A symmetric matrix A is positive definite if any of the following equivalent conditions hold: all the eigenvalues of A are positive; the product xTAx > 0 for all nonzero vectors x; or there exists a nonsingular matrix W such that A = WTW.
A symmetric matrix A is positive semidefinite if all the eigenvalues of A are nonnegative, is negative definite if all the eigenvalues are negative, and is negative semidefinite if all the eigenvalues are nonpositive.
Matrices + Definiteness Tests
, eigenvalues: 3, 1,
![]()
, eigenvalues: 0, 2,