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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