Matrix analysis: Difference between revisions

Revision as of 19:53, 26 July 2014

Revision as of 16:44, 13 August 2015

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For all matrices '''A''' and '''B''' in ''M''_''mn''(''F''), and all numbers ''α'' in ''F'', a matrix norm, delimited by double vertical bars || ... ||, fulfills:<ref group="note">Some authors, e.g. Horn and Johnson, use triple vertical bars instead of double: |||'''A'''|||.</ref>

*[[Nonnegative]]:

::<math>\| \mathbf{A} \| \ge 0</math>

:with equality only for '''A''' = '''0''', the [[zero matrix]].

*[[Scalar multiplication]]:

::<math>\|\alpha \mathbf{A}\|=|\alpha| \|\mathbf{A}\|</math>

*The [[triangular inequality]]:

::<math>\|\mathbf{A}+\mathbf{B}\| \leq \|\mathbf{A}\|+\|\mathbf{B}\|</math>

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It is defined for matrices of any dimension (i.e. no restriction to square matrices).

==Positive definite and semidefinite matrices ==

{{main|Positive definite matrix}}

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Matrix elements are not restricted to constant numbers, they can be [[mathematical variable]]s.

===Functions of matrices ===

A functions of a matrix takes in a matrix, and return something else (a number, vector, matrix, etc...).

===Matrix-valued functions ===

A matrix valued function takes in something (a number, vector, matrix, etc...) and returns a matrix.

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*{{cite book|title=Matrix Analysis and Applied Linear Algebra Book and Solutions Manual|author=C. Meyer|year=2000 |publisher=SIAM|isbn=089-871-454-0|volume=2|series=Matrix Analysis and Applied Linear Algebra|url=http://books.google.co.uk/books?id=Zg4M0iFlbGcC&printsec=frontcover&dq=Matrix+Analysis&hl=en&sa=X&ei=SCd1UryWD_LG7Aag_4HwBg&ved=0CGoQ6AEwCQ#v=onepage&q=Matrix%20Analysis&f=false}}

*{{cite book|title=Applied Linear Algebra and Matrix Analysis|author=T. S. Shores|year=2007|publisher=Springer|isbn=038-733-195-6|series=[[Undergraduate Texts in Mathematics]]|url=http://books.google.co.uk/books?id=8qwTb9P-iW8C&printsec=frontcover&dq=Matrix+Analysis&hl=en&sa=X&ei=SCd1UryWD_LG7Aag_4HwBg&ved=0CGQQ6AEwCA#v=onepage&q=Matrix%20Analysis&f=false}}

*{{cite book|title=Matrix Analysis|author=Rajendra Bhatia|year=1997|volume=169|series=Matrix Analysis Series|publisher=Springer|isbn=038-794-846-5|url=http://books.google.co.uk/books?id=F4hRy1F1M6QC&printsec=frontcover&dq=matrix+analysis&hl=en&sa=X&ei=_SR1UpbnNarA7AaPjIHIDA&redir_esc=y#v=onepage&q=matrix%20analysis&f=false}}