en>Dthomsen8: clean up, typo(s) fixed: , → , (2) using AWB

2014-01-23T13:27:51Z

clean up, typo(s) fixed: , → , (2) using AWB

← Older revision		Revision as of 15:27, 23 January 2014
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	~~Nice to satisfy you~~, ~~my title~~ is ~~Ling~~ and ~~I totally dig~~ that ~~name~~. The ~~factor she adores most~~ is ~~to perform handball but she can't make it her occupation~~. ~~He currently life in Arizona~~ and ~~his mothers~~ and ~~fathers live nearby~~. ~~Interviewing~~ is ~~what she does but quickly she'll be on her own~~.<br><br>~~Here~~ is ~~my web page~~ - ~~auto warranty;~~ [http://~~Frag5~~.com/~~index~~.~~php?mod~~=~~users&action~~=~~view&id~~=~~13770 try this website~~],		In [[mathematics]], a '''<math>P</math>-matrix''' is a complex square [[matrix (mathematics)\|matrix]] with every principal [[Minor (linear algebra)\|minor]] > 0. A closely related class is that of <math>P_0</math>-matrices, which are the closure of the class of <math>P</math>-matrices, with every principal minor <math>\geq</math> 0.

			== Spectra of <math>P</math>-matrices ==
			By a theorem of Kellogg, the [[eigenvalue]]s of <math>P</math>- and <math>P_0</math>- matrices are bounded away from a wedge about the negative real axis as follows:

			:If <math>\{u_1,...,u_n\}</math> are the eigenvalues of an <math>n</math>-dimensional <math>P</math>-matrix, then
			::<math>\|arg(u_i)\| < \pi - \frac{\pi}{n}, i = 1,...,n</math>
			:If <math>\{u_1,...,u_n\}</math>, <math>u_i \neq 0</math>, <math>i = 1,...,n</math> are the eigenvalues of an <math>n</math>-dimensional <math>P_0</math>-matrix, then
			::<math>\|arg(u_i)\| \leq \pi - \frac{\pi}{n}, i = 1,...,n</math>

			== Remarks ==
			The class of nonsingular [[M-matrix\|''M''-matrices]] is a subset of the class of <math>P</math>-matrices. More precisely, all matrices that are both <math>P</math>-matrices and [[Z-matrix (mathematics)\|''Z''-matrices]] are nonsingular <math>M</math>-matrices. The class of [[sufficient matrix\|sufficient matrices]] is another generalization of <math>P</math>-matrices.<ref>{{cite journal\|first1=Zsolt\|last1=Csizmadia\|first2=Tibor\|last2=Illés\|title=New criss-cross type algorithms for linear complementarity problems with sufficient matrices\|journal=Optimization Methods and Software\|volume=21\|year=2006\|number=2\|pages=247–266\|doi=10.1080/10556780500095009\|
			url=http://www.cs.elte.hu/opres/orr/download/ORR03_1.pdf\|format=pdf\|url2=http://www.tandfonline.com/doi/abs/10.1080/10556780500095009\|eprint=http://www.tandfonline.com/doi/pdf/10.1080/10556780500095009\|mr=2195759\|<!-- ref=harv -->}}</ref>

			If the [[Jacobian matrix and determinant\|Jacobian]] of a function is a <math>P</math>-matrix, then the function is injective on any rectangular region of <math>\mathbb{R}^n</math>.

			A related class of interest, particularly with reference to stability, is that of <math>P^{(-)}</math>-matrices, sometimes also referred to as <math>N-P</math>-matrices. A matrix <math>A</math> is a <math>P^{(-)}</math>-matrix if and only if <math>(-A)</math> is a <math>P</math>-matrix (similarly for <math>P_0</math>-matrices). Since <math>\sigma(A) = -\sigma(-A)</math>, the eigenvalues of these matrices are bounded away from the '''positive''' real axis.

			== See also ==
			* [[Z-matrix (mathematics)]]
			* [[M-matrix]]
			* [[Perron–Frobenius theorem]]
			* [[Hurwitz matrix]]

			== Notes ==

			<references/>

			==References==

			* {{cite journal\|first1=Zsolt\|last1=Csizmadia\|first2=Tibor\|last2=Illés\|title=New criss-cross type algorithms for linear complementarity problems with sufficient matrices\|journal=Optimization Methods and Software\|volume=21\|year=2006\|number=2\|pages=247–266\|doi=10.1080/10556780500095009\|
			url=http://www.cs.elte.hu/opres/orr/download/ORR03_1.pdf\|format=pdf\|url2=http://www.tandfonline.com/doi/abs/10.1080/10556780500095009\|eprint=http://www.tandfonline.com/doi/pdf/10.1080/10556780500095009\|mr=2195759\|ref=harv}}
			* [[David Gale]] and [[Hukukane Nikaido]], The Jacobian matrix and global univalence of mappings, ''Math. Ann.'' 159:81-93 (1965) {{doi\|10.1007/BF01360282}}
			* Li Fang, On the Spectra of <math>P</math>- and <math>P_0</math>-Matrices, ''Linear Algebra and its Applications'' 119:1-25 (1989)
			* R. B. Kellogg, On complex eigenvalues of <math>M</math> and <math>P</math> matrices, ''Numer. Math.'' 19:170-175 (1972)

			[[Category:Matrix theory]]
			[[Category:Matrices]]


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en>Fuhghettaboutit: Fuhghettaboutit moved page Standard Linear Solid model to Standard linear solid model: Requested move; see talk page

2012-08-08T12:51:40Z

Fuhghettaboutit moved page Standard Linear Solid model to Standard linear solid model: Requested move; see talk page

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Nice to satisfy you, my title is Ling and I totally dig that name. The factor she adores most is to perform handball but she can't make it her occupation. He currently life in Arizona and his mothers and fathers live nearby. Interviewing is what she does but quickly she'll be on her own.<br><br>Here is my web page - auto warranty; [http://Frag5.com/index.php?mod=users&action=view&id=13770 try this website],

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en>Dthomsen8: clean up, typo(s) fixed: , → , (2) using AWB

en>Fuhghettaboutit: Fuhghettaboutit moved page Standard Linear Solid model to Standard linear solid model: Requested move; see talk page