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Topic: Vandermonde determinant

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	Determinant - LoveToKnow 1911
		To indicate the method of proof, observe that the determinant on the left-hand side, qua linear function of its columns, may be I The reason is the connexion with the corresponding theorem for the multiplication of two matrices.
		The germ of the theory of determinants is to be found in the writings of Gottfried Wilhelm Leibnitz (1693), who incidentally discovered certain properties when reducing the eliminant of a system of linear equations.
		Determinants composed of binomial coefficients have been studied by V. von Zeipel; the expression of definite integrals as determinants by A. Tissot and A. Enneper, and the expression of continued fractions as determinants by Jacobi, V. Nachreiner, S. Gunther and E. Furstenau.
	www.1911encyclopedia.org /Determinant (1102 words)

	Determinant - Exampleproblems
		Determinants are used to characterize invertible matrices (namely as those matrices, and only those matrices, with non-zero determinants), and to explicitly describe the solution to a system of linear equations with Cramer's rule.
		Determinants are used to calculate volumes in vector calculus: the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors.
		The Pfaffian is an analog of the determinant for
	www.exampleproblems.com /wiki/index.php/Determinant (1830 words)

	Кафедра вычислительных технологий и моделирования BMиК MГУ
		Vandermonde's four mathematical papers, with their dates of publication by the Académie des Sciences, were Mйmoire sur la rйsolution des йquations (1771), Remarques sur des problиmes de situation (1771), Mйmoire sur des irrationnelles de diffйrens ordres avec une application au cercle (1772), and Mйmoire sur l'йlimination (1772).
		Vandermonde's real and unrecognised claim to fame was lodged in his first paper, in which he approached the general problem of the solubility of algebraic equations through a study of functions invariant under permutations of the roots of the equation.
		Vandermonde considers the intertwining of the curves generated by the moving knight and his work in this area marks the beginning of ideas which would be extended first by Gauss and then by Maxwell in the context of electrical circuits.
	www.inm.ras.ru /vtm/lection/direct1/vandermonde.htm (1332 words)

	Determinant
		Determinants are mathematical objects that are very useful in the analysis and solution of systems of linear equations.
		The determinant of the similarity transformation of a matrix is equal to the determinant of the original
		and the determinant of a complex conjugate is equal to the complex conjugate of the determinant
	ai.korea.ac.kr /~kaizer/math/Determinant.htm (657 words)

	PlanetMath: Vandermonde matrix
		Vandermonde matrices usually arise when considering systems of polynomials evaluated at specific points (i.e.
		Vandermonde matrices also appear in the computation of FFTs (Fast Fourier Transforms).
		This is version 1 of Vandermonde matrix, born on 2002-09-28.
	planetmath.org /encyclopedia/VandermondeMatrix.html (135 words)

	PlanetMath: proof of determinant of the Vandermonde matrix
		So all that remains is the determine the value of this constant.
		"proof of determinant of the Vandermonde matrix" is owned by rspuzio.
		This is version 7 of proof of determinant of the Vandermonde matrix, born on 2006-03-08, modified 2006-11-03.
	planetmath.org /encyclopedia/PrrofOfDeterminantOfTheVandermondeMatrix.html (205 words)

	Matrices and determinants
		In the 1812 paper the multiplication theorem for determinants is proved for the first time although, at the same meeting of the Institut de France, Binet also read a paper which contained a proof of the multiplication theorem but it was less satisfactory than that given by Cauchy.
		These were important in that for the first time the definition of the determinant was made in an algorithmic way and the entries in the determinant were not specified so his results applied equally well to cases were the entries were numbers or to where they were functions.
		An axiomatic definition of a determinant was used by Weierstrass in his lectures and, after his death, it was published in 1903 in the note On determinant theory.
	www-groups.dcs.st-andrews.ac.uk /~history/HistTopics/Matrices_and_determinants.html (2630 words)

	Vandermonde (Site not responding. Last check: )
		Vandermonde's real and unrecognised claim to fame was lodged in his first paper, in which he approached the general problem of the solubility of algebraic equations through a study of functions invariant under permutations of the roots of the equation.
		Vandermonde considers the intertwining of the curves generated by the moving knight and his work in this area marks the beginning of ideas which would be extended first by Gauss and then by Maxwell in the context of electrical circuits.
		Vandermonde, however, thought of the determinant as a function and gave properties of the determinant function.
	www-gap.dcs.st-and.ac.uk /~history/Mathematicians/Vandermonde.html (1323 words)

	Vandermonde matrix - Wikipedia, the free encyclopedia
		The above determinant is sometimes called the discriminant, although many authors, including this encyclopedia, refer to the discriminant as the square of this determinant.
		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.
	en.wikipedia.org /wiki/Vandermonde_matrix (422 words)

	Mathematics Preprints
		The square of the Vandermonde determinant and its q-generalisation,
		Herein the properties of the square of the Vandermonde determinant as a symmetric function are explored in detail.
		Important properties satisfied by the coefficients arising in the expansion of the square of the Vandermonde determinant in terms of Schur functions are developed and generalized to q-dependent coefficients via the q-discriminant.
	www.maths.soton.ac.uk /research/viewabstract.phtml?entry=1787&table=applied (133 words)

	Determinant -- from Wolfram MathWorld
		Determinants are mathematical objects that are very useful in the analysis and solution of systems of linear equations.
		The determinant of the similarity transformation of a matrix is equal to the determinant of the original
		and the determinant of a complex conjugate is equal to the complex conjugate of the determinant
	mathworld.wolfram.com /Determinant.html (675 words)

	Alexandre Théophile Vandermonde Biography / Biography of Alexandre Théophile Vandermonde World of ...
		Vandermonde was born in Paris, France on February 28, 1735, the son of a physician.
		Vandermonde went along with this plan, since he loved music, for all of his childhood and well into his adult life.
		Vandermonde was reportedly the first mathematician to prepare a systematic treatment of the theory of determinants.
	www.bookrags.com /biography-alexandre-theophile-vandermonde-wom (466 words)

	Vandermonde biography
		Like Monge, Vandermonde was a strong supporter of the Revolution which began with the storming of the Bastille on 14 July 1789.
		Cauchy states quite clearly that Vandermonde had priority over Lagrange for this remarkable idea which eventually led to the study of group theory.
		The reason for this strong claim by Muir is that, although mathematicians such as Leibniz had studied determinants earlier than Vandermonde, all earlier work had simply used the determinant as a tool to solve linear equations.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Vandermonde.html (1353 words)

	Search Results for Determinant
		This paper, On the reduction of arithmetical binary cubics which have a negative determinant, was written jointly with Mathews and it was in fact the only paper Berwick jointly authored throughout his career.
		These were important in that for the first time the definition of the determinant was made in an algorithmic way and the entries in the determinant were not specified so his results applied equally well to cases were the entries were numbers or to where they were functions.
		An axiomatic definition of a determinant was used by Weierstrass in his lectures and, after his death, it was published in 1903 in the note On determinant theory.
	www-gap.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=Determinant&CONTEXT=1 (1226 words)

	Fermat's Last Theorem: Alexandre-Theophile Vandermonde
		Instead, Vandermonde argued that there should be no theory of music as mathematical theory, but, instead, musicians should judge music solely based on their trained ear.
		In Vandermonde's first paper on mathematics, he presented a solution to the eleventh root of unity using ideas of symmetrical polynomials and permutations.
		Still, the last of his four math papers did focus on determinants in which for the first time, a determinant is presented as a mathematical function and Vandermonde proceeds to investigate its mathematical properties.
	fermatslasttheorem.blogspot.com /2008/01/alexandre-theophile-vandermonde.html (491 words)

	[No title]
		The transpose of V is called row-wise Vandermonde matrix and it arises in the polynomial interpolation problem on particular points distribution.
		We introduce the rectangular Vandermonde matrix (6.4) Apq={aji-1}i=1,p;j=1,q and use the Kronecker product in a similar way as in the square matrix case, in order to express the matrix V by means of the compact expression (6.5) V=(Apq(Ik)D, where D is the block diagonal matrix of order n=kq defined in ((1.8).
		The rowwise rectangular Vandermonde matrix VH is the conjugate transpose of the matrix V and is given by (6.6) VH = DH(ApqH(Ik) In many applications the points defining the Vandermonde matrix are symmetric with respect to the origin of the complex plane.
	www.dm.unibo.it /~tommasin/lavori/VandermondeH3R-2004-5.doc (2338 words)

	Determinant (Site not responding. Last check: )
		PDA_DGEDI - Determinant and inverse of a matrix using the factors from PDA_DGEFA...
		Symmetry as A Compositional Determinant: Index & Abstract...
		Social Capital as a Health Determinant: How is it defined?...
	www.scienceoxygen.com /math/136.html (129 words)

	Lecture Summary in Linear Algebra for FED. (Site not responding. Last check: )
		In particular, we observe that there are 6 terms in the determinant of order 3: among which 3 of them have positive signs and 3 of them have negative signs.
		Determinant is linear in each row and in each column.
		The determinant remains the same if the rows and columns are interchanged in the same order.
	www.sftw.umac.mo /~fstitl/linweb/summary.html (722 words)

	Section VM Vandermonde Matrix
		Alexandre-Théophile Vandermonde was a French mathematician in the 1700’s who was among the first to write about basic properties of the determinant (such as the effect of swapping two rows).
		Vandermonde matrices are not very interesting as numerical matrices, but instead appear more often in proofs and applications where the scalars
		As a bonus, the determinant of a Vandermonde matrix has an especially pleasing formula.
	linear.ups.edu /xml/0108/fcla-xml-1.08li99.xml (631 words)

	commalg.org - the center for commutative algebra
		Since it is known that the order complexes of finite intervals in the poset of monomials in $k[\Lambda]$ ordered by divisibility in $k[\Lambda ]$ govern the $\Tor$-groups, the newly developed tools are applicable and serve as the main ingredients for the proof of the bounds and the construction of the resolution.
		Abstract: We show that, for generic bihomogeneous polynomials, the determinant of the matrix of moving planes is irreducible.
		Motivated by a question of Herbert Hauptman, we consider the problem of determining phases by direct algebraic means in the case of crystal structures with $n$ equal atoms in the unit cell, with $n$ small.
	www.commalg.org /preprints/2003_12.shtml (1983 words)

	ipedia.com: Alexandre-Théophile Vandermonde Article (Site not responding. Last check: )
		Alexandre-Théophile Vandermonde (28 February 1735 -1 January 1796) was a French musician and chemist who worked with Bezout and Lavoisier; his name is now principally associated with determinant theory in mathematics.
		Mémoire sur des irrationnelles de différens ordres avec une application au cercle (1772) was on combinatorics, and Mémoire sur l'élimination (1772) on the foundations of determinant theory.
		A special class of matrices, the Vandermonde matrices are named after him.
	www.ipedia.com /alexandre_theophile_vandermonde.html (213 words)

	Matrices and Determinants - Numericana
		Wendt's Determinant: The circulant of the binomial coefficients.
		Alexandre-Théophile Vandermonde (1735-1796) who is the rightful founder of the modern theory of determinants...
		The determinant of a circulant matrix is commonly called the circulant determinant or, for short, the circulant of the n coefficients involved.
	home.att.net /~numericana/answer/matrix.htm (2845 words)

	Vandermonde matrix and determinant from Interactive Mathematics Miscellany and Puzzles
		Vandermonde matrix and determinant from Interactive Mathematics Miscellany and Puzzles
		I failed to mention the Vandermonde matrix because I couldn't see how it fit in with the story.
		Hence the Vandermonde matrix, whose columns are these vectors, is nonsingular.
	www.cut-the-knot.org /proofs/Vandermonde.shtml (47 words)

	MIT OpenCourseWare, Department Template
		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.
		This is also true of the determinant, all of whose terms are products of n factors, each having one term in the denominator and none in the numerator, for a net excess of n in the denominator.
		be the determinant of what is left, which in the two dimensional case is nothing, whose determinant we define to be 1.
	www.cs.cityu.edu.hk /~luoyan/mirror/mit/ocw.mit.edu/18/18.013a/f01/required-readings/chapter_b/contents.html (1221 words)

	Determinants
		The determinants of a matrix and its transpose are equal,
		Multiplying all the elements of the row (column) of the determinant by the same number the determinant will be multiplied by the same number.
		The determinant will not change if add to the row (column) an arbitrary row (column) multiplied by an arbitrary number.
	www.cs.ut.ee /~toomas_l/linalg/lin1/node14.html (638 words)

	Constrained, non-linear, derivative-free parallel optimization of continuous, high computing load, noisy objective ...
		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''.
		These kind of systems are very often badly conditioned (determinant near zero) and can't be resolved directly.
	www.applied-mathematics.net /mythesis/node14.html (267 words)

	Determinant: Encyclopedia II - Determinant - Properties
		The determinant is a multiplicative map in the sense that for all n-by-n matrices A and B. This is generalized by the Cauchy-Binet formula to products of non-square matrices.
		Because of this, the determinant of some linear transformation T : V → V for some finite dimensional vector space V is independent of the basis for V.
		The determinant of real square matrices is a polynomial function from to, and as such is everywhere differentiable.
	www.experiencefestival.com /a/Determinant_-_Properties/id/1321020 (634 words)

	18.013A Calculus with Applications, Fall 2001, Online Textbook
		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.
		This is also true of the determinant, all of whose terms are products of n factors, each having one term in the denominator and none in the numerator, for a net excess of n in the denominator.
		be the determinant of what is left, which in the two dimensional case is nothing, whose determinant we define to be 1.
	ocw.mit.edu /ans7870/18/18.013a/textbook/chapter_b/contents.html (1229 words)

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