Definite quadratic form

In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every nonzero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite. In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every nonzero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite. A semidefinite (or semi-definite) quadratic form is defined in the same way, except that 'positive' and 'negative' are replaced by 'not negative' and 'not positive', respectively. An indefinite quadratic form is one that takes on both positive and negative values. More generally, the definition applies to a vector space over an ordered field. Quadratic forms correspond one-to-one to symmetric bilinear forms over the same space. A symmetric bilinear form is also described as definite, semidefinite, etc. according to its associated quadratic form. A quadratic form Q and its associated symmetric bilinear form B are related by the following equations: The latter formula arises from expanding Q ( x + y ) = B ( x + y , x + y ) {displaystyle Q(x+y)=B(x+y,x+y)} . As an example, let V = R 2 {displaystyle V=mathbb {R} ^{2}} , and consider the quadratic form where x = (x1, x2) ∈ V {displaystyle in V} and c1 and c2 are constants. If c1 > 0 and c2 > 0, the quadratic form Q is positive-definite, so Q evaluates to a positive number whenever ( x 1 , x 2 ) ≠ ( 0 , 0 ) . {displaystyle (x_{1},x_{2}) eq (0,0).} If one of the constants is positive and the other is 0, then Q is positive semidefinite and always evaluates to either 0 or a positive number. If c1 > 0 and c2 < 0, or vice versa, then Q is indefinite and sometimes evaluates to a positive number and sometimes to a negative number. If c1 < 0 and c2 < 0, the quadratic form is negative-definite and always evaluates to a negative number whenever ( x 1 , x 2 ) ≠ ( 0 , 0 ) . {displaystyle (x_{1},x_{2}) eq (0,0).} And if one of the constants is negative and the other is 0, then Q is negative semidefinite and always evaluates to either 0 or a negative number. In general a quadratic form in two variables will also involve a cross-product term in x1x2: This quadratic form is positive-definite if c 1 > 0 {displaystyle c_{1}>0} and c 1 c 2 − c 3 2 > 0 , {displaystyle c_{1}c_{2}-{c_{3}}^{2}>0,} negative-definite if c 1 < 0 {displaystyle c_{1}<0} and c 1 c 2 − c 3 2 > 0 , {displaystyle c_{1}c_{2}-{c_{3}}^{2}>0,} and indefinite if c 1 c 2 − c 3 2 < 0. {displaystyle c_{1}c_{2}-{c_{3}}^{2}<0.} It is positive or negative semidefinite if c 1 c 2 − c 3 2 = 0 , {displaystyle c_{1}c_{2}-{c_{3}}^{2}=0,} with the sign of the semidefiniteness coinciding with the sign of c 1 . {displaystyle c_{1}.}