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Topic: Companion matrix

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	Companion matrix - Wikipedia, the free encyclopedia
		In linear algebra, the companion matrix of the monic polynomial
		The characteristic polynomial as well as the minimal polynomial of C(p) are equal to p; in this sense, the matrix C(p) is the "companion" of the polynomial p.
		But every matrix is similar to a matrix made up of blocks of companion matrices.
	en.wikipedia.org /wiki/Companion_matrix (210 words)

	List of matrices - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-10-08)
		Companion matrix - the companion matrix of a polynomial is a special form of matrix, whose eigenvalues are equal to the roots of the polynomial.
		Permutation matrix - matrix representation of a permutation.
		Toeplitz matrix - a matrix with constant diagonals.
	encyclopedia.learnthis.info /l/li/list_of_matrices.html (622 words)

	Companions (Site not responding. Last check: 2007-10-08)
		Companion planting was widely touted in the 1970s as part of the organic gardening movement.
		Companion planting and use of nurse crops are proscribed in Leviticus 19:19.
		In linear algebra, the companion matrix of the monic polynomial :
	www.wwwtln.com /finance/42/companions.html (837 words)

	Matrix Manual: Special Matrices (Site not responding. Last check: 2007-10-08)
		[22]: A 22 matrix is a Givens reflection iff it is a Householder matrix.
		A symmetric or Hermitian Hessenberg matrix is tridiagonal.
		A is symmetric iff it is congruent to a diagonal matrix.
	www.psi.toronto.edu /matrix/special.html (2192 words)

	Matrix Reference Manual: Special Matrices
		A 3#3 orthogonal matrix is either a rotation matrix or else a rotation matrix plus a reflection in the plane of the rotation according to whether it is proper or improper.
		A Rotation matrix is orthogonal with a determinant of +1.
		A signature matrix is a diagonal matrix whose diagonal entries are all +1 or -1.
	www.ee.uwa.edu.au /~roberto/teach/matrix/special.html (4269 words)

	jordan.html (Site not responding. Last check: 2007-10-08)
		The multipliciation matrix is known as the companion matrix:
		The roots of the polynomial are eigenvalues of the companion matrix:
		Because we have a triple root, the companion matrix cannot be diagonalized.
	www.math.uic.edu /~jan/mcs563/jordan1.html (226 words)

	Linear Phase Portraits: Matrix Entry -- Help (Site not responding. Last check: 2007-10-08)
		Choose between a companion matrix or a general matrix using the [Companion matrix] key.
		When [Companion Matrix] is not selected, only the entries from the bottom rows can be selected.
		When [Companion matrix] is selected, the matrix can also be controlled by rolling over the plane at upper left or by grabbing the [tr] or [det] sliders alongside it.
	www-math.mit.edu /daimp/LinPhasePorMatrixHelp.html (137 words)

	canon (Function Reference)
		The reduction to companion form uses a state similarity transformation based on the controllability matrix [1].
		The companion transformation requires that the system be controllable from the first input.
		The companion form is often poorly conditioned for most state-space computations; avoid using it when possible.
	www-rohan.sdsu.edu /doc/matlab/toolbox/control/ref/canon.html (116 words)

	The Guild Companion: Matrix Spell Casting (Site not responding. Last check: 2007-10-08)
		A long time ago I wrote an article describing my view on The Ecology of Magic and while it was written for another system, the concepts developed in it are easily converted to this system's framework.
		Some casters use a focus, such as a holy symbol, while others use material items that are consumed in the casting, as the core of the matrix formed.
		The caster may then use any of his spell slots (number of spells per day) to cast any spell that he has prepared of an equal or lesser level (equal or less than the spell slot that is).
	www.guildcompanion.com /scrolls/2002/apr/matrix.html (1949 words)

	A companion matrix resultant for Bernstein (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		Abstract: A closed form expression for a companion matrix M of a Bernstein polynomial is obtained, and this is used to derive an expression for a resultant matrix of two Bernstein polynomials.
		A measure of the numerical condition of a resultant matrix, for polynomials in an arbitrary basis, is reviewed and this is used...
		1 A Bezoutian matrix for Chebyshev polynomials (context) - Barnett - 1988
	citeseer.ist.psu.edu /686440.html (488 words)

	Polynomial Roots from Companion Matrix Eigenvalues - Edelman, Murakami (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		Abstract: In classical linear algebra, the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial.
		An algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition.
		17 Pseudozeros of polynomials and pseudospectra of companion ma..
	citeseer.ist.psu.edu /25171.html (499 words)

	MuPAD documentation (Site not responding. Last check: 2007-10-08)
		If p is a polynomial, then the component ring of the returned matrix is the coefficient ring of p, except in two cases for built-in coefficient rings: if the coefficient ring of p is Expr then the domain
		Dom::ExpressionField() is the component ring of the companion matrix.
		The companion matrix of a univariate polynomial p of degree n is an
	www.mupad.de /doc/31/eng/linalg_companion.html (331 words)

	Linear Algebra Glossary (Site not responding. Last check: 2007-10-08)
		A border banded matrix is a 2 by 2 block matrix comprising a (large) leading block which is a square banded matrix, two dense rectangular side strips, and a (small) trailing block which is a square dense matrix.
		The inverse of a (nonsingular) circulant matrix is a circulant matrix.
		The transpose of a circulant matrix is a circulant matrix.
	www.csit.fsu.edu /~burkardt/papers/linear_glossary.html (13644 words)

	: Class JElmat
		compan(JElmat poly) is a companion matrix of the polynomial with the coefficients as elements of poly matrix vector.
		The poly matrix has to be a vector (either a column or row).
		matrix with all ones on the main diagonal and zeros everywhere.
	www.cs.utexas.edu /~kane/Java_Docs_HTML/jamlab/JElmat.html (2440 words)

	A note on parameter differentiation of matrix exponentials, with applications to continuous-time modelling, Henghsiu ...
		The new formula expresses the derivatives of a matrix exponential in terms of minors, polynomials, the exponential of the matrix as well as matrix inversion, and hence is algebraically more manageable.
		For the companion matrix of a continuous-time autoregress\-ive moving average process, the derivatives of the exponential of the companion matrix can be computed recursively.
		The second example concerns the exponential of the tri\-diagonal transition intensity matrix of a finite-state-space continuous-time Markov chain whose instantaneous transitions must be between adjacent states.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.bj/1066418883 (300 words)

	A General Test for the Cointegrating Rank in Vector Autoregressive Models
		It is shown that the limiting distribution of the eigenvalues closest to one of the companion matrix is free of nuisance parameters, a useful result in its own right.
		The test is based on the eigenvalues of the companion matrix, more precisely on the sum of the real parts of those closest to one.
		The roots of the companion matrix are often inspected as a diagnostic tool.
	ideas.repec.org /p/hhb/hanken/0499.html (800 words)

	linalg::companion -- Companion matrix of a univariate polynomial (Site not responding. Last check: 2007-10-08)
		returns the companion matrix associated with the polynomial
		is a polynomial, then the component ring of the returned matrix is the coefficient ring of
		is the component ring of the companion matrix.
	www.mupad.de /doc/25/eng/linalg/companion.shtml (247 words)

	Octave - Polynomial Manipulations
		Compute the companion matrix corresponding to polynomial coefficient vector
		The companion matrix is The eigenvalues of the companion matrix are equal to the roots of the polynomial.
		matrix multiplication is used instead of element by element multiplication as is used in polyval.
	www.math.utah.edu /docs/info/octave_11.html (353 words)

	Cryptography
		To compute the determinant of a 2 x 2 matrix, step across the entries in the first row of the matrix, multiply each entry by the determinant of the 1 x 1 matrix that appears when the row and the column the entry appears in are eliminated from the matrix.
		The determinant of a 3 x 3 matrix is produced in the same way: step across the first row, multiply each entry by the determinant of the 2 x 2 matrix that appears when the entry's row and column are crossed out, and alternately add and subtract the resulting numbers.
		Next, the matrix is flipped over the diagonal from the upper left corner to the lower right corner so that the first row be comes the first column, the second rows becomes the second column, and so on.
	www.threaded.com /cryptography8.htm (13300 words)

	Comments on the Mid Term
		Let H be a nonsingular matrix such that Hx1 = cx1 - a multiple of the unit vector e1.
		Also please use the suggested matrix M (Elimination matrix) in Problem 23 text for H, and not the Householder reflection.
		While there are various choices you could use for H to effect deflation on the 4x4 matrix C4 (any H such that Hx1 = ce1 will do), each may give a different 3x3 matrix C3, although of course the eigenvalues of C3 will always be the same by similarity.
	www.cs.colorado.edu /~mcbryan/3656.04/mail/70.htm (884 words)

	Math 60 -- Notes A2: Linear Algebra (Site not responding. Last check: 2007-10-08)
		It is not always obvious what the correct eigenvalues and eigenvectors are for a matrix.
		+ dx + e = 0, the companion matrix for this polynomial is the matrix with all zeroes except for (1) ones in the superdiagonal (diagonal above the major diagonal), and (2) the negatives of the coefficients in the last row (with the constant in the first column).
		THEOREM: The roots of the polynomial are the same as the eigenvalues of the companion matrix.
	math.scu.edu /~dsmolars/ma60/notesa2.html (238 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		18.03 Class 35, May 6 The companion matrix and its phase portraits; The matrix exponential: initial value problems.
		Recall that the first column of any matrix B is the product B[1;0].
		Such a matrix will be denoted by Phi(t); so here Phi(t) = [ e^t e^{2t} ; 0 e^{2t} ] Phi(t) behaves very much like we want e^{At} to behave; its columns are independent solutions.
	web.mit.edu /18.03/spr02/lec/5.6.02.html (675 words)

	Linear Algebra Glossary
		Each elemental matrix is represented by a list of the row/column indices (variables) associated with the element and by a small dense matrix giving the numerical values by columns, or in the symmetric case, only the lower triangular part.
		Eigenvalue programs typically transform a matrix into upper Hessenberg form, and then carry out the QR method on this matrix, which converges rapidly to a matrix which is diagonal except for 2 by 2 blocks corresponding to complex eigenvalues.
		The identity matrix, usually denoted I, is a square matrix with 1's on the main diagonal and 0's elsewhere.
	orion.math.iastate.edu /burkardt/papers/linear_glossary.html (13535 words)

	An #tex2html_wrap_inline165# Algorithm for Frobenius Normal Form (Site not responding. Last check: 2007-10-08)
		Near optimal randomized (Las Vegas) algorithms which use fast matrix mutliplication techniques are given in [2].
		where E is the elementary matrix corresponding to an elementary row operation: interchanging two rows; multiplying a row by an invertible element; adding a multiple of a row to a different row.
		After stage one the work matrix is transposed and the three step process applied recursively to the trailing block.
	www-lmc.imag.fr /cathode2/Cirm/abstract/abs_storjohann/abs_storjohann.html (347 words)

	Matrix Market: Spectral Portraits
		This page was kindly contributed to the Matrix Market by Alan McCoy, Vincent Toumazou, and Valerie Fraysse of the Qualitative Computing Group at CERFACS.
		Perturbations on the computed eigenvalues are the consequence of perturbations E on the matrix A.
		Suppose that the matrix A is known to the normwise relative precision
	math.nist.gov /MatrixMarket/spectral.html (503 words)

	Stata help for vecstable
		amat(matrix_name) specifies a valid Stata matrix name by which the companion matrix can be saved.
		The companion matrix is referred to as the A matrix in Lüutkepohl and varstable.
		graph causes vecstable to draw a graph of the eigenvalues of the companion matrix.
	www.stata.com /help.cgi?vecstable (296 words)

	Talk:Vandermonde matrix - Wikipedia, the free encyclopedia
		I switched to the transpose, since the equation for polynomial interpolation was incorrect and the transposed matrix works better in companion matrix as well.
		Something is wrong with the formula for the confluent Vandermonde matrix; the index surely cannot be 0?
		Not knowing much of anything about these matrices, the latter seems more naturally to me, seeing that k=0 is the case of an ordinary Vandermonde matrix, and j=k gives a (-1)!
	en.wikipedia.org /wiki/Talk:Vandermonde_matrix (484 words)

	Companion matrix at opensource encyclopedia (Site not responding. Last check: 2007-10-08)
		In linear algebra, a companion matrix is of form
		The eigenvalues of a companion matrix can be solved from
		Furthermore, the eigenvectors are of form $[1\,\lambda_i\,\lambda_i^2\,\lambda_i^3\dots\lambda_i^{n-1}]^T$ or, in other words, the eigenmatrix is a Vandermonde matrix.
	www.wiki.tatet.com /Companion_matrix.html (229 words)

	ECS EPrints Service - Evaluation of matrix polynomials in the state companion matrix of linear time invariant systems (Site not responding. Last check: 2007-10-08)
		ECS EPrints Service - Evaluation of matrix polynomials in the state companion matrix of linear time invariant systems
		Evaluation of matrix polynomials in the state companion matrix of linear time invariant systems
		Harris, C. Evaluation of matrix polynomials in the state companion matrix of linear time invariant systems.
	eprints.ecs.soton.ac.uk /2052 (82 words)

	PlanetMath:
		Cauchy matrices (in Cauchy matrix) owned by kshum
		characteristic matrix of diagonal element cross-section owned by lars_h
		condition number (=matrix condition number) owned by stevecheng
	planetmath.org /encyclopedia/C (3726 words)

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