to the matrix $A$ is orthogonal to itself (original subspace).

$$Ax = y$$ $$ x^Ty = 0 \iff x \perp y $$

The question is how to do it? I know that I could construct 4 quadratic equations directly from $x^TAx=0$ but I believe that there has to be an easier way of doing things.

EDIT: Ok, I actually can't, at best I'm able to construct one equation which doesn't help me much.

Answer 1

They may or may not have mentioned the name Hermite, or Witt, in this area. The idea is to start at the upper left corner, choose coefficients in a square, perhaps with a constant coefficient, that wipes out the first row and column. I ordered my variables $(x,y,z,w),$ the coefficient of $x^2$ is one, to take care of the top row and left column I used $(x + 2 * y + w)^2.$ After that, the variable $x$ no longer appears anywhere, and we begin the same process with the letter $y,$ where we needed a coefficient of $-1$ to continue. The quadratic form indicated by your matrix is

$$ (x + 2 y + w)^2 - ( y - z + w)^2 + z^2 - w^2. $$ This task is fairly quick, no eigenvalues... more typing in a minute

This is ad hoc, I wrote the null cone as $$ (x + 2 y + w)^2 - w^2 = ( y - z + w)^2 - z^2, $$ or factoring differences of squares, $$ ( x + 2 y) (x+2y+2w) = (y+w)(y-2z+w). $$ One way to make this true is to require $$ y+w = 0; \; \; x=0 $$ which specifies a $2$-plane in $\mathbb R^4,$ because $(x+2y+2w)$ also becomes zero.

EDIT: a few hours later. I scribbled some things. There are infinitely many 2-planes contained in this null cone. Take any four real numbers $A,B,C,D$ such that $$ A^2 + B^2 = C^2 + D^2. $$ There is a 2-plane contained in the null cone spanned by the $(x,y,z,w)$ vectors $$ (A-2B-2C+D, \; B+C-D, \; B, \; D ), $$ $$ (-2A-B+C+2D, \; A-C-D, \; A, \; C). $$

There is a differential geometry phrase that goes with this: there are infinitely many 2-planes passing through the origin and contained in the cone over the Clifford torus.