The Schur product theorem

ProofAdvanced⏱ 30 minClassical MLschur producthadamardpsdkernels

Theory: Kernels, RKHS and the Representer Theorem →

Problem

Prove that if $A,B\succeq0$ then the Hadamard (entrywise) product $A\circ B\succeq0$ . Deduce that the product of two positive definite kernels is positive definite, and that $K_1^m$ and $e^{K_1}$ are positive definite for PD $K_1$ .

Write a complete argument. Every inequality you use should be one you could state precisely.

Solution

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

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