Definite matrix: Difference between revisions

Revision as of 15:32, 4 September 2003

Revision as of 15:52, 5 September 2003

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TheIn [[linear algebra]], the '''positive definite''' [[matrix (mathematics)|matrices]] (or more generally, a [[linear transformation]] of a [[vector space]] V to its [[Dual vector space|dual]], V*) are (in several sensesways) analogous to the positive [[real number]]s. An ''n'' × ''n'' [[Hermitian]] [[matrix (mathematics)|matrix]] ''M'' is said to be '''positive definite''' if it has one (and therefore all) of the following 6 equivalent properties:

'''(1)''' For all non-zero vectors ''z'' in '''C'''^''n'' we have

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'''(6)''' All the following matrices have positive [[determinant]]: the upper left 1-by-1 corner of ''M'', the upper left 2-by-2 corner of ''M'', the upper left 3-by-3 corner of ''M'', ..., and ''M'' itself.

This concept extends to the more general setting of a [[linear transformation]] of a real [[vector space]] V to its [[Dual vector space|dual]], V*).

== Further properties ==