Positive (or negative) (semi-)definite matrix

Consider the quadratic form of some real-valued symmetric matrix and real-valued non-zero vector . Then the following definitions hold:

Relation	Name
	Positive-definite
	Positive semi-definite
	Negative-definite
	Negative semi-definite
In practice, we are most often interested in positive-definite and positive semi-definite matrices, because they have many properties that are useful in proofs.