On the equivalence of two traces

Question

If we are given $$\rm{Trace}\{ G \: a \: a^T\} = \rm{Trace}\{H \: w \: w^T\}$$ where $a$ is $N \times 1$ vector, $G$ is $N \times N$ symmetrical matrix, and $w^T = [a^T \: t^T \: 1]$ and $t$ is another $N \times 1$ vector, how can I find $H$ in terms of $G, a, t$?

Answer 1

You can't. For given $a$ and $t$, $\phi: H \to \text{Trace}(H w w^T)$ is a linear functional on the space of $(2N+1) \times (2N+1)$ matrices. The matrices $H$ for which $\phi(H)$ takes a given value $\text{Trace}(G a a^T)$ form a hyperplane in this space. You have given us no reason to favour any one member of this hyperplane over the others.