Definite quadratic form: Difference between revisions

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In [[mathematics]], a '''definite quadratic form''' is a [[quadratic form]] over some [[Real number|real]] [[vector space]] {{math|''V''}} that has the same [[positive and negative numbers|sign]] (always positive or always negative) for every nonzero vector of {{math|''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 much the same way, except that "always positive" and "always negative" are replaced by "notalways negativenonnegative" and "notalways positivenonpositive", respectively. In Another '''indefinite'''words, quadraticit formmay is one that takestake on both positive and negativezero values.

An '''indefinite''' quadratic form takes on both positive and negative values.

More generally, the definition applies to a vector space over an [[ordered field]].<ref>Milnor & Husemoller (1973) p.&nbsp;61</ref>

More generally, thethese definitiondefinitions appliesapply to aany vector space over an [[ordered field]].<ref>{{harvnb|Milnor & |Husemoller (|1973) p|page=61}}.&nbsp;61</ref>

==Associated symmetric bilinear form==