# SPD Matrices and a False Inequality

This one is from my research, and it’s a doozy. Given two vectors $x, y$ such that each element in $x$ is less in absolute value than the corresponding element in $y$, show that for any SPD matrix $A$ that $x^TAx \le y^T A y$.

After spending a good amount of timing trying to prove this, I realized that this is in general not true (in fact, the result I was suppose to be chasing would’ve been disproven if the above statement was true). As a counter example, consider the following counterexample from a Bernstein basis application:

Let $x = [1, 0, -1/3], y = [1, -1, 1/3]$. Let the matrix be

`[2/7, 1/7, 2/35; 1/7, 6/35, 9/70 ; 2/35, 9/70, 6/35].`

Then the quadratic forms will be 4/15 and 1/7 respectively.

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