Can we prove that this equation is positive or negative definite by factorizating it ?

Question

p(x,y,z) = 3x² + y² +2z² - 2xy +2xz - 2yz

Answer 1

By factorizing, yes. First note that p(x,y,z) = (x-y+z)² + x² + x² + z². We have written p as the sum of four squares. Suppose that not all of x,y,z are zero. If at least one of x or z is nonzero, then at least one of the last two squares is positive. If x and z are both zero, then y must be nonzero, so that the first square is positive.

Answer 2

I get p(x, y, z) = (x, y, z)A(x, y, z)^T where

A = [3 -1 1]

[-1 1 -1]

[1 -1 2]

There's a theorem that says the quadratic form p is positive definite if and only if the principal minors of A all have positive determinants. The principal minors are:

S₁ = the 1,1 entry of A; determinant is 3 > 0

S₂ = [3 -1] (i.e., the 2x2 matrix in the upper left-hand corner); determinant of S₂ is 3 - 1 = 2 > 0

[-1 1]

S₃ = the matrix A itself. det(A) = 3(2 - 1) - (-1) (-2 + 1) + 1(0) = 3 - 1 > 0

Therefore p is positive definite