Suppose that $\A$ is an $n \times n$ symmetric matrix. $\A$ is said to be positive definite if for all nonzero $n$-dimensional vectors $\x$, we have

Similarly, $\A$ is said to be:

• Positive semidefinite (or non-negative definite) if $\x \Tr \A \x \ge 0$
• Negative definite if $\x \Tr \A \x < 0$
• Negative semidefinite (or non-positive definite) if $\x \Tr \A \x \le 0$

See also:

• Concave function
• Eigenvalue
• Generalized inequality
• Random vector
• Regularity conditions