Quadratic form: Difference between revisions

Revision as of 14:40, 17 October 2014

Revision as of 19:55, 24 January 2015

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Any ''n''×''n'' real [[symmetric matrix]] ''A'' determines a quadratic form ''q''_''A'' in ''n'' variables by the formula

: <math>q_A(x_1,\ldots,x_n) = \sum_{i,=1}^{n}\sum_{j=1}^{n}a_{ij}{x_i}{x_j}. </math>

Conversely, given a quadratic form in ''n'' variables, its coefficients can be arranged into an ''n''×''n'' symmetric matrix. One of the most important questions in the theory of quadratic forms is how much can one simplify a quadratic form ''q'' by a homogeneous linear change of variables. A fundamental theorem due to [[Carl Gustav Jacobi|Jacobi]] asserts that ''q'' can be brought to a '''diagonal form'''

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An ''n''-ary '''quadratic form''' over a field ''K'' is a [[homogeneous polynomial]] of degree 2 in ''n'' variables with coefficients in ''K'':

: <math>q(x_1,\ldots,x_n) = \sum_{i,=1}^{n}\sum_{j=1}^{n}a_{ij}{x_i}{x_j}, \quad a_{ij}\in K. </math>

This formula may be rewritten using matrices: let ''x'' be the [[column vector]] with components ''x''₁, ..., ''x''_''n'' and {{nowrap|1=''A'' = (''a''_''ij'')}} be the ''n''×''n'' matrix over ''K'' whose entries are the coefficients of ''q''. Then