Linear algebra question, quadratic forms.

[enfield]

A is a square matrix. x, b are vectors.

I know for Ax=b, that given b, there are an infinite number of pairs (A, x) which satisfy the equation.

I'm wondering if the same is true for xAx=b.

in particular, what if (x, A, b) are all stochastic vectors/matrices (i.e the entries of b and x add to 1, and so do each column of A). Would that make it so there was a single solution (A,x) for a given b?

Thanks.

 

[HallsofIvy]

First, b cannot be a vector: $xAx$ will be a number, not a vector. And it is easy to find examples showing there is not a unique solution: Let b= 1. Then
$$\begin{pmatrix}1 & 0\end{pmatrix}\begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix}\begin{pmatrix}1\\ 0\end{pmatrix}= 1$$

Or
$$\begin{pmatrix}1 & 1\end{pmatrix}\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}\begin{pmatrix}1\\ 1\end{pmatrix}= 1$$