Does this matrix equality hold? How to prove it? $B \bullet A=B\bullet U\Lambda U^T=U^TBU\bullet A$?

Question

Suppose $A,B$ are all symmetric matrices, How tp prove that $B \bullet A=B\bullet U\Lambda U^T=U^TBU\bullet A$, in which $\bullet$ is the inner product of a matrix defined as $A\bullet B=\sum_{i,j=1}A_{ij}B_{ij}$. $\Lambda $ is a diagonal matrix, $U$ is a orthognal matrix.

Answer 1

Presumably, we're given that $A = U\Lambda U^T$.

Note that $A \bullet B = \operatorname{Tr}(A^TB)$. With that in mind, we note that $$ B \bullet A = \operatorname{Tr}(B^TA) = \operatorname{Tr}([B^TU\Lambda] U^T) = \operatorname{Tr}(U^T[B^TU \Lambda]) = \operatorname{Tr}([U^TBU]^T\Lambda) = [U^TBU] \bullet \Lambda $$