# Singular values of Hadamard Product

For two Matrices $A,B \in \mathbb{R}^{m \times n}$ the Hadamard Product is defined as $(A \circ B)_{i,j} = A_{i,j}B_{i,j}$.

For a proof of convergene I require an upper (and ideally a lower) bound on \begin{align} \sum_{i>k}^{\min(m,n)}{\sigma_i^2(A \circ B)} \end{align} for Matrices with $|A_{i,j}| < 1$ and $|B_{i,j}| < 1$ for all $i,j$ and an arbitrary $1 \leq k < \min(m,n)$

From tests with randomly generated matrices I suspect that the following strong pointwise property, that could be used for a good upper bound, holds:

\begin{align} \sigma_i(A \circ B) < \text{max}(\sigma_i(A),\sigma_i(B)) \end{align}

Generally a decrease in the singular values is expected due to the well known fact that $\lVert A \rVert_F = \sqrt{\sum_i{\sigma_i^2(A)}}$ and by definition $\lVert A \circ B \rVert_F < \min(\lVert A \rVert_F,\lVert B \rVert_F)$, but this does not imply either of the inequalities stated above.

Is there some known upper bound I could use for this problem? Does someone see a proof or counterargument to the strong bound I provided?

• Obviously false even for the norm: take $A=B=\begin{bmatrix}1&1\\-1&1\end{bmatrix}$. Computers spoiled us entirely: we run hundreds of random tests on 10 by 10 instead of thinking for 5 minutes about 2 by 2 :-( Jun 5, 2018 at 14:07
• It seems that the inequalities in your question (both for the singular values and for the Frobenius norm) do not scale correctly: If $A = B = t I$ (where $I$ is the $n\times n$-identity matrix and $t \in [0,\infty)$), then the singular values and the Frobenius norm of $A$ and $B$ grow linearly in $t$, while the singular values and the Frobenius norm of $A \circ B$ grow quadratically in $t$. Maybe you forgot a square root somewhere? Jun 5, 2018 at 15:13
• As i presumed $|A_{i,j}|,|B_{i,j}|<1$ this does not matter since for $t<1$ obviously $t^2<t$ holds. @fedja is right that this does not work for their example, though I wonder if this only happens for some null subset of $\mathbb{R}^{m \times n}$. I would still be interested in a good upper bound for the first sum. Jun 5, 2018 at 15:41
• I would still be interested in a good upper bound for the first sum. Define "good". In other words, state precisely what quantities you know, in terms of which the estimate should be given. Jun 5, 2018 at 15:47
• I see. Apparently, I overlooked your condition on the entries of $A$ and $B$ (still, the scaling behaviour of the inequalities seems strange to me, but this might be due to the fact that I don't have much experience with Hadamard products). Considering your question on whether @fedja's counterexample only occurs on a nullset: By a perturbation argument, the same problem occurs on an open neighbourhood of fedja's matrices (and this neighbourhood intersects the set of matrices which fulfil your condition on the entries in a nonempty open set). Jun 5, 2018 at 15:56

Here is something related: 