I have nonnegative matrices $A,B,C$ such that $A = B \circ C$. It seems that $A^N \leq B^N \circ C^N$ (i.e. $(B\circ C)^N \leq B^N \circ C^N$) is true element-wise. I would like to show that this it true or provide a counterexample otherwise. I've tried writing out the matrix multiplication as a sum without much success. Any help is appreciated.
Edit: I should also note that my matrices are all substochastic, in particular, the elements of matrices $A,B,C$ are bounded between $0$ and $1$.
Best Answer
I was able to prove this by induction: (Trivially satisfied when $N=1$, so let base case be $N=2$): $$ [(A \circ B)^2]_{ji} = \sum_m (A\circ B)_{jm} (A \circ B)_{mi}=\sum_mA_{jm} A_{mi} B_{jm}B_{mi} \leq \left(\sum_m A_{jm}A_{mi}\right) \left(\sum_mB_{jm}B_{mi}\right)=(A^2 \circ B^2)_{ji}. $$
Next, assume that $(A\circ B)^N \leq A^N \circ B^N$. Multiply both sides by an extra term of $A \circ B$: $$ [(A\circ B)^{N+1}]_{ji} \leq [(A\circ B)(A^N \circ B^N)]_{ji} $$ $$ [(A\circ B)^{N+1}]_{ji} \leq \sum_m (A\circ B)_{jm} (A^N \circ B^N)_{mi} $$ $$ [(A\circ B)^{N+1}]_{ji} \leq \sum_m A_{jm} (A^N)_{mi} \cdot B_{jm} (B^N)_{mi} $$ $$ [(A\circ B)^{N+1}]_{ji} \leq \left(\sum_m A_{jm} (A^N)_{mi})\right) \cdot \left(\sum_m B_{jm} (B^N)_{mi}\right) $$ $$ [(A\circ B)^{N+1}]_{ji} \leq [A^{N+1} \circ B^{N+1}]_{ji} $$