Consider the following two quadratic forms on $\mathbb{R}^3$:
$Q(x, y, z)=\lambda x^2+4y^2+16z^2$, $R(x, y, z)=2xy+2yz$
Determine precisely those values of $\lambda\in\mathbb{R}$ such that there's a linear transformation $T$ on $\mathbb{R}^3$ so that $R=Q\circ T$
By some direct computation, I've found out that if such a linear $T$ exists then
$\begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0 \end{bmatrix} =T^t\begin{bmatrix} \lambda & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & 16 \end{bmatrix}T$
But then I don't know how to proceed.