Definite quadratic form: Difference between revisions

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Quadratic forms correspond one-to-one to [[symmetric bilinear form]]s over the same space.<ref>This is true only over a field of [[characteristic (algebra)|characteristic]] other than 2, but here we consider only [[ordered field]]s, which necessarily have characteristic 0.</ref> A symmetric bilinear form is also described as '''definite''', '''semidefinite''', etc. according to its associated quadratic form. A quadratic form {{math|''Q''}} and its associated symmetric bilinear form {{math|''B''}} are related by the following equations:

:<math>\, Q(x) = B(x,x) </math>begin{align}

Q(x) &= B(x, x) \\

:<math>\, B(x,y) &= B(y,x) = \tfracfrac{1}{2} (Q(x + y) - Q(x) - Q(y)) </math>

\end{align}</math>

==Examples==