Proving Inequalities using Linear Algebra

We prove the inequality 6x^2 - 4xy + 9y^2 ≥ 0 by expressing 6x^2 - 4xy + 9y^2 in terms of matrices, and then using diagonalisation. It turns out that this method can be generalised.

[cacostaangulo]

Excellent. I've just discovered your channel and I like the variety of your solving problems and tricks. Thanks a lot.

[martinepstein]

Nice. My approach was to show that the matrix A can be written as S^T S for some matrix S, meaning
(x, y)^T A (x, y) = (x, y)^T S^T S (x, y) = (S (x, y))^T S (x, y) = ||S (x, y)||^2 >= 0

[evankalis]

This was incredible. Had no idea you could use linear algebra this way

[harrikarri]

General method is fine, but I think it was far too complicated for this particular example. I would simply write 6x^2-4xy+9y^2 = 2x^2+4x^2-4xy+y^2+8y^2 = 2x^2+(2x-y)^2+8y^2 which is clearly non-negative.