(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
(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.
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2 Negative-definite, semidefinite and indefinite matrices 3 Generalizations |
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.
The Hermitian matrix M is said to be negative-definite if
A Hermitian matrix which is neither positive- nor negative-semidefinite is called indefinite.
Suppose K denotes the field R or C, V is a vector space over K, and B : V × V → K is a bilinear map which is Hermitian in the sense that B(x,y) is always the complex conjugate of B(y,x). Then B is called positive definite if B(x,x) > 0 for every nonzero x in V.Further properties
Negative-definite, semidefinite and indefinite matrices
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).Generalizations