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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)

	Augustin Louis Cauchy - Wikipedia, the free encyclopedia
		In 1833 the deposed king Charles X of France summoned Cauchy to be tutor to his grandson, the duke of Bordeaux, an appointment which enabled Cauchy to travel and thereby become acquainted with the favourable impression which his investigations had made.
		Returning to Paris in 1838, Cauchy refused a proffered chair at the Collège de France, but in 1848, the oath having been suspended, he resumed his post at the École Polytechnique, and when the oath was reinstituted after the coup d'état of 1851, Cauchy and François Arago were exempted from it.
		Cauchy had two brothers: Alexandre Laurent Cauchy (1792–1857), who became a president of a division of the court of appeal in 1847, and a judge of the court of cassation in 1849; and Eugène François Cauchy (1802–1877), a publicist who also wrote several mathematical works.
	en.wikipedia.org /wiki/Cauchy (1169 words)

	Determinant Summary
		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.bookrags.com /Determinant (3042 words)

	CAUCHY, A.L.(1789-1857)
		Cauchy was born in Paris in 1789 and recceived his early education from his father.
		Cauchy wrote extensively and profoundly in both pure and applied mathe matics, and he can probably be ranked next to Euler in volume of output.
		Cauchy's work exhibits great attention to rigor, and as such was largely reaponsible for inspiring other mathematicians to attempt the banishment of blind formal manipulation and of intuitive proofs from analysis.
	library.thinkquest.org /22584/temh3041.htm (493 words)

	DETERMINANT - Online Information article about DETERMINANT
		article, consider the determinant ax +by +cz -d b,c ; a'x+b'y+c'z-d', b', c' a"x+b"y+c"z—d", b", c" it appears that this is =xla,b,c I+y b,b,cI+zlc,b-,c,d,b,c I; r r ' ' r '.
		Multiplication of two Determinants of the same Order.—The theorem is obtained very easily from the last preceding definition of a determinant.
		Lagrange, in his memoir on Pyramids, used determinants of the third order, and proved that the square of a determinant was also a determinant.
	encyclopedia.jrank.org /DEM_DIO/DETERMINANT.html (1922 words)

	My Love for Mathematics
		In 1815, at the age of 26, Cauchy was made Professor of Mathematics at the Ecole Polytechnique and was recognized as the leading mathematician in France.
		Cauchy and his contemporary Gauss were the last men to know the whole of mathematics as known at their time, and both made important contributions to nearly every branch, both pure and applied, as well as to physics and astronomy.
		Cauchy was the founder of complex function theory and a pioneer in the theory or permutation groups and determinants.
	pages.intnet.mu /rasputin/mathppl.htm (5162 words)

	PlanetMath: Cauchy matrix
		The determinant of a square Cauchy matrix is
		Any submatrix of a rectangular Cauchy matrix has full rank.
		This is version 6 of Cauchy matrix, born on 2004-07-30, modified 2006-03-15.
	planetmath.org /encyclopedia/CauchyMatrix.html (68 words)

	Lattice Paths and the Flagged Cauchy Determinant (ResearchIndex)
		Abstract: We obtain a flagged form of the Cauchy determinant and establish a correspondence between this determinant and nonintersecting lattice paths, from which it follows that Cauchy identity on Schur functions.
		By choosing di#erent origins and destinations for the lattice paths, we are led to an identity of Gessel on the Cauchy sum of Schur functions in terms of the complete symmetric functions in the full variable sets.
		1 ans and determinants for Schur Q-functions (context) - Hamel - 1996
	citeseer.ist.psu.edu /631463.html (485 words)

	Read This: Conflicts Between Generalization, Rigor, and Intuition
		As great as Cauchy was, the second half of his career, after his return to Paris in 1838, would have been more successful if there had been someone he felt comfortable discussing mathematical ideas with on level terms; he would have published less, and better.
		This, Cauchy's first great paper, was also the first work to use the word determinant in the modern sense.) But neither Ampère nor Binet was quite a strong enough mathematician to be what the mature Cauchy really needed.
		The author discusses at some length whether the work of Cauchy and others with infinitely small quantities ought to be considered non-standard analysis, in the sense of Abraham Robinson and the calculus textbook of Jerome Keisler.
	www.maa.org /reviews/GenRigorIntuition.html (2016 words)

	Chapter B: Fun with Determinants
		We also present a curious formula involving determinants that was discovered by Lewis Carroll, the author of Alice in Wonderland, and a magical algorithm for computing determinants and even inverses based upon it, with almost no effort.
		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.
		And the determinant in one or all of a9 b9 and c9.
	www-math.mit.edu /18.013A/HTML/chapter_b/contents.html (1870 words)

	Springer Online Reference Works
		A Gram determinant is equal to the square of the
		Gram determinants were introduced by J.P. Gram [1] and, independently, by K.A. Andreev [2] in the context of problems of expansion of functions into orthogonal series and the best quadratic approximation to functions.
		The Gram determinant is used in many problems of linear algebra and function theory: studies of linear dependence of systems of vectors or functions, orthogonalization of systems of functions, construction of projections, and also in studies on the properties of systems of functions.
	eom.springer.de /g/g044740.htm (292 words)

	The flagged Cauchy determinant (Site not responding. Last check: 2007-10-11)
		We consider a flagged form of the Cauchy determinant, for which we provide a combinatorial interpretation in terms of nonintersecting lattice paths.
		In combination with the standard determinant for the enumeration of nonintersecting lattice paths, we are able to give a new proof of the Cauchy identity for Schur functions.
		Moreover, by choosing different starting and end points for the lattice paths, we are led to a lattice path proof of an identity of Gessel which expresses a Cauchy-like sum of Schur functions in terms of the complete symmetric functions.
	www.mat.univie.ac.at /~kratt/artikel/cauchy.html (146 words)

	Hyundai matrix (Site not responding. Last check: 2007-10-11)
		Gauss gave a determinant for large matrices are hyundai matrix similar to the total yield is bushels.
		Cauchy also introduced by cramer and bezout were impractical and, in an appendix to the coefficient matrix.
		Cauchys work is found by matrix b, the element at row vector a i row.
	matrices.pisecure.net /hyundai-matrix.html (3227 words)

	Lascoux, SLC42p (Site not responding. Last check: 2007-10-11)
		Starting with plane partitions possessing certain type of symmetries, many combinatorial objects came to the fore, the enumeration of which was the subject of intensive studies during the last twenty years, with of course, seminal contributions of George Andrews.
		Dividing this determinant by some straightforward factors, one is reduced to studying a symmetric polynomial in two sets of variables.
		In the same run, we reduce the dimension by 1 and factorize the determinant associated to the Bethe model of a 1-dimensional gas of bosons.
	www.maths.tcd.ie /EMIS/journals/SLC/wpapers/s42lascoux.html (156 words)

	Chen's paper - Lattice paths and the flagged Cauchy determiant (Site not responding. Last check: 2007-10-11)
		We obtain a flagged form of the Cauchy determinant and establish a correspondence between this determinant and nonintersecting lattice paths, from which it follows that Cauchy identity on Schur functions.
		By choosing different origins and destinations for the lattice paths, we are led to an identity of Gessel on the Cauchy sum of Schur functions in terms of the complete symmetric functions in the full variable sets.
		We also present an evaluation of the Cauchy determinant by the Jacobi symmetrizer.
	www.yongchuan.org /papers/cauchydet.html (123 words)

	Jacobi
		He also worked on determinants and studied the functional determinant now called the Jacobian.
		Jacobi was not the first to study the functional determinant which now bears his name, it appears first in a 1815 paper of Cauchy.
		He proves, among many other things, that if a set of n functions in n variables are functionally related then the Jacobian is identically zero, while if the functions are independent the Jacobian cannot be identically zero.
	library.wolfram.com /examples/quintic/people/Jacobi.html (274 words)

	Cauchy Riemann
		Let c be a point on the unit circle, whence z/c rotates the complex plane, pulling c back to 1 and ci back to i.
		We can now derive the Cauchy Riemann condition when u and v are given in polar coordinates.
		Let's use the Cauchy Riemann condition to show an analytic function f with derivative 0 is constant.
	www.mathreference.com /cx,crc.html (680 words)

	[No title] (Site not responding. Last check: 2007-10-11)
		Then the classical Cauchy identity on Schur functions is stated as follows: \begin{theo} \label{CauchyIdentity} For $n\geq 1$, we have \begin{equation} \prod_{i,j=1}^{n}\frac{1}{1-x_iy_j}=\sum_{\lambda}s_{\lambda}(\x)s_{\lambda}(\y), \end{equation} where the sum ranges over all partitions with length $\leq n$.
		The key ingredient in our lattice path construction is a flagged form of the Cauchy determinant with respect to the variable sets.
		Therefore, we have accomplished an algebraic proof of the equivalence of flagged Cauchy determinant \eqref{latticecauchy} and the determinant \eqref{seconddet} in the full variable sets.
	www.yongchuan.org /papers/cauchydet/cauchydet.tex (1847 words)

	Cauchy matrix - Wikipedia, the free encyclopedia
		In mathematics, the Cauchy matrix is an m×n matrix A, whose elements are given by
		When m=n and the matrix is square, the determinant, known as a Cauchy determinant, is given explicitly by
		Every submatrix of a Cauchy matrix is itself a Cauchy matrix.
	en.wikipedia.org /wiki/Cauchy_determinant (270 words)

	PlanetMath: Cauchy-Binet formula
		In both steps above, we have used the property that the determinant is multilinear in the colums of a matrix.
		But the last sum is none other than the determinant
		Cross-references: permutation group, parity, permutation, vanish, terms, multilinear, property, formula, rank, order, minors, sum, product, determinant, matrix
	planetmath.org /encyclopedia/CauchyBinetFormula.html (138 words)

	Linear Algebra Glossary (Site not responding. Last check: 2007-10-11)
		Once the form is computed, it is easy to compute the determinant, inverse, the solution of linear systems (even for underdetermined or overdetermined systems), the rank, and solutions to linear programming problems.
		In numerical work, the determinant is not a reliable indicator of singularity, and other data, such as the size of the matrix elements encountered during pivoting, are preferred.
		The determinant of A is equal to the product of the determinants of the factors, and hence is easily computed: the determinant of P is plus or minus 1, and that of L is 1, and that of U is simply the product of its diagonal elements.
	www.csit.fsu.edu /~burkardt/papers/linear_glossary.html (13553 words)

	KARL JACOBI (Site not responding. Last check: 2007-10-11)
		Jacobi's name is probably best known to undergraduates from the Jacobian, an n×n determinant formed from a set of n functions in n unknowns.
		Jacobi was certainly not the first to use it; the "Jacobian" already appears in an 1815 paper of Cauchy.
		But Jacobi did write a long memoir about it in 1841, and proved that the Jacobian of n functions vanishes if and only if the functions are related (Cauchy had only proved the "if" part).
	www.nadn.navy.mil /Users/math/meh/jacobi.html (597 words)

	Proper DB Design
		If a determinant (activity in the above case) cannot be a key in the relation, those attributes that depend on the determinant will repeat each time the value for the determinant repeats (see swimming in the original relation above).
		To prevent determinant values from repeating it is best to create a separate relation where the determinant is a key (remember keys can't repeat).
		A key of a relation is made up of one or more attributes which functionally determine a tuple (all attributes in a relation), but no subset of the key should also determine the tuple.
	csc.noctrl.edu /f/kwt/460/normal.htm (1964 words)

	[No title] (Site not responding. Last check: 2007-10-11)
		The determinant is calculated by computing det X11, and det S during the recursion.
		ALG is a recursive function that will update the global variables det (determinant), inv (inverse), rk (rank), and sltn (solution).
		Output: Determinant, inverse and rank of X We consider resetting values of the global variables as the outputs of ALG.
	www.cs.caltech.edu /~joyjoy/CSReport.doc (2247 words)

	AMERICAN MATHEMATICAL MONTHLY -November 2004
		In fact, the determinant and characteristic polynomial can be defined for any finite-dimensional algebra over a field (e.g., n-by-n matrices, the quaternions, a finite-degree field extension).
		The attempt failed, but only through inadvertence—it is not difficult to derive a correct generalization along the same lines, using two other results in Cauchy’s paper.
		I try to argue that if Cauchy had thought more carefully about what he was doing, he might have found the 1-psi-1 formula some seventy years before Ramanujan probably did.
	www.maa.org /pubs/monthly_nov04_toc.html (529 words)

	Earliest Known Uses of Some of the Words of Mathematics (D) (Site not responding. Last check: 2007-10-11)
		Descartes' rule of signs is found in English in 1855 in An elementary treatise on mechanics, embracing the theory of statics and dynamics, and its application to solids and fluids.
		DETERMINANT (discriminant of a quantic) was introduced in 1801 by Carl Friedrich Gauss in his Disquisitiones arithmeticae Werke Bd.
		Cauchy employed the word in “Memoire sur les fonctions qui ne peuvent obtenir que deux valeurs egales et des signes contraires par suite des transpositions operees entre les variables qu'elles renferment”, addressed on November 30, 1812, and first published in Journal de l'Ecole Poytechnique, XVIIe Cahier, Tome X, Paris, 1815 Oeuvres (2) i: p.
	members.aol.com /jeff570/d.html (6295 words)

	Clyde Davenport's Commutative Hypercomplex Math Page
		It is remarkable that any 4 X 4 real matrix would yield its determinant, eigenvalues, and eigenvectors by inspection.
		Because of the way that the 4-D function is defined as a pair of classical complex functions, the 4-D Cauchy-Riemann equations are immediate [but messy to develop; see Davenport(6), 1991].
		The eigenvalues, determinant, and vector norm, of course, are invariant.
	home.usit.net /~cmdaven/hyprcplx.htm (4469 words)

	geoms05 (Site not responding. Last check: 2007-10-11)
		The talk should be accessible to anyone possessing rudimentary notions of (linear) algebra, calculus and geometry.
		The determinant of an operator acting in an infinite dimensional space is difficult to define.
		Last time I presented Quillen's construction of the Determinant line bundle over Fredholm operators and outlined how Quillen used this to give a construction of a determinant function for Cauchy Riemann operators.
	math.arizona.edu /~foth/geoms05.html (1665 words)

	Atlas: Variations on Cauchy's determinant and Schur's Pfaffian by Soichi Okada (Site not responding. Last check: 2007-10-11)
		Atlas: Variations on Cauchy's determinant and Schur's Pfaffian by Soichi Okada
		We present several identities of Cauchy-type determinants and Schur-type Pfaffians involving generalized Vandermonde determinants.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caql-59.
	atlas-conferences.com /cgi-bin/abstract/caql-59 (97 words)

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