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Topic: Characteristic polynomial

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	Polynomial - Wikipedia, the free encyclopedia
		Because of their simple structure, polynomials are very easy to evaluate, and are used extensively in numerical analysis for polynomial interpolation or to numerically integrate more complex functions.
		In linear algebra, the characteristic polynomial of a square matrix encodes several important properties of the matrix.
		As there is no general closed formula to calculate the roots of a polynomial of degree 5 and higher, root-finding algorithms are used in numerical analysis to approximate the roots.
	www.wikipedia.org /wiki/Polynomial (2063 words)

	Characteristic polynomial (Site not responding. Last check: 2007-10-07)
		In the case of a diagonal matrix, the characteristic polynomial is easy to define: if the diagonal entries are ''a, b, c the characteristic polynomial will be (t''-''a)(t''-''b)(t''-''c)...
		That is, the diagonal entries become the root s of the characteristic polynomial.
		Polynomial Toolbox A package for polynomials, polynomial matrices and their application in systems, signals and control.
	www.serebella.com /encyclopedia/article-Characteristic_polynomial.html (801 words)

	Characteristic polynomial - Wikipedia, the free encyclopedia
		In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial or secular equation.
		This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace.
		The latter is the characteristic polynomial of A.
	en.wikipedia.org /wiki/Characteristic_polynomial (573 words)

	Polynomial - Wikipedia, the free encyclopedia
		Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics.
		A polynomial with one, two or three terms is called monomial, binomial or trinomial respectively.
		Which algorithm is used for a given polynomial depends on the form of the polynomial and the chosen x.
	en.wikipedia.org /wiki/Polynomial (2063 words)

	Kids.net.au - Encyclopedia Polynomial -
		Polynomials are important because they are the simplest functions: their definition involves only addition and multiplication (since the powers are just shorthands for repeated multiplications).
		The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Weierstrass approximation theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial.
		In order to determine function values of polynomials for given values of the variable x, one does not apply the polynomial as a formula directly, but uses the much more efficient Horner scheme instead.
	www.kids.net.au /encyclopedia-wiki/po/Polynomial (1464 words)

	Eigenvalue, eigenvector and eigenspace - Wikipedia, the free encyclopedia
		This is the characteristic polynomial of A: the eigenvalues of a matrix are the zeros of its characteristic polynomial.
		Computing the polynomial becomes expensive in itself, and exact (symbolic) roots of a high-degree polynomial can be difficult to compute and express: the Abel–Ruffini theorem implies that the roots of high-degree (5 and above) polynomials cannot be expressed simply using nth roots.
		The algebraic multiplicity of an eigenvalue λ of A is the order of λ as a zero of the characteristic polynomial of A; in other words, if t is one root of the polynomial, it is the number of factors (t − λ) in the characteristic polynomial after factorization.
	en.wikipedia.org /wiki/Eigenvector (4159 words)

	Characteristic polynomial -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-07)
		In (The part of algebra that deals with the theory of linear equations and linear transformation) linear algebra, one associates a (A mathematical expression that is the sum of a number of terms) polynomial to every square matrix, its characteristic polynomial.
		This polynomial encodes several important properties of the matrix, most notably its (Click link for more info and facts about eigenvalue) eigenvalues, its (A determining or causal element or factor) determinant and its (Either of two lines that connect a horse's harness to a wagon or other vehicle or to a whiffletree) trace.
		The matrix A and its (A matrix formed by interchanging the rows and columns of a given matrix) transpose have the same characteristic polynomial.
	www.absoluteastronomy.com /encyclopedia/c/ch/characteristic_polynomial.htm (719 words)

	Definition of Matrix eigenvalue problem
		The characteristic polynomial, defined as $det(A - \lambda I), is a$ polynomial in $\lambda whose roots are the eigenvalues of A .$
		However, finding the roots of the characteristic polynomial may be an ill-conditioned problem even when the underlying eigenvalue problem is well-conditioned.
		It can be shown that for any polynomial, there exists a matrix (see companion matrix) having that polynomial as its characteristic polynomial (actually, there are infinitely many).
	www.wordiq.com /definition/Matrix_eigenvalue_problem (434 words)

	Minimal polynomial (Site not responding. Last check: 2007-10-07)
		The minimal polynomial of an n -by- n matrix A over a field F is the monic polynomial p (x) over F of least degree such that p (A)=0.
		λ is a root of the characteristic polynomial of A
		In field theory given a field extension E / F and an element α of E which is algebraic over F the minimal polynomial of α is the monic polynomial p with coefficients in F of least degree such that p (α) = 0.
	www.freeglossary.com /Minimal_polynomial (250 words)

	Characteristic polynomial (Site not responding. Last check: 2007-10-07)
		In the case of a diagonal matrix, the characteristic polynomial iseasy to define: if the diagonal entries are a, b, c the characteristic polynomial will be(t-a)(t-b)(t-c)...
		AB should have the same characteristic polynomial, it essentially forces the definition given later.If M and N are similar matrices, then they also have the same characteristic polynomial.
		This is indeed a polynomial, since determinants are defined interms of sums of products.
	www.therfcc.org /characteristic-polynomial-152483.html (452 words)

	Characteristic polynomial (Site not responding. Last check: 2007-10-07)
		For a diagonal matrix A the characteristic polynomial is easy to if the diagonal entries are a b c the characteristic polynomial will be
		converse however is not true in general: matrices with the same characteristic polynomial need be similar.
		A is similar to a triangular matrix if and only if its characteristic can be completely factored into linear factors K.
	www.freeglossary.com /Characteristic_polynomial (483 words)

	Characteristic polynomial - Term Explanation on IndexSuche.Com (Site not responding. Last check: 2007-10-07)
		The characteristic polynomial of ''A'', denoted by ''p''''A''(''t''), is the element of the polynomial ring ''K''[''t''] defined by :''p''''A''(''t'') = det(''A'' - ''tI'') where ''I'' denotes the ''n''-by-''n'' identity_matrix.
		The degree of the polynomial ''p'A''(''t'') is ''n''.
		For 2×2 matrices, the characteristic polynomial of A is nicely expressed then as : ''t''2-tr(A)''t''+det(A) where tr(A) represents the matrix trace of A and det(A) the determinant of A. The of ''A'' divides the characteristic polynomial of ''A''.
	www.indexsuche.com /Characteristic_polynomial.html (451 words)

	charpoly
		It is fairly easy to figure out the characteristic polynomial of each one when such tools are available.
		I've noticed that the characteristic polynomials of symmetric figures are more factorable than asymmetric figures.
		The roots of a characteristic polynomial for a matrix A are also the eigenvalues for that matrix.
	www.mathpuzzle.com /charpoly.htm (585 words)

	Element Operations
		The minimal polynomial of the element a of the field F, relative to the ground field of F. This is a polynomial with coefficients in the ground field, having a as a root.
		The minimal polynomial of the element a of the field F, relative to the subfield E of F. This is a polynomial with coefficients in E, having a as a root.
		Given an element a of a finite field F, return the characteristic polynomial of a with respect to the subfield E of F. (This polynomial is the characteristic polynomial of the companion matrix of a written as a polynomial over E, and is a power of the minimal polynomial over E.)
	www.math.ufl.edu /help/magma/text330.html (755 words)

	Element Operations
		Given a univariate polynomial f in F[x], over a finite field F, such that the degree of f is greater than or equal to 1, this function returns true if and only if f defines a primitive extension G=F[x]/f of F (that is, x is primitive in G).
		The minimal polynomial of the element a of the field F, relative to the ground field of F. This is the unique minimal-degree monic polynomial with coefficients in the ground field, having a as a root.
		The minimal polynomial of the element a of the field F, relative to the subfield E of F. This is the unique minimal-degree monic polynomial with coefficients in E, having a as a root.
	www.math.wisc.edu /help/magma/text372.html (1180 words)

	Introduction
		For the purpose of this discussion, the characteristic polynomial of an n x n matrix A is the determinant of (A — xI), where I is the n-dimensional identity matrix.
		We will define the characteristic polynomial of a simple graph to be the characteristic polynomial of its adjacency matrix.
		The characteristic polynomial of G is the determinant of M
	www.mathpuzzle.com /characteristic.html (2118 words)

	Non-Periodic Tilings With N-fold Symmetry
		The determinant of the coefficient array is 1, and the characteristic polynomial is
		Since the characteristic polynomial is irreducible, we know the roots are irrational, so the asymptotic proportions of the tiles are also irrational, which implies that the tiling is non-periodic.
		Combining this with the known inflation scale factor and the edge length ratios given by the square roots of the roots of the characteristic polynomial, we can partition a slice of a regular 14-gon into three triangles to satisfy these requirements, and arrive at the tiling discussed previously.
	www.mathpages.com /home/kmath539/kmath539.htm (3213 words)

	6- (Site not responding. Last check: 2007-10-07)
		Thus, the characteristic polynomial does not characterize a single matrix, but a class of similar matrices, although even these classes are not uniquely determined by their characteristic polynomials.
		E is a linear polynomial in L and belongs to the commutative ring R(, L) (cf.
		The ring of polynomials K[x] over the field K is a Euclidean domain; its unities are the polynomials of degree 0, that is, the elements of K which are non-zero (cf.
	kr.cs.ait.ac.th /~radok/math/mat5/algebra62.htm (5933 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		Date: Wed, 22 Sep 1999 15:49:27 +0100 Newsgroups: sci.math.research Hello everybody, I am looking for a hermitean (or symmetric) (N by N) matrix which has a prescribed characteristic polynomial P(x) of degree N. It is assumed that the given characteristic polynomial has real roots only.
		The elements of the matrix should be given explicitly in terms of the coefficients of the polynomial P(x).
		One solution to this problem (using only square roots and arithmetic operations) is given in the article MR 94i:15006 15A18 Schmeisser, Gerhard(D-ERL-MI) A real symmetric tridiagonal matrix with a given characteristic polynomial.
	math.niu.edu /~rusin/known-math/99/charpoly (248 words)

	Polynomial (Site not responding. Last check: 2007-10-07)
		Polynomial (adjective): Of, relating to, or consisting of more than two names or terms.
		The polynomials up to degree n form a vector space of dimension n + 1, which is sometimes called
		A root or zero of a polynomial f is a number ζ so that f(ζ) = 0.
	www.worldhistory.com /wiki/P/Polynomial.htm (1801 words)

	PDA ( Palm Pilot ) synchronization using characteristic polynomial interpolation
		Characteristic Polynomial Interpolation-based Synchronization (CPIsync), that relies on recent information-theoretic research results.
		The most salient property of this scheme is that the amount of communication needed for synchronizing the databases of the PDA and the PC relates only to their mutual differences.
		The basic approach taken by CPIsync is a hashing of the data into a certain type of polynomial known as characteristic polynomial.
	ipsit.bu.edu /nislab/cpisync.html (323 words)

	Learn more about Polynomial in the online encyclopedia. (Site not responding. Last check: 2007-10-07)
		Where the leading coefficient is 1, we describe the polynomial as monic.
		Note that the polynomials of degree ≤ n are precisely those functions whose (n+1)st derivative is identically zero.
		From the definition of O-notation above, the polynomial is in O(x
	www.onlineencyclopedia.org /p/po/polynomial.html (1534 words)

	characteristic_polynomial (Site not responding. Last check: 2007-10-07)
		Given a square matrix ''A'', we want to find a polynomial whose roots are precisely the eigenvalues of ''A''.
		For a diagonal matrix ''A'', the characteristic polynomial is easy to define: if the diagonal entries are ''a'', ''b'', ''c'' the characteristic polynomial will be
		''A'' is similar to a triangular matrix if and only if its characteristic polynomial can be completely factored into linear factors over ''K''.
	q-basic.xodox.de /characteristic_polynomial (503 words)

	Graphs, Matrices, Isomorphism
		If two graphs are isomorphic, they have the same eigenvalues (and the same characteristic polynomial).
		The idiosyncratic polynomial is the characteristic polynomial of the matrix that results from replacing all zeroes in the adjacency matrix by some variable, y.
		The group of isomorphisms from a graph G to itself is the automorphism group of G, Aut(G).
	www.math.fau.edu /locke/Graphmat.htm (1673 words)

	The characteristic polynomial and determinant are not ad hoc constructions (Site not responding. Last check: 2007-10-07)
		There are analogues of the determinant and characteristic polynomial for quaternions, octonions, Jordan algebras, finite-dimensional field extensions, etc., and the two "good" definitions do not cover those cases.
		This paper gives a definition that handles all the cases simultaneously, including deriving the usual formulas for the determinant and characteristic polynomials of matrices from the general definition.
		One can find a similar notion of characteristic polynomial in the literature (e.g., in Bourbaki and in Section 10 of Hilbert's Zahlbericht), where instead of specializing the minimal polynomial of the generic element, one specializes the characteristic polynomial of left multiplication by the generic element.
	www.mathcs.emory.edu /~skip/dets/dets.html (466 words)

	Characteristic Polynomial Equation
		The characteristic polynomial equation for a linear PDE with constant coefficients is obtained by taking the 2D Laplace transform of the PDE with respect to
		In more general PDEs, propagation may be dispersive, in which case the phase velocity depends on frequency (see §D.6 for an analysis of stiff vibrating strings, which are dispersive).
		Moreover, wave propagation may be damped in a frequency-dependent way, in which case one or more roots of the characteristic polynomial equation will have negative real parts; if any roots have positive real parts, we say the initial-value problem is ill posed since is has exponentially growing solutions in response to initial conditions.
	www-ccrma.stanford.edu /~jos/waveguide/Characteristic_Polynomial_Equation.html (290 words)

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