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	Kronecker Product
		This is a collection of pointers to work related to the Kronecker product (or tensor product, direct
		The Kronecker product notation arises in many different areas of science and engineering,
		Kronecker Product of Matrices and Applications, Wissenschaftsverlag, 1991.
	www.cs.duke.edu /~nikos/KP/home.html (147 words)

	Tensor product - Wikipedia, the free encyclopedia
		The tensor product inherits all the indices of its factors.
		With matrices this operation is usually called the Kronecker product, a term used to make clear that the result has a particular block structure imposed upon it, in which each element of the first matrix is replaced by the second matrix, scaled by that element.
		In J the tensor product is the dyadic form of / (for example a / b or a / b / c).
	en.wikipedia.org /wiki/Tensor_product (992 words)

	PlanetMath: Kronecker product
		The Kronecker product is also known as the direct product or the tensor product [1].
		This is version 2 of Kronecker product, born on 2003-04-06, modified 2004-10-11.
		invariants of the tensor product by mcintosh on 2003-10-04 19:51:25
	planetmath.org /encyclopedia/DirectProduct5.html (165 words)

	Leopold Kronecker - Wikipedia, the free encyclopedia
		Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on "whole numbers", saying, "God made the integers; all else is the work of man" (Bell 1986, p.
		Kronecker was a student and lifelong friend of Ernst Kummer.
		The Kronecker delta and Kronecker product are named after Kronecker, as are the Kronecker-Weber theorem, Kronecker's theorem in number theory and Kronecker's lemma.
	www.wikipedia.org /wiki/Leopold_Kronecker (313 words)

	Kronecker product - Wikipedia, the free encyclopedia
		The Kronecker product should not be confused with the usual matrix multiplication, which is an entirely different operation.
		The Kronecker product of matrices corresponds to the abstract tensor product of linear maps.
		The Kronecker product is named after Leopold Kronecker, even though there is little evidence that he was the first to define and use it.
	en.wikipedia.org /wiki/Kronecker_product (474 words)

	Leopold Kronecker - Wikipedia, the free encyclopedia
		In 1845, Kronecker wrote his dissertation at the University of Berlin on number theory, giving special formulation to units in certain algebraic number fields.
		Kronecker also contributed to the concept of continuity, reconstructing the form of irrational numbers in real numbers.
		Kronecker's finitism made him a forerunner of intuitionism in foundations of mathematics.
	en.wikipedia.org /wiki/Leopold_Kronecker (313 words)

	Kronecker, Leopold - Hutchinson encyclopedia article about Kronecker, Leopold (Site not responding. Last check: 2007-10-11)
		Kronecker was born at Liegnitz (now Legnica, Poland), and studied at Berlin, Bonn, and Breslau.
		Kronecker was obsessed with the idea that all branches of mathematics (apart from geometry and mechanics) should be treated as parts of arithmetic.
		Kronecker published a system of axioms in 1870, later shown to govern finite Abelian groups.
	encyclopedia.farlex.com /Kronecker,+Leopold (220 words)

	Matrix multiplication - Wikipedia, the free encyclopedia
		The ordinary matrix product can be thought of as a dot product of a column-list of vectors and a row-list of vectors.
		This notion of multiplication is important because if A and B are interpreted as linear transformations (which is almost universally done), then the matrix product AB corresponds to the composition of the two linear transformations, with B being applied first.
		The Hadamard product is studied by matrix theorists, but it is virtually untouched by linear algebraists.
	en.wikipedia.org /wiki/Matrix_multiplication (994 words)

	Matrix multiplication (Site not responding. Last check: 2007-10-11)
		This notion of multiplication is important because A and B are interpreted as linear transformations (which is almost universally done) then matrix product AB corresponds to the composition of the linear transformations with B being applied first.
		The Hadamard product of two m -by- n matrices A and B denoted by A · B is an m -by- n matrix given by (A andmiddot B)[ i j ]= A [ i j ] * B [ i j ].
		product is studied by matrix theorists but is virtually untouched by linear algebraists.
	www.freeglossary.com /Matrix_multiplication (1023 words)

	[No title] (Site not responding. Last check: 2007-10-11)
		Here, by tensor product of representations, the usual meaning is intended: the result should be a linear representation on the product vector space.
		The direct product for vector spaces (not to be confused with the tensor product">tensor product) is very similar to the one defined for groups above, using the product">cartesian product with the operation of addition being componentwise, and the scalar multiplication just distributing over all the components.
		Functors are often defined by universal properties; examples are the tensor product">tensor product, the direct sum and product">direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits.
	www.worldhistory.com /wiki/t/tensor-product.htm (660 words)

	Talk:Tensor product - InformationBlast
		there seems to be a section under matrix multiplication on Kronecker product/direct product, which is the same thing as tensor product as far as i know, with a slightly different definition.(the rank is neglected) I know that my definition is correct.
		Since matrix multiplication is a more general subject, the Kronecker product section should be kept brief, perhaps indicating that more information is to be found here.
		The way it is defined in matrix multiplication, there is a slight difference between the matrix direct product/kronecker product and the tensor product, namely regarding the rank as previously discussed.
	www.informationblast.com /Talk:Tensor_product.html (1245 words)

	[No title]
		The direct product of (symmetric) association schemes was first studied by Kusumoto [7], in connection with the construction of new incomplete block designs from a family of given designs.
		Although the wreath product is a well-known standard construction in group theory, the concept of the wreath product is different from that of the wreath product in group theory and has almost no relevance.
		It is well known that the character table of the direct product of two groups is the tensor product of the character tables of the direct factors (as representations of direct product are tensor products of representations of the factors).
	orion.math.iastate.edu /sysong/Preprints/FR-Prod.doc (3400 words)

	Apparatus for incorporating multiple data rates in an orthogonal direct sequence code division multiple access ... (Site not responding. Last check: 2007-10-11)
		The data symbols of a low rate user at rate f.sub.s /2.sup.r, where r may range from 0 to q, are spread using a longer code of length 2.sup.r N, which is constructed by the Kronecker product of a length 2.sup.r orthogonal code and the N chip codeword allocated to this user.
		Since the chip rate, which is the product of the spreading sequence length and the symbol rate, is fixed, it follows from f.sub.c =(N/2)(2f.sub.s), that the spreading sequence length per symbol must be reduced to N/2.
		The Kronecker product of the j.sup.th row of G(2.sup.r) with the codeword c.sub.k produces a member L.sub.jk, j=0,.
	patents.nimblewisdom.com /patent/6563808-Apparatus-for-incorporating-multiple-data-rates-in-an-orthogonal-direct- (4520 words)

	kronecker.nb (Site not responding. Last check: 2007-10-11)
		The Kronecker product (often called tensor product) is entered with the "CircleTimes" operator.
		It is important to mention that the Kronecker product has a lower precedence than the composition product, donoted by a dot ".".
		Most QuCalc datatypes may be parameters to the Kronecker product, as long as the operation is meaningful.
	crypto.cs.mcgill.ca /QuCalc/exemples/kronecker (192 words)

	Optimization Online - Strengthened Existence and Uniqueness Conditions for Search Directions in Semidefinite Programming
		The first is given by Shida-Shindoh-Kojima and is based on the semidefiniteness of the symmetrization of the product $SX$ at the current iterate.
		As well, we present new results on the relationship between the Kronecker product and the {\em symmetric Kronecker product} that arise from these matrix equations.
		We conclude with a proof that the existence and uniqueness of the AHO direction is a generic property for every SDP problem and extend the results to the general Monteiro-Zhang family of search directions.
	www.optimization-online.org /DB_HTML/2003/07/695.html (298 words)

	wht_prog.html (Site not responding. Last check: 2007-10-11)
		The Kronecker product of two matrices is the block matrix obtained by replacing each element of the first matrix, by that element multiplied by the second matrix.
		It is not difficult to show that the Kronecker product is associative but not generally commutative.
		Since the WHT is defined by the tensor product, the WHT matrix can be factored using the decomposition rule for the Kronecker product.
	www.mcs.drexel.edu /~jjohnson/wi04/cs680/lectures/wht_prog1.html (1698 words)

	[No title] (Site not responding. Last check: 2007-10-11)
		Applications of the Kronecker Product Charles Van Loan Department of Computer Science Cornell University Just about every fast linear transform corresponds to a ``sparse'' factorization of the the underlying transform matrix and the factors usually involve Kronecker products.
		Extensions of the factorization framework to wavelets will be discussed leading to some interesting generalizations of the Kronecker product.
		Currently, he is interested in applications of the Kronecker product.
	www.cs.toronto.edu /colloq/1996/abs_vanloan (191 words)

	Citations: Kronecker Products and matrix calculus in system theory - Brewer (ResearchIndex)
		: A 2) A 3# A 4) A 1 A 3 (A 2 A 4) and the corresponding Hermitian property of Kronecker products [2] A 2) A A 2.
		If is the vector of nonzero frequency components of C D F, and W is the vector of corresponding indices, the resulting frequency components and indices are given by a a a (12) a a a (13) The use of this technique reduces the computation time by a factor 10 compared to a....
		21 Appendix We brie y recall the de nition of Kronecker sum and Kronecker product.
	citeseer.ist.psu.edu /context/94178/0 (1333 words)

	Matrix_multiplication (Site not responding. Last check: 2007-10-11)
		It's also easy to see why the number of columns in the proportions matrix has to be the same as the number of rows in the vectors matrix: they have to represent the same number of vectors.
		The Hadamard product of two ''m''-by-''n'' matrices ''A'' and ''B'', denoted by ''A'' · ''B'', is an ''m''-by-''n'' matrix given by
		For any two arbitrary matrices ''A'' and ''B'', we have the direct product or Kronecker product ''A'' {{dirprod}} ''B'' defined as
	q-basic.xodox.de /Matrix_multiplication (893 words)

	[No title]
		The Kronecker product of A:m#n and B:p#q is equal to the mp#nq matrix [a(1,1)B … a(1,n)B ; … ; a(m,1)B … a(m,n)B ].
		The Kronecker Product operation is denoted by a × sign enclosed in a circle.
		[A:n#n, B:m#m] The mn eigenvalues of KRON(A,B) are equal to the mn possible products of an eigenvalue of A multiplied by an eigenvalue of B with the appropriate algebraic multiplicities.
	www.psi.toronto.edu /matrix/relation.html (312 words)

	SMU Department of Mathematics Technical Reports (Site not responding. Last check: 2007-10-11)
		If this kernel is known, then discretizations lead to a blurring matrix which is a Kronecker product of two matrices of smaller dimension.
		In this paper we describe an interpolation scheme to construct a Kronecker product approximation to the blurring matrix from a set of observed point spread functions for separable, or nearly separable, spatially variant blurs.
		An image restoration problem from the Hubble Space Telescope is used to illustrate the effectiveness of an approximate SVD preconditioner constructed from the Kronecker product decomposition.
	www.smu.edu /math/techreports/1998abs.html (985 words)

	SC Seminar (Site not responding. Last check: 2007-10-11)
		The structure of the "blurring" matrix determines whether it is feasible to use matrix factorizations, or whether iterative methods are more appropriate for solving the linear systems.
		In this talk we describe some aspects of image restoration algorithms, and show that the blurring matrix can often be represented in terms of Kronecker products.
		We also show how to exploit the Kronecker product structure to improve computational efficiency.
	crd.lbl.gov /SCG/SCSeminars/20040205.html (115 words)

	PlanetMath:
		direct product (=categorical direct product) owned by djao
		direct product and restricted direct product of groups owned by yark
		direct product of groups (in direct product and restricted direct product of groups) owned by yark
	planetmath.org /encyclopedia/D (1598 words)

	MS26 Software Involving Kronecker Products (Site not responding. Last check: 2007-10-11)
		Kronecker products of matrices arise in a wide variety of applications and have been exploited in numerous algorithms to significantly reduce operation counts.
		Huge systems of equations involving millions of variables frequently arise and are routinely solved using Kronecker product methods in the engineering community.
		Frequently, the normal equations are utilized to solve Kronecker Product least squares problems, a technique which is known to be unstable.
	www.siam.org /meetings/archives/an95/ms26.htm (132 words)

	MS33 Numerical Analysis and Applications Associated with Kronecker Products (Site not responding. Last check: 2007-10-11)
		For least squares problems, parallel solution of huge linear systems involving millions of variables become possible when the Kronecker product structure is present, owing to fact that the order of the systems is reduced from N to sqrt N. Applications in photogrammetry, signal processing, and fast transform methods are commonplace.
		Kronecker product pre-conditioners will afford the same reduction in operation counts to a much larger class of problems.
		The speakers will address new numerical methods and applications where Kronecker products arise.
	www.siam.org /meetings/an99/ms33.htm (143 words)

	Glossary of terms for the maximal determinant problem (Site not responding. Last check: 2007-10-11)
		The order of a Hadamard matrix must be 1, 2 or 4n with n an integer.
		Kronecker product: If A is a p×q matrix and B is an r×s matrix, then the Kronecker product, A⊗B, is the pr×qs matrix
		Sylvester construction [Sy]: If A and B are Hadamard matrices of orders p and q, then a Hadamard matrix of order pq is formed by taking the Kronecker product A ⊗ B. Sylvester matrix [Sy]: Let S
	www.indiana.edu /~maxdet/glossary.html (226 words)

	LISREL: Explanation of covariance matrices (Site not responding. Last check: 2007-10-11)
		Under normality, it is well known that the ACM can be expressed as the so-called Kronecker product of the population covariance matrix with itself.
		It is also well known that the inverse of the ACM in the case of multivariate normality is the Kronecker product of inverse(SIGMA) and inverse(SIGMA).
		By specifying that the method of maximization is normal maximum likelihood, one avoids the necessity of inverting the usually large k x k ACM matrix obtained by PRELIS.
	www.assess.com /Support/730-19991027.htm (1085 words)

	Mercedes H. Rosas's publications (Site not responding. Last check: 2007-10-11)
		(That is for those pairs of shapes such that the coefficients in the Schur expansion of its products are multiplicity free, as described by John Stembridge in his article "Multiplicity-Free Products of Schur Functions".
		The Kronecker product of two Schur functions, is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group.
		A combinatorial overview of the theory of MacMahon symmetric functions and a study of the Kronecker product of Schur functions,
	www.ma.usb.ve /~mrosas/articles/articulos.html (1160 words)

	LACIM : Laboratoire de Combinatoire et d'Informatique Mathématique (Site not responding. Last check: 2007-10-11)
		Abstract: The Kronecker product of two Schur functions, is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group.
		The coefficients of the Kronecker product of two Schur functions in the Schur basis are the multiplicity of the irreducible representations of the symmetric group is such a tensor product.
		We use Sergeev's Formula for a Schur function of a difference of two alphabets and the comultiplication expansion for a Schur function to find closed formulas for the Kronecker coefficients corresponding hook shapes or two-row shapes.
	www.lacim.uqam.ca /semin.cbn/semindetail/semin.cbn.031205.html (111 words)

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