Quadratic forms and definiteness
General · Mathematics
Study notes
Q: Is Q(x,y) = 2x²+2xy+2y² positive definite? Classify. Matrix: [[2,1],[1,2]]. Leading minors: 2 > 0, det = 3 > 0. Sylvester: POSITIVE DEFINITE! Complete square: 2x²+2xy+2y² = (x+y)²+x²+y² > 0 (nonzero!) ✓. Eigenvalues 3, 1 > 0 ✓! (Minimum at origin!)