Definite matrix: Difference between revisions

Revision as of 21:18, 29 April 2005

Revision as of 21:21, 29 April 2005

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A real matrix ''M'' may have the property that ''x''^T''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''^T''Mx'' > 0 for all real nonzero vectors ''x'' if and only if the symmetric part, (''M'' + ''M''^<subsup>T</subsup>) / 2, is positive definite.

The situation for complex matrices may be different, depending on how one generalizes the inequality ''z''^*''AzMz'' > 0. If ''z''^*''AzMz'' is real for all complex vectors ''z'', then the matrix ''AM'' is necessarily Hermitian. So, if we require that ''z''^*''AzMz'' be real and positive, then ''AM'' is automatically Hermitian. On the other hand, we have that Re(''z''^*''AzMz'') > 0 for all complex nonzero vectors ''z'' if and only if the Hermitian part, (''AM'' + ''AM''^_*) / 2, is positive definite.

There is no agreement in the literature on the proper definition of ''positive-definite'' for non-Hermitian matrices.