# Does Convolution preserve positive definiteness?

Let $A$ be a symmetric positive semi definite matrix, $A \in SPD^n$, and $K \in \mathbb{R}^{k\times k}$ a kernel.

What are the necessary and sufficient conditions over $K$ that will make the convolution of $A$ and $K$ $$(A * K)_{ij} = \sum_u \sum_v A_{uv}K_{i-u,j-v}$$ Positive semi definite ?

• From the numerical tests I ran, a PSD kernel seems to work. I can't figure out how to prove it though !
• Do you convolve on the matrix? You can probably express it with Kronecker products in a higher dimension (vectorized) space. – mathreadler Aug 11 '17 at 20:28
• @mathreadler I don't see how to express it as a Kronecker product. There is a way to express it as a Hadamard product which involves submatrices of $A$, but I don't find it useful. (Both products preserve PSD though). – Tool Aug 14 '17 at 17:49