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 × 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 positivedefinite 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 selfadjoint operator A on an inner product space is positivedefinite if  (x, Ax) > 0 for every nonzero vector x.
See in particular positivedefinite matrix. See also:  positivedefinite function
 restricted negativedefinite function.
