I'm trying to relate the Frobenius Norm of a Hadamard Product to a trace that does not include another Hadamard Product, if possible. In other words, if A and B are (sxr) matrices, with not all positive values,
$\left||A\circ B \right||^2_F = $?
I'm trying to relate the sum of the squares of all the entries of the Hadamard product. I know that the sum of all the entries of the Hadamard product are
$\sum_i \sum_j (A\circ B)_{ij} = tr(A B^T) $
and also
$\left||A\circ B \right||^2_F = tr((A\circ B)^T(A\circ B)) $
But I am trying to get $\left||A\circ B \right||^2_F $ in the form of the trace of some combination of A and B, without a Hadamard product. Even an inequality would help. I've tried various identities and inequalities related to Hadamard Product and Frobenius Norm, but I am not having any luck.
Any suggestions would be appreciated.
Best Answer
Let $$\eqalign{ D_a &= {\rm Diag}({\rm vec}(A)) \cr D_b &= {\rm Diag}({\rm vec}(B)) \cr }$$ Then $$\eqalign{ \|A\circ B\|_F^2 &= \|D_aD_b\|_F^2 = {\rm tr}(D_a^2D_b^2) \cr }$$