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			In [[linear algebra]], the [[Ferdinand Georg Frobenius\|Frobenius]] '''companion matrix''' of the [[monic polynomial]]
			:<math>
			p(t)=c_0 + c_1 t + \cdots + c_{n-1}t^{n-1} + t^n ~,
			</math>
			is the [[square matrix]] defined as

			:<math>C(p)=\begin{bmatrix}
			0 & 0 & \dots & 0 & -c_0 \\
			1 & 0 & \dots & 0 & -c_1 \\
			0 & 1 & \dots & 0 & -c_2 \\
			\vdots & \vdots & \ddots & \vdots & \vdots \\
			0 & 0 & \dots & 1 & -c_{n-1}
			\end{bmatrix}.</math>

	~~Hi there~~. ~~The author~~'s ~~name is Eusebio but he never really beloved that name~~. The [http://~~Data~~.~~Gov~~.uk/~~data/search~~?q=~~desired+hobby desired hobby~~] ~~for him~~ ~~his kids~~ is to ~~actually drive and he~~'~~s already doing it~~ for ~~some time~~. ~~Auditing~~ is ~~how he supports your man~~'~~s family~~. ~~Massachusetts~~ is ~~where or even~~ and ~~his wife survive~~. He's ~~been working around his website for individuals time now. Check it out here~~: ~~http~~://~~circuspartypanama~~.~~com~~<br><br>~~My blog post; [http~~://~~circuspartypanama~~.~~com clash~~ of ~~clans hacks no survey~~]		With this convention, and on the [[Basis (linear algebra)\|basis]] {{math\| ''v''<sub>1</sub>, ... , ''v''<sub>''n''</sub>}}, one has
			:<math>Cv_i = C^{i}v_1 = v_{i+1}</math>
			(for {{math\|''i'' < ''n''}}), and {{math\| ''v''<sub>1</sub>}} generates {{mvar\|V}} as a {{math\| ''K''[''C'']}}-module: {{mvar\|C}} cycles basis vectors.

			Some authors use the [[transpose]] of this matrix, which (dually) cycles coordinates, and is more convenient for some purposes, like linear [[recursive relation]]s.

			==Characterization==
			The [[characteristic polynomial]] as well as the [[minimal polynomial (linear algebra)\|minimal polynomial]] of {{math\|''C''(''p'')}} are equal to {{mvar\|p}}.<ref>
			{{Cite book \| last = Horn \| first = Roger A. \| coauthors = Charles R. Johnson
			\| title = Matrix Analysis \| publisher = Cambridge University Press \| year = 1985
			\| location = Cambridge, UK \| pages = 146–147
			\| url = http://books.google.com/books?id=f6_r93Of544C&pg=PA147&dq=%22companion+matrix%22&cd=1#v=onepage&q=%22companion%20matrix%22&f=false
			\| accessdate = 2010-02-10 \| isbn = 0-521-30586-1}}</ref>

			In this sense, the matrix {{math\|''C''(''p'')}} is the "companion" of the polynomial {{mvar\|p}}.

			If {{mvar\|A}} is an ''n''-by-''n'' matrix with entries from some [[field (mathematics)\|field]] {{mvar\|K}}, then the following statements are equivalent:
			* {{mvar\|A}} is [[similar (linear algebra)\|similar]] to the companion matrix over {{mvar\|K}} of its characteristic polynomial
			* the characteristic polynomial of {{mvar\|A}} coincides with the minimal polynomial of {{mvar\|A}}, equivalently the minimal polynomial has degree {{mvar\|n}}
			* there exists a [[cyclic permutation\|cyclic vector]] {{math\|'''v'''}} in <math>V=K^n</math> for {{mvar\|A}}, meaning that {'''v''', ''A'''''v''', ''A''<sup>2</sup>'''v''', ..., ''A''<sup>''n''−1</sup>'''v'''} is a [[basis (linear algebra)\|basis]] of ''V''. Equivalently, such that ''V'' is cyclic as a <math>K[A]</math>-module (and <math>V=K[A]/(p(A))</math>); one says that {{mvar\|A}} is ''regular''.

			Not every square matrix is similar to a companion matrix. But every matrix is similar to a matrix made up of blocks of companion matrices. Furthermore, these companion matrices can be chosen so that their polynomials divide each other; then they are uniquely determined by {{mvar\|A}}. This is the [[Frobenius normal form\|rational canonical form]] of {{mvar\|A}}.

			==Diagonalizability==
			If {{math\|''p''(''t'')}} has distinct roots {{math\|''λ''<sub>1</sub>, ..., ''λ''<sub>''n''</sub>}} (the [[eigenvalue]]s of ''C''(''p'')), then ''C''(''p'') is [[diagonalizable]] as follows:
			:<math>V C(p) V^{-1} = \operatorname{diag}(\lambda_1,\dots,\lambda_n)</math>
			where {{mvar\|V}} is the [[Vandermonde matrix]] corresponding to the {{mvar\|λ}}'s.

			In that case,<ref>[[Richard E. Bellman\|Bellman]], Richard (1987), ''Introduction to Matrix Analysis'', SIAM, ISBN 0898713994 .</ref> traces of powers ''m'' of {{mvar\|C}} readily yield sums of the same powers ''m'' of all roots of ''p''(''t''),
			:<math>\mathrm{Tr} C^m = \sum_{i=1}^{n} \lambda_i^m ~. </math>

			In general, the companion matrix may be non-diagonalizable.

			==Linear recursive sequences==
			Given a [[linear recursive sequence]] with characteristic polynomial
			:<math>p(t)=c_0 + c_1 t + \cdots + c_{n-1}t^{n-1} + t^n \, </math>
			the (transpose) companion matrix
			:<math>C^T(p)=\begin{bmatrix}
			0 & 1 & 0 & \cdots & 0\\
			0 & 0 & 1 & \cdots & 0\\
			\vdots & \vdots & \vdots & \ddots & \vdots \\
			0 & 0 & 0 & \cdots & 1\\
			-c_0 & -c_1 & -c_2 & \cdots & -c_{n-1}
			\end{bmatrix}</math>
			generates the sequence, in the sense that
			:<math>C^T\begin{bmatrix}a_k\\
			a_{k+1}\\
			\vdots \\
			a_{k+n-1}
			\end{bmatrix}
			= \begin{bmatrix}a_{k+1}\\
			a_{k+2}\\
			\vdots \\
			a_{k+n}
			\end{bmatrix}.</math>
			increments the series by 1.

			The vector {{math\|(1,''t'',''t''<sup>2</sup>, ..., ''t''<sup>''n''-1</sup>)}} is an eigenvector of this matrix for eigenvalue {{mvar\|t}}, when {{mvar\|t}} is a root of the characteristic polynomial {{math\|''p''(''t'')}}.

			For {{math\|''c''<sub>0</sub> {{=}} −1}}, and all other {{math\|''c<sub>i</sub>''{{=}}0}}, i.e., {{math\|''p''(''t'') {{=}} ''t<sup>n</sup>''−1}}, this matrix reduces to Sylvester's cyclic [[Generalizations_of_Pauli_matrices#Construction:_The_clock_and_shift_matrices\|shift matrix]], or [[circulant matrix]].

			==See also==
			*[[Frobenius endomorphism]]
			*[[Cayley-Hamilton theorem]]

			==Notes==
			{{reflist}}

			{{DEFAULTSORT:Companion Matrix}}
			[[Category:Matrices]]
			[[Category:Matrix theory]]

en>Jim1138: Undid revision 503970441 by 24.205.142.154 (talk) rv unexplained removal of author's last name

2012-07-24T17:35:57Z

Undid revision 503970441 by 24.205.142.154 (talk) rv unexplained removal of author's last name

New page

Hi there. The author's name is Eusebio but he never really beloved that name. The [http://Data.Gov.uk/data/search?q=desired+hobby desired hobby] for him his kids is to actually drive and he's already doing it for some time. Auditing is how he supports your man's family. Massachusetts is where or even and his wife survive. He's been working around his website for individuals time now. Check it out here: http://circuspartypanama.com<br><br>My blog post; [http://circuspartypanama.com clash of clans hacks no survey]

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en>Jim1138: Undid revision 503970441 by 24.205.142.154 (talk) rv unexplained removal of author's last name