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!