Is there a relation between trace of congruent matrices?

Question

If $A$ and $\bar{A}$ are two congruent positive definite matrices such that $A=V\bar{A}V^T$ (det$(V) \neq 0$), then is there an expression for trace$(A)$ in terms of trace$(\bar{A})$? I know trace$(\bar{A})$ and I want to find trace$(A)$. Thanks.

Answer 1

No. E.g. when $0<t<1,\, \bar{A}=\operatorname{diag}(t,1-t)$ and $V=\operatorname{diag}(1,2)$, we have $\operatorname{tr}(\bar{A})=1$ for all $t$, but $\operatorname{tr}(A)=\operatorname{tr}(V\bar{A}V^T)=\operatorname{tr}(\operatorname{diag}(t, 4-4t))=4-3t$ varies with $t$.