en>Andyjsmith: Reverted 1 edit by 82.141.96.3 (talk) to last revision by Dravecky. (TW)

2013-10-28T10:49:49Z

Reverted 1 edit by 82.141.96.3 (talk) to last revision by Dravecky. (TW)

← Older revision		Revision as of 12:49, 28 October 2013
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	~~The name~~ of the ~~author~~ is ~~Jayson~~. ~~Alaska~~ is ~~where he~~'~~s usually been residing~~. ~~Office supervising is where my main earnings arrives from but I~~'~~ve usually needed my own company~~. ~~To play lacross~~ is the ~~thing I adore most~~ of ~~all~~.<br><br>~~Also visit my web page~~ ... ~~tarot card readings (~~[~~http:~~//~~cartoonkorea.com~~/~~ce002~~/~~1093612 cartoonkorea~~.~~com~~])		In [[algebra]], if <math>A_0, A_1,\dots,A_l</math> are <math>n\times n</math> complex [[matrix (mathematics)\|matrices]] for some nonnegative integer <math>l</math>, and <math>A_l \ne 0</math> (the zero matrix), then the '''matrix [[Pencil (mathematics)\|pencil]]''' of degree <math>l</math> is the matrix-valued function defined on the complex numbers
			:<math>L(\lambda) = \sum_{i=0}^l \lambda^i A_i. </math>

			A particular case is a linear matrix pencil:
			:<math>A-\lambda B \, </math>
			with
			:<math>\lambda \in \mathbb C\text{ (or }\mathbb R\text{),}</math>
			where <math>A</math> and <math>B</math> are complex (or real) <math>n \times n</math> [[matrix (mathematics)\|matrices]]. We denote it briefly with the notation <math>(A,B)</math>

			A pencil is called ''regular'' if there is at least one value of <math>\lambda</math> such that <math>\det(A-\lambda B)\neq 0</math>. We call ''eigenvalues'' of a matrix pencil <math>(A,B)</math> all complex numbers <math>\lambda</math> for which <math>\det(A-\lambda B)=0</math> (see [[eigenvalue]] for comparison). The set of the eigenvalues is called the ''spectrum'' of the pencil and is written <math>\sigma(A,B)</math>.
			Moreover, the pencil is said to have one or more eigenvalues at infinity if <math>B</math> has one or more 0 eigenvalues.

			==Applications==
			Matrix pencils play an important role in [[numerical linear algebra]]. The problem of finding the eigenvalues of a pencil is called the [[generalized eigenvalue problem]]. The most popular algorithm for this task is the [[QZ algorithm]], which is an implicit version of the [[QR algorithm]] to solve the associated eigenvalue problem <math>B^{-1}Ax=\lambda x</math> without forming explicitly the matrix <math>B^{-1}A</math> (which could be impossible or ill-conditioned if <math>B</math> is singular or near-singular)


			==Pencil generated by commuting matrices==
			If <math>AB = BA</math>, then the pencil generated by <math>A</math> and <math>B</math> (Marcus & Minc, 1969, p. 79):

			1) consists only of matrices similar to a diagonal matrix

			or

			2) has no matrices in it similar to a diagonal matrix

			or

			3) has exactly one matrix in it similar to a diagonal matrix.


			== See also ==
			* [[Generalized eigenvalue problem]]
			* [[Nonlinear eigenproblem]]
			* [[Quadratic eigenvalue problem]]
			* [[Rayleigh quotient#Generalization\|Generalized Rayleigh quotient]]

			==References==
			{{Reflist}}
			* {{Citation
			\| last = Marcus & Minc
			\| date = 1969
			\| title = A survey of matrix theory and matrix inequalities
			\| publisher = Courier Dover Publications
			}}

			[[Category:Linear algebra]]


			{{Linear-algebra-stub}}

en>Dravecky: stub sorting

2012-07-01T04:40:46Z

stub sorting

New page

The name of the author is Jayson. Alaska is where he's usually been residing. Office supervising is where my main earnings arrives from but I've usually needed my own company. To play lacross is the thing I adore most of all.<br><br>Also visit my web page ... tarot card readings ([http://cartoonkorea.com/ce002/1093612 cartoonkorea.com])

Electrically short - Revision history

en>Andyjsmith: Reverted 1 edit by 82.141.96.3 (talk) to last revision by Dravecky. (TW)

en>Dravecky: stub sorting