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Encyclopedia > Positive definite

In mathematics, a definite bilinear form B is one for which

B(v,v)

has a fixed sign (positive or negative) when it is not 0.


To give a formal definition, let K be one of the fields R (real numbers) or C (complex numbers,. Suppose that V is a vector space over K, and

B : V VK

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 positive-definite if

B(x,x) > 0

for every nonzero x in V. If it is greater than or equal to zero, we say B is positive semidefinite. Similarly for negative definite and negative semidefinite. If it is otherwise unconstrained, we say B is indefinite.


A self-adjoint operator A on an inner product space is positive-definite if

(x, Ax) > 0 for every nonzero vector x.

See in particular positive-definite matrix.


See also:

  • positive-definite function
  • restricted negative-definite function.

  Results from FactBites:
 
PlanetMath: positive definite (237 words)
The definiteness of a matrix is an important property that has use in many areas of mathematics and even physics.
Thus the determinant of a positive definite matrix is positive, and a positive definite matrix is always invertible.
This is version 6 of positive definite, born on 2002-02-15, modified 2004-06-17.
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