SPD Matrices and a False Inequality

This one is from my research, and it’s a doozy. Given two vectors such that each element in is less in absolute value than the corresponding element in , show that for any SPD matrix that .

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 . 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.