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Topic: Vandermonde matrix

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Vandermonde determinant

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	Seminars \| Signals and Systems Group, Uppsala University
		In this seminar, the focus is on the use of the Davies Transformation, which is essentially a spatial-DFT, to map the UCA to a virtual array.
		This non-robust behaviour is the result of the Davies transformation matrix having a large norm for certain array parameters.
		The robust matrix is found by posing and solving a quadratic semi-infinite optimization problem which trades-off the Vandermonde approximation error with a matrix of lower norm.
	www.signal.uu.se /Courses/Semabstracts/s0212.html (298 words)

	Vandermonde matrix - Wikipedia, the free encyclopedia
		The Vandermonde determinant plays a central role in the Frobenius formula, which gives the character of conjugacy classes of representations of the symmetric group.
		Confluent Vandermonde matrices are used in Hermite interpolation.
		A commonly known special Vandermonde matrix is the discrete Fourier transform matrix.
	en.wikipedia.org /wiki/Vandermonde_matrix (422 words)

	List of matrices
		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.
	www.brainyencyclopedia.com /encyclopedia/l/li/list_of_matrices.html (644 words)

	Simplified construction of erasur code for three of fewer erasures
		The key observation was that a Vandermonde matrix has the property that any m by m submatrix is invertible, which allows the data elements to be reconstructed from any m of the m+e result words.
		A Vandermonde matrix is a matrix whose columns consist of consecutive powers of distinct elements of the ground field over which we are working.
		In a standard Vandermonde matrix, the first row contains the zero'th powers, the second row the first powers, etc. Invertibility follows from computation of the determinant; which (up to sign) is equal to the product of the differences of the elements used to generate the matrix.
	research.microsoft.com /research/sv/Kohinoor/ReedSolomon3.htm (845 words)

	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 (200 words)

	PlanetMath: Vandermonde matrix (Site not responding. Last check: )
		Vandermonde matrices usually arise when considering systems of polynomials evaluated at specific points (i.e.
		This may happen, for example, when trying to solve for constants from initial conditions in systems of differential equations or recurrence relations.
		This is version 1 of Vandermonde matrix, born on 2002-09-28.
	planetmath.org /encyclopedia/VandermondeMatrix.html (134 words)

	Vandermonde matrix -- Facts, Info, and Encyclopedia article (Site not responding. Last check: )
		In that case one may use a generalization called confluent Vandermonde matrices, which makes the matrix (Click link for more info and facts about positive definite) positive definite while retaining most properties.
		The above formula for confluent Vandermonde matrices can be readily derived by letting two parameters and go arbitrarily close to each other.
		When the values range over powers of a (Click link for more info and facts about finite field) finite field, then the determinant is more commonly known as the Moore determinant, which has a number of interesting properties.
	www.absoluteastronomy.com /encyclopedia/v/va/vandermonde_matrix.htm (355 words)

	[No title] (Site not responding. Last check: )
		Use linalg::vandermonde([v[1],..., v[n]], R) to define the n × n Vandermonde matrix over the field R.
		We construct a 3 × 3 Vandermonde matrix:
		The inverses of large floating point Vandermonde matrices are subject to severe round-off effects.
	www.sciface.com /STATIC/DOC30/eng/linalg_vandermonde.html (164 words)

	Constrained, non-linear, derivative-free parallel optimization of continuous, high computing load, noisy objective ... (Site not responding. Last check: )
		The matrix above is called the ``Vandermonde Matrix''.
		The determinant of the Vandermonde Matrix (called here after the ``Vandermonde determinant'') will be null.
		If the Vandermonde determinant is not null for this set of points, the problem is ``well poised''.
	iridia.ulb.ac.be /~fvandenb/work/thesis/node14.html (267 words)

	MAT520 - Project 1
		grcar Grcar matrix -- a Toeplitz matrix with sensitive eigenvalues.
		hanowa Matrix whose eigenvalues lie on a vertical line in the complex plane.
		ris Ris matrix -- a symmetric Hankel matrix.
	math.la.asu.edu /~bdw/MAT520/Fa98/P1 (175 words)

	[No title]
		A Toeplitz matrix is defined by EMBED Equation.3 , which means that all entries on the same downward diagonal are required to be the same.
		Here is an example of a Hankel matrix: EMBED Equation.3 A Vandermonde matrix begins with a column vector, and raises the entries of the column to the c-1 power, where c is the number of the column in the matrix.
		MNEG: Input: Matrix M Output: Additive inverse of M The sum of M and the additive inverse of M is 0.
	www.cs.caltech.edu /~joyjoy/CSReport.doc (2247 words)

	HPCLab University of Patras - Courses
		Analysis of the D&C approach, Examples, The Algorithm of Strassen for Matrix Multiplication, Sorting arrays using D&C methods.
		Memory storage and solution of special linear systems: Banded matrices, Hessenberg matrices, block tridiagonal matrices, Vandermonde, Toeplitz, Hankel matrices.
		Polynomials and the irrelation to matrices of special structure.
	www.hpclab.ceid.upatras.gr /home.php?action=courses&language=2 (860 words)

	[No title] (Site not responding. Last check: )
		The Vandermonde points v and the right hand side y of the linear system are entered as vectors:
		The Vandermonde points v and the right hand side y of the linear system are entered as 2 × 1 matrices:
		We construct the Vandermonde matrix V and verify the result:
	www.sciface.com /STATIC/DOC30/eng/linalg_vandermondeSolve.html (302 words)

	A Generalized Hilbert Matrix Problem and Confluent Chebyshev--Vandermonde Systems
		One is for the case where the points involved in the generalized Hilbert matrices satisfy a TH-relation introduced in the present paper, which include equidistant points, clustered points, and Chebyshev points.
		The other method is to reduce the generalized Hilbert matrix problem to products of confluent Vandermonde-like matrices and dual confluent Vandermonde-like matrices with vectors by using J-matches and links of polynomials.
		The solution of Chebyshev--Vandermonde systems is also reduced to the generalized Hilbert matrix problem by using J-matches, links of Chebyshev polynomials, and the inversion of a class of generalized Hilbert matrices.
	epubs.siam.org /sam-bin/dbq/article/30722 (376 words)

	Vandermonde Matrix
		This multiplies the determinant by the same constant, and the matrix remains nonsingular.
		The exponential functions, represented by the rows, are multiplied by a particular polynomial, and the result is evaluated at n successive integers in the domain.
		Since the matrix remains nonsingular, the functions are linearly independent.
	www.mathreference.com /la-det,vander.html (388 words)

	[No title] (Site not responding. Last check: )
		% K = 4: Helmert matrix: a permutation of a lower Hessenberg matrix, % whose first row is ONES(1:N)/SQRT(N).
		% Symmetric matrix arising in the Hartley transform.
		% D. Bini and P. Favati, On a matrix algebra related to the discrete % Hartley transform, SIAM J. Matrix Anal.
	www.cs.unc.edu /~dm/UNC/COMP205/Program/Matrix/orthog.m (245 words)

	[No title] (Site not responding. Last check: )
		# # The Vandermonde matrix and determinant were discussed in class.
		You can # however factor this expression using a simple maple command, and the # expression that you get is much more informative: > factor(det(A)); # In particular, you see from this expression that the determinat is # zero if and only if at least two of x, y, z are equal.
		For instance, try # the 4 by 4 hilbert matrix, call it C: # > C:=hilbert(4); # Problem 2c: With A is in problem 2a and C as just defined, use maple # to verify that det(AC) = det(A)det(C), and also verify that AC is # different from CA.
	www.math.utah.edu /~toledo/2270proj03.txt (638 words)

	215 Assignments
		Let A be the matrix K+J, where K is the nxn identity matrix and J is the nxn matrix entirely of 1's.
		Special Problem #3 (Due Tuesday, October 22, 12 noon): The "trace" of a square matrix is the sum of the entries along the main diagonal (i.e.
		A Vandermonde matrix is a square matrix whose entries in each row are successive powers of a scalar, starting with the power 0.
	www.haverford.edu /math/jtecosky/215/215assignments.html (1062 words)

	PlanetMath: determinant of the Vandermonde matrix (Site not responding. Last check: )
		Anyone with an account can edit this entry.
		"determinant of the Vandermonde matrix" is owned by GeraW.
		This is version 5 of determinant of the Vandermonde matrix, born on 2004-08-31, modified 2004-09-02.
	planetmath.org /encyclopedia/DeterminantOfTheVandermondeMatrix.html (73 words)

	18.013A Calculus with Applications, Fall 2001, Online Textbook
		When a matrix has elements that are monomials or even polynomials in some set of variables, then its determinant will in general be a polynomial in those variables, and this is sometimes useful in evaluating it.
		By the basic property of a determinant, that it is 0 if two of its rows are the same, we can deduce that determinant of a VanderMonde matrix will be 0 when any two of its rows are the same.
		be the determinant of the matrix obtained from A by removing its i-th row and j-th column.
	ocw.mit.edu /ans7870/18/18.013a/textbook/chapter_b/contents.html (1229 words)

	MATH2070: LAB 6: Polynomial Interpolation (Site not responding. Last check: )
		The Vandermonde matrix method yields apparently poor coefficients but pretty smooth results, the Lagrange polynomial method gives good coefficients but a rough approximation, and Newton's divided difference method gives reasonable results.
		It is characterized by the fact that for each row (sometimes column) of the coefficient matrix, the succesive entries are generated by a decreasing (sometimes increasing) set of powers of a set of variables.
		Both the Vandermonde matrix approach and the Lagrange interpolating polynomials did well on some examples and poorly on another.
	www.math.pitt.edu /~sussmanm/2070Fall03/lab_06 (2709 words)

	: Class Pure16Code
		Notes about large N support: you can just generate the top kk vandermonde* matrix, call it V, then invert it, then when you have k blocks generate a matrix M with the k rows you need
		and E = M* V^{-1} is the encoding matrix for the systematic code which you then need to invert to perform the decoding.
		Probably there is a fast way to invert E given that M is also a vandermonde matrix so it is "easy" to compute M^{-1}
	onionnetworks.com /fec/javadoc/com/onionnetworks/fec/Pure16Code.html (726 words)

	How to Integrate a Banana
		First, I construct a projector to obtain a cross-section of the volume, then I interpolate 55 data points for both the function describing the top half of the banana’s projection and the bottom half.
		I use a Vandermonde matrix to fit a polynomial of degree 2 to each function.
		The big matrix on the left is “the” Vandermonde matrix, if you haven’t seen it before.
	www.caam.rice.edu /~jnsabino/banana2.htm (652 words)

	The Vandermonde Matrix
		Definition 9 An alternating function is a function that changes sign when we transpose any two of the variables.
		The proof of Proposition 1 is a direct consequence of the following lemma.
		The Vandermonde matrix plays an important role when proving certain bounds on the distances of cyclic codes.
	www-math.cudenver.edu /~rrosterm/crypt_proj/node6.html (268 words)

	[No title]
		If you got a matrix in the stack of the HP42 and then press 1/x, it doesn't calculate the invers of that matrix, but it it calculates the invers of each element of the matrix.
		This is used just to dimension auxiliar matrix T (term) to have the same dimensions as A (and B), thus no need to pass N (i.e: they are NxN matrices) as an extra parameter to the subprogram.
		I was familiar with the Taylor-series definition of the matrix exponential from a graduate-level linear algebra course, but was unaware until now of any closed-form expression.
	www.hpmuseum.org /cgi-sys/cgiwrap/hpmuseum/archv014.cgi?read=66689 (3026 words)

	Computer Lab 4 (Site not responding. Last check: )
		The routine it does use has test for the condition number of the matrix and other checks to decide if a propagating roundoff error may have resulted in an incorrect answer.
		The later indicates the rank of the matrix is 2 hence A is singular.
		Vander.m Vander(n) creates the n by n Vandermonde matrix discussed on page 248-49 of the text.
	www.math.montana.edu /~pernarow/M441/Comp_Tut_4.html (468 words)

	[No title] (Site not responding. Last check: )
		That is, we find: %A'A=LL' L=chol(A'A)' %(Octave's convention is for chol() to return an upper %triangular matrix*.
		Two important properties of Q (or any orthogonal %matrix): % 1.
		Qx(2-norm)=x(2-norm) %R is an upper triangular matrix* %Note the dimensions of the resulting matrix R! %The text only considers R as the first n rows of this matrix %and calls the rest "0" as they are zeroed-out rows.
	www.cs.toronto.edu /~arnold/350/04f/tutorials/09/lls.m (452 words)

	A General Vandermonde Factorization of a Hankel Matrix - Boley, Luk, Vandevoorde (ResearchIndex) (Site not responding. Last check: )
		It is shown that an infinite Hankel matrix of a finite rank (or a finite Hankel matrix) admits a generalized Vandermonde decomposition H = V T DV, where V is a generalized Vandermonde matrix, and D is a block diagonal matrix.
		The full structure of this decomposition was first fully discussed by Vandevoorde [9], but the development here is based solely on linear algebra considerations, specifically the use of the Jordan Canonical Form.
		Boley, F. Luk, and D. Vandevoorde, A general Vandermonde factorization of a Hankel matrix, preprint, University of Minnesota, 1998.
	citeseer.ist.psu.edu /boley98general.html (393 words)

	My Netscape FAQ
		It is well known that the ratio of the determinants of a generalized Vandermonde matrix and the corresponding Vandermonde matrix gives the Schur symmetric functions.
		We prove that the generalized matrix factors, one of the factors being the Vandermonde matrix, and thus give a new derivation of the determinantal form for the Schur functions.
		The material in the talk will be completely elementary: we will prove all facts we need about symmetric functions and Vandermonde matrices from scratch.
	www.math.clemson.edu /~sgao/WEB/ADMSeminar/adms99.html (1523 words)

	Fast Fourier Transform
		Then you just need to insert those terms cleverly in appropriate places in the Vandermonde matrix - also, observe that matrix (2.10) is symmetric.
		is a Vandermonde matrix for the 4-point discrete Fourier transform, and that multiplications in equation (2.12) can be expressed as:
		When it comes to coding this algorithm, you can either start from the top and then write a recursive procedure, or you can start at the bottom and evaluate four 2-point transforms first, then two 4-point transforms and finally the single 8-point transform.
	beige.ucs.indiana.edu /B673/node12.html (358 words)

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