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	Matrix (mathematics) - Wikipedia, the free encyclopedia
		The entry of a matrix A that lies in the i -th row and the j-th column is called the i,j entry or (i,j)-th entry of A.
		The rank of a matrix A is the dimension of the image of the linear map represented by A; this is the same as the dimension of the space generated by the rows of A, and also the same as the dimension of the space generated by the columns of A.
		The trace of a square matrix is the sum of its diagonal entries, which equals the sum of its n eigenvalues.
	en.wikipedia.org /wiki/Matrix_(mathematics) (1851 words)

	Construction of Matrix Algebras and their Elements
		Given a matrix algebra R defined as a subalgebra of M_n(S).
		This function creates a n by n matrix over the finite field K of cardinality q specified in a "Cambridge" format in the general matrix algebra of degree n over K. The parameter t specifies the type of the format.
		the matrix having the one of the coefficient ring of R in position (i, j) and zeros elsewhere.
	www.umich.edu /~gpcc/scs/magma/text912.htm (936 words)

	Presentations for matrix algebras
		The element t_i is conjugate to a permutation matrix of degree n_i in A_i.
		This algebra is Morita equivalent to A, and hence shares many of the same homological properties of the algebra A. In the course of obtaining the presentation, several aspects of the algebra are computed.
		The algebra is block upper triangular, where the upper left block is a field extension of degree three and the lower block is a filed extension of degree 2.
	www.math.lsu.edu /magma/text851.htm (2081 words)

	GAP Manual: 38 Algebras (Site not responding. Last check: 2007-10-13)
		This holds for example for all finitely presented algebras (see chapter Finitely Presented Algebras) and all those matrix algebras whose generators are the generators of a finite group.
		The images of unital algebras under operation homomorphisms are either unital or trivial, since the identity of the source acts trivially, so its image under the homomorphism is the identity of the image.
		The image of a subalgebra under a algebra homomorphism is computed by computing the images of a set of generators of the subalgebra, and the result is the subalgebra generated by those images.
	www.math.jussieu.fr /~jmichel/htm/CHAP038.htm (3037 words)

	What ARE Clifford Algebras and Spinors?
		If the algebra A is also generated as a ring by the copies of R and X or, equivalently, as a real algebra by {1} and X, then A is said to be a (real) Clifford algebra for X (Clifford's term...
		Since odd dimensional Clifford algebras are the sum of two matrix algebras, spinors for odd dimensional Clifford algebras are rows (or columns) of one of the matrix algebras, the matrix subalgebras of the even-grade elements of the graded Clifford algebra.
		The non-isomorphism of the octonions O with the Clifford algebra Cl(3) is due to the nonassociativity of the 7-dimensional vector cross-product.
	www.valdostamuseum.org /hamsmith/clfpq.html (5336 words)

	Allan Donsig's research page
		This invariant is complete for a certain family of limit algebras: inductive limits of digraph algebras (a.k.a.\ finite dimensional CSL algebras) satisfying two conditions: (1) the inclusions of the digraph algebras respect the order-preserving normalisers, and (2) the digraph algebras have chordal digraphs.
		If the algebra is also a CSL algebra, we scharacterize when the first homology group of the algebra is contained in the first homology group of the (4,4) entry; in these cases, the only obstruction to a derivation being inner arises from the (4,4) entry.
		In the present paper, we weamine the clase of nest algebras T in AF -algebras which share the distinctive properties of the refinement algebra: (1) T is a nest algebra in which the nest generated the diagonal, (2) T admits a locally constant cocycle.
	www.math.unl.edu /~adonsig1/research.html (2559 words)

	Draft: Geometry and Lie Groups
		The model for this application was Galois' use of finite groups to solve algebraic equations of degree two, three, and four, and to show that the general polynomial equation of degree greater than four could not be solved by radicals.
		The properties of a Lie algebra are identified with the properties of the original Lie group in the neighborhood of the origin.
		A typical Lie algebra is a semidirect sum of a semisimple Lie algebra and a solvable subalgebra that is invariant.
	www.physics.drexel.edu /~bob/LieGroups.html (1676 words)

	Research - UNL - Department of Mathematics (Site not responding. Last check: 2007-10-13)
		Operator Theory and Operator Algebras are concerned with the study of linear operators, usually on vector spaces whose elements are functions.
		Limit algebras are limits, in a suitable sense, of matrix algebras and share characteristics of both finite-dimensional and infinite-dimensional operator algebras.
		David Pitts has interests in nest algebras, limit algebras and free semigroup algebras, which are a non-commutative analog of analytic functions in several variables.
	www.math.unl.edu /pi/research/operator (156 words)

	GAP Manual: 40.1. More about Matrix Algebras (Site not responding. Last check: 2007-10-13)
		A matrix algebra is an algebra (see More about Algebras) the elements of which are matrices.
		There is a canonical isomorphism of a matrix algebra onto a row space (see chapter Row Spaces) that maps a matrix to the concatenation of its rows.
		If one is interested in matrix algebras as domains themselves then one should think of this algebra as of a row space that admits a multiplication.
	www.math.uiuc.edu /Software/GAP-Manual/More_about_Matrix_Algebras.html (272 words)

	Title page for ETD etd-10272004-164622
		It is this problem that we consider for associative algebras, matrix groups, and Lie algebras.
		For associative algebras with 1 generator we also get the same result as the general associative algebra case.
		Finally we consider Lie algebras with 2 generators and here n is larger than in the general case.
	www.lib.ncsu.edu /theses/available/etd-10272004-164622 (173 words)

	Midwest Lie Algebras and Related Topics Conference
		Among non-trivial consequences is the connection between the exponent of the growth of the identity component of an associative algebra graded by a finite group and that of the whole algebra (with M.Zaicev).
		Abstract: We obtain a presentation of Schur algebras (and q-Schur algebras) by generators and relations, one which is compatible with the usual presentation of the enveloping algebra (quantized enveloping algebra) corresponding to the Lie algebra gl_n of n x n matrices.
		However, a certain subcategory of vertex algebras, obtained by restricting the order of locality of generators, has a universal object, which would be the natural candidate for a free vertex algebra.
	condor.depaul.edu /~scatoiu/conference/abstracts.html (909 words)

	Presentations for Matrix Algebras
		The initial data for a decomposition of the matrix algebra A. The output is a sequence of records, one for each simple quotient algebra of A, each consisting of the following fields.
		: The matrix of the primitive idempotent on the quotient algebra.
		Each entry in the list of lists is a tuple consisting of a matrix in the condensed algebra and the monomial which expresses this matrix as a product of the generators.
	magma.maths.usyd.edu.au /magma/htmlhelp/text877.htm (2054 words)

	Primitive Idempotents, Spinors, Cl(8)
		The simplest example in a matrix algebra is the matrix consisting entirely of zeros, except for a single entry of 1 somewhere in the main diagonal.
		They are where the matix algebra consists of 2x2 matrices over R, C or H, but in the case of 4x4 matrix algebras the primitive idempotents are products of commuting pairs of such idempotents.
		Though as minimal left ideals of matrix algebras any two spinor spaces are equivalent, they may lie differently when the Clifford algebra structure of the matrix algebra is taken into account...".
	www.valdostamuseum.org /hamsmith/8idempotents.html (2805 words)

	GAP Manual: 68 The MeatAxe
		For example, for GAP a MeatAxe matrix is a record containing at least information about the file name, the underlying finite field, and the dimensions of the matrix (see MeatAxe Matrices).
		is an element of a free algebra (see chapter Finitely Presented Algebras) that represents the matrix in terms of generators.
		MeatAxe matrix algebras are used to construct and describe MeatAxe modules and their structure (see MeatAxe Modules).
	www.mathematik.uni-kassel.de /gap3r4p4/CHAP068.htm (2928 words)

	Clifford algebras
		It turns out that the Clifford algebras are all full matrix algebras or the direct sums of two full matrix algebras:
		The link between differential equations and homotopy groups first came about as a result of the realization that ellipticity of a differential operator can be defined in terms of the symbol of the differential operator.
		The paper [33] shows that the minimal-dimensional representations of the Clifford algebras give rise to Dirac operators whose symbols generate the stable homotopy groups of the orthogonal group.
	www.math.harvard.edu /history/bott/bottbio/node14.html (245 words)

	Intelligent Systems :: Staff :: Zolotykh Andrej Alexeevich
		His Ph.D. thesis "The Theory of Trace Identities with Involution of Matrix Algebras and Superalgebras" was successfully defended in 1989.
		Mikhalev A.A., Zolotykh A.A. Endomorphism of the free Lie algebra preserving primitivity of elements is an automorphism.
		Zolotykh A.A. The Theory of Trace Identities with Involution of Matrix Algebras and Superalgebras.
	intsys.msu.ru /en/staff/zolotykh (1052 words)

	GAP Manual: 40.2. Bases for Matrix Algebras (Site not responding. Last check: 2007-10-13)
		As stated in section More about Matrix Algebras, the implementation of bases for matrix algebras follows that of row space bases, see Row Space Bases for the details.
		Due to the multiplicative structure that allows to view a matrix algebra A as an A-module with action via multiplication from the right, there is additionally the notion of a standard basis for A, which is essentially described in StandardBasis for Row Modules.
		The default way to compute a vector space basis of a matrix algebra from a set of generating matrices is to compute this standard basis and a semi-echelonized basis in parallel.
	www.math.uiuc.edu /Software/GAP-Manual/Bases_for_Matrix_Algebras.html (140 words)

	[ref] 60 Lie Algebras
		is the matrix of the adjoint representation of the element
		is the associative matrix algebra (with 1) generated by the matrices of the adjoint representation of the subalgebra
		Representatios of Lie algebras are delat with in the same way as representations of ordinary algebras (see Representations of Algebras).
	www.math.temple.edu /computing/gap/ref/CHAP060.htm (4374 words)

	6.2 Algebraic Number Fields as Matrix Algebras
		Let n be any positive integer and consider a sub-algebra K of the algebra of n×n matrices with rational entries; by this we mean that K contains scalar multiples of the identity matrix and is closed under matrix addition, subtraction and multiplication.
		To handle division we also insist that non-zero matrices in K are invertible (this actually implies that the inverses are also in K but it is not entirely trivial to prove this).
		To summarise, we will henceforth think of an algebraic number field as a sub-algebra of the ring of n×n matrices which is commutative, with all non-zero elements being invertible.
	www.imsc.res.in /~kapil/crypto/notes/node28.html (518 words)

	Advances in Computation: Theory and Practice, Book Series, Volume 4, Preface
		Matrix patterns and structural properties provide a uniform means for describing apparently different features of the problems that they model.
		In this case exploiting the specific features (in terms of structure and sparsity) is a mandatory step for the design of efficient algorithms (in terms of storage and time).
		This double interest is one of the motivation that has led many researchers to work on these problems and that has made very active the research in the area of structured matrix computations as it results from the many international conferences and workshops that have been recently organized on these topics.
	www.cs.okstate.edu /~actp/volumes/vol04pref.html (391 words)

	16: Associative rings and algebras
		There is a long FAQ on sets with products (rings), a particular emphasis of which is the study of division rings over the reals.
		This includes the study of matrix rings, division rings such as the quaternions, and rings of importance in group theory.
		What are the (other) roots of p(X)=0 in the ring M_n(F) where p is the characteristic polynomial of a matrix A? Family of division algebras over a field (plus Milnor conjecture, K-theory)
	www.math.niu.edu /~rusin/known-math/index/16-XX.html (541 words)

	DoubleMatrix
		Constructs a matrix from an array of vectors (columns).
		Returns the division of this matrix by a scalar.
		Returns the multiplication of this matrix by a scalar.
	i21www.ira.uka.de /motris/javadoc/JSci/maths/DoubleMatrix.html (349 words)

	Clifford Algebras
		Nonetheless, there is a profound relation between Clifford algebras and normed division algebras.
		is a division algebra, we can find an operator of this form mapping any point on the unit sphere to any other point.
		But as mentioned in Section 2.2, one can check explicitly that the sedenions are not a division algebra.
	math.ucr.edu /home/baez/octonions/node6.html (698 words)

	PlanetMath: ideals in matrix algebras
		"ideals in matrix algebras" is owned by mathcam.
		This is version 7 of ideals in matrix algebras, born on 2003-10-10, modified 2005-03-25.
		(Linear and multilinear algebra; matrix theory :: Algebraic systems of matrices)
	planetmath.org /encyclopedia/MatrixIdealsOverCommutativeRings.html (74 words)

	GAP Manual: 68.16 Functions for MeatAxe Matrix Algebras (Site not responding. Last check: 2007-10-13)
		The following functions are overlaid in the operations record of MeatAxe matrix algebras.
		be a MeatAxe module acted on by the MeatAxe matrix algebra
		returns a MeatAxe matrix algebra of n times n matrices (where n is the dimension of
	www-groups.dcs.st-and.ac.uk /gap/Gap3/Manual3/C068S016.htm (210 words)

	Research
		Characterize all ideals of PI-algebras which are the kernels of homomorphisms into matrix algebras over some field.
		Say if a polynomial f vanishes at all zeroes (in the algebraic closure) of polynomials g1, g2,...,gk, then there exists some positive integer m such that f^m is in the ideal generated by the gi's.
		some power of f is in the ideal generated by the gi's plus the ideal of identities of some matrix ring.
	www.math.ucsd.edu /~jnowacki/math (403 words)

	Matrix Algebras
		While a matrix algebra may be defined over any ring
		Centralizer of a subalgebra in the complete matrix algebra
		The order of a unit over a finite field is found using the very efficient algorithm of Leedham-Green.
	magma.maths.usyd.edu.au /magma/Features/node180.html (81 words)

	CMB - Facial Structures for the Positive Linear Maps Between Matrix Algebras
		CMB - Facial Structures for the Positive Linear Maps Between Matrix Algebras
		Facial Structures for the Positive Linear Maps Between Matrix Algebras
		Let $\scr P$ denote the convex set of all positive linear maps from the matrix algebra $M_n(\bbd C)$ into itself.
	journals.cms.math.ca /cgi-bin/vault/view/kye6613 (101 words)

	IngentaConnect On Contractibility of Matrix Algebras (Site not responding. Last check: 2007-10-13)
		We show first that for each C*-algebra A, contractibility of A implies contractibility of M
		We next prove that an incidence algebra A of upper triangular matrices, defined by a partially ordered set
		q, is a contractible Banach algebra iff there is no discordant couple of D-transitive triples of elements of
	www.ingentaconnect.com /content/nisc/qm/2002/00000025/00000003/art00007 (106 words)

	Math arXiv: Search results (Site not responding. Last check: 2007-10-13)
		math.QA/0508506 The GL(mn) type quantum matrix algebras II: the structure of the characteristic subalgebra and its spectral parameterization.
		hep-th/0003210 Quantum matrix algebra for the SU(n) WZNW model.
		math.QA/9912063 Modified Affine Hecke Algebras and Drinfeldians of Type A. Tolstoy, O. Ogievetsky, P. Pyatov, A. Isaev.
	front.math.ucdavis.edu /author/Pyatov-P* (213 words)

	Isometries between matrix algebras
		As an attempt to understand linear isometries between
		-algebras without the surjectivity assumption, we study linear isometries between matrix algebras.
		We prove that the converse is true if
	www.austms.org.au /Publ/JAustMS/V77P1/l28.html (51 words)

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