Positive definite and semidefinite matrices

A matrix is classified by the signs of its eigenvalues:

• all positive: positive definite
• all positive or zero: positive semidefinite
• all negative: negative definite
• all negative or zero: negative semidefinite

Positive semidefinite matrices guarantee that xᵀAx ≥ 0 for every x. Positive definite matrices also guarantee that xᵀAx = 0 only when x = 0.