Redirected from Positive semidefinite
(1) For all non-zero vectors z in Cn we have
(2) For all non-zero vectors x in Rn we have
(3) For all non-zero vectors u in Zn (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 1-by-1 corner of M, the upper left 2-by-2 corner of M, the upper left 3-by-3 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 N2 = M.
Negative definite, semidefinite and indefinite matrices
The Hermitian matrix M is said to be negative definite if
for all non-zero x in Rn (or, equivalently, all non-zero x in Cn). It is called positive semidefinite if
for all x in Rn (or Cn) and negative semidefinite if
for all x in Rn (or Cn).
A Hermitian matrix which is neither positive nor negative semidefinite is called indefinite.
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