(1) For all nonzero vectors z in C^{n} we have
(2) For all nonzero vectors x in R^{n} we have
(3) For all nonzero vectors u in Z^{n} (all components being integers), we have
(4) All eigenvalues of M are positive.
(5) The form
(6) All the following matrices have positive determinant: the upper left 1by1 corner of M, the upper left 2by2 corner of M, the upper left 3by3 corner of M, ..., and M itself.
Every positive definite matrix is invertible and its inverse is also positive definite. If M is positive definite and r > 0 is a real number, then rM is positive definite. If M and N are positive definite, then M + N is also positive definite, and if MN = NM, then MN is also positive definite. To every positive definite matrix M, there exists precisely one square root: a positive definite matrix N with N^{2} = M.
Negative definite, semidefinite and indefinite matrices
The Hermitian matrix M is said to be negative definite if
for all nonzero x in R^{n} (or, equivalently, all nonzero x in C^{n}). It is called positive semidefinite if
for all x in R^{n} (or C^{n}) and negative semidefinite if
for all x in R^{n} (or C^{n}).
A Hermitian matrix which is neither positive nor negative semidefinite is called indefinite.
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