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A Hermitian matrix which is neither positive- nor negative-semidefinite is called '''indefinite'''.
== Non-Hermitian matrices ==
A real matrix ''M'' may have the property that ''x''<sup>T</sup>''Mx'' > 0 for all nonzero real vectors ''x'' without being symmetric. The matrix
:<math> \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} </math>
provides an example. In general, we have ''x''<sup>T</sup>''Mx'' > 0 for all real nonzero vectors ''x'' if and only if the symmetric part, (''M'' + ''M''^<sub>T</sub>) / 2, is positive definite.
The situation for complex matrices may be different, depending on how one generalizes the inequality ''z''<sup>*</sup>''Az'' > 0. If ''z''<sup>*</sup>''Az'' is real for all complex vectors ''z'', then the matrix ''A'' is necessarily Hermitian. So, if we require that ''z''<sup>*</sup>''Az'' be real and positive, then ''A'' is automatically Hermitian. On the other hand, we have that Re(''z''<sup>*</sup>''Az'') > 0 for all complex nonzero vectors ''z'' if and only if the Hermitian part, (''A'' + ''A''^<sub>*</sub>) / 2, is positive definite.
There is no agreement in the literature on the proper definition of ''positive-definite'' for non-Hermitian matrices.
== Generalizations ==
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Then <math>B</math> is called ''positive definite'' if <math>B(x, x) > 0</math> for every nonzero <math>x</math> in <math>V</math>.
== References ==
* Roger A. Horn and Charles R. Johnson. ''Matrix Analysis,'' Chapter 7. Cambridge University Press, 1985. ISBN 0-521-30586-1 (hardback), ISBN 0-521-38632-2 (paperback).
[[Category:Matrices]]
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