# matrices – Finding quadratic forms of a certain kind that are the composition of a given quadractic norm and some linear operator?

We have quadratic forms $$Q = lambda x^2 +4y^2+16z^2$$ and $$R=2xy+2yz$$. For what real values of $$lambda$$ does there exist a real linear operator $$T$$ on $$mathbb{R}^3$$ such that $$R=Q circ T$$?

From what I know about quadratic forms, this problem simplifies to finding $$lambda$$ such that there exists $$T$$ with
$$T^t begin{pmatrix} lambda & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 16 end{pmatrix} T = begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 1 \ 0 & 1 & 0 end{pmatrix}$$.

However, I don’t know how to proceed from here. Just by writing out the elements of T and doing the multiplication longhand I’ve determined that $$lambda < 0$$, but I don’t think that approach will yield anything more helpful.

I also saw this: How to solve quadratic matrix equations of the form \$A^T B A=C\$?

Unfortunately, the solution to that question seemed somewhat specific to that exact problem, and that approach won’t exactly work here because his answer for $$A$$ was the inverse of some matrix, and my $$T$$ must not be invertible.

Finally, I’ve considered diagonalizing the matrix for $$R$$ or writing it alternatively as $$begin{pmatrix} 0 & 2 & 0 \ 0 & 0 & 2 \ 0 & 0 & 0 end{pmatrix}$$, but I don’t see how those will help. Thanks!