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Morita.txt The bimodules which induce the equivalences of module categories ca* *n both be taken to be Rn, but viewed as `row vectors' (or 1 x n matrices) and `column vec* *tors' (or n x 1 matrices) respectively. Since the category of right Rop-modules is isomorphic to the cate* *gory of left R-modules, we can view M as an Sop-Rop-bimodule and N as an Rop-Sop-bimodule, a* *nd then they provide the equivalence of categories between Mod-Ropand Mod-Sop. So the S-mod* *ule F R is a small projective generator of the category of S-modules. hopf.math.purdue.edu /Schwede/Morita.txt   (4235 words)

PlanetMath [ simultaneous block-diagonalization of upper triangular commuting matrices ] [ simultaneous upper triangular block-diagonalization of commuting matrices ] [ simultaneous triangularisation of commuting matrices over any field ] planetmath.org /browse/categories   (4235 words)

15: Linear and multilinear algebra; matrix theory In this category we might include 15A06: Linear equations, 15A09: Matrix inversion and generalized inverses, 15A18: Eigenvalues and singular values, 15A23: Factorization of matrices (SVD, LU, QR, etc.), 15A12: Conditioning of matrices, as well as applications to physics (15A90), Control Theory (93) and Statistics (62) such as what is there known as Principal Component Analysis. Also crowded near the center of the diagram are several fields concerned with linear spaces and linear transformations, and in some cases the reflection of those ideas in the corresponding matrices. topological) spaces is classified separately in the fields of functional analysis including 46: Function Analysis proper, 43: Abstract harmonic analysis, and 47: Operator theory. www.math.niu.edu /~rusin/known-math/index/15-XX.html   (1605 words)

Morita.txt Since the category of right Rop-modules is isomorphic to the cate* *gory of left R-modules, we can view M as an Sop-Rop-bimodule and N as an Rop-Sop-bimodule, a* *nd then they provide the equivalence of categories between Mod-Ropand Mod-Sop. The bimodules which induce the equivalences of module categories ca* *n both be taken to be Rn, but viewed as `row vectors' (or 1 x n matrices) and `column vec* *tors' (or n x 1 matrices) respectively. So the S-mod* *ule F R is a small projective generator of the category of S-modules. hopf.math.purdue.edu /Schwede/Morita.txt   (1605 words)

Free Module, Basis The module automorphisms correspond to the nonsingular matrices. For now, we'll call this the definition of a "free module", but if you study category theory, you'll find that a free object has a more general definition, and when that definition is applied to the category of unitary modules, the "direct sum" definition falls out. These module homomorphisms are called linear transformations in the world of linear algebra, and they can be represented by matrices, where matrix multiplication implements function composition. www.mathreference.com /mod,basis.html   (512 words)

Morita.txt Since the category of right Rop-modules is isomorphic to the cate* *gory of left R-modules, we can view M as an Sop-Rop-bimodule and N as an Rop-Sop-bimodule, a* *nd then they provide the equivalence of categories between Mod-Ropand Mod-Sop. The bimodules which induce the equivalences of module categories ca* *n both be taken to be Rn, but viewed as `row vectors' (or 1 x n matrices) and `column vec* *tors' (or n x 1 matrices) respectively. Acknowledgments: The Morita theory in stable model categories which I descri* *be in Section 4 is based on joint work with Brooke Shipley spread over many years and* * several papers; I would like to take this opportunity to thank her for the pleasant and* * fruitful collaboration. hopf.math.purdue.edu /Schwede/Morita.txt   (4235 words)

Skew-symmetric matrix - Wikipedia, the free encyclopedia Skew-symmetric matrices fall into the category of normal matrices and are thus subject to the spectral theorem, which states that any real or complex skew-symmetric matrix can be diagonalized by a unitary matrix. If matrices A and B are both skew-symmetric, then their product AB is a symmetric matrix. Another way of saying this is that the space of skew-symmetric matrices forms the Lie algebra o(n) of the Lie group O(n). en.wikipedia.org /wiki/Skew-symmetric_matrix   (626 words)

psi^3 as an upper triangular matrix, by Jonathan Barker and Victor Snaith In the 2-local stable homotopy category the group of left-bu-module automorphisms of bu \wedge bo which induce the identity on mod 2 homology is isomorphic to the group of infinite upper triangular matrices with entries in the 2-adic integers. We identify the conjugacy class of the matrix corresponding to 1 \wedge \psi^3, where \psi^3 is the Adams operation. psi^3 as an upper triangular matrix, by Jonathan Barker and Victor Snaith www.math.uiuc.edu /K-theory/0728   (626 words)

ACT's CAAP Tests: Mathematics Test Items in this category are based on advanced algebra concepts including rational exponents, exponential and logarithmic functions, complex numbers, matrices, inverses of functions, and domains and ranges. Knowledge and skills assessed in this category may include graphing in the standard coordinate plane or the real number line, graphing conics, linear equations in two variables, graphing systems of equations, and similar types of skills. The CAAP Mathematics Test is a 35-item, 40-minute test designed to measure students' proficiency in mathematical reasoning. www.act.org /caap/tests/math.html   (348 words)

Product When matrices or members of various other associative algebras are multiplied the product usually depends on the order of the factors; in other words, matrix multiplication, and the multiplications in those other algebras, are non-commutative. In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The dot product and cross product are forms of multiplication of vectors. www.city-search.org /pr/product.html   (348 words)

ipedia.com: Linear algebra Article In the spectral theory of operators control of infinite-dimensional matrices is gained, by applying mathematical analysis in a theory that isn't purely algebraic. In module theory one replaces the field of scalars by a ring. Since linear algebra is a successful theory, its methods have been developed in other parts of mathematics. www.ipedia.com /linear_algebra_1.html   (851 words)

CMUC Lawvere's presentation of a metric space as a V-category is included in our setting, via the Betti-Carboni-Street-Walters interpretation of a V-category as a monad in the bicategory of V-matrices, and so are Barr's presentation of topological spaces as lax algebras, Lowen's approach spaces, and Lambek's multicategories, which enjoy renewed interest in the study of n-categories. For a symmetric monoidal-closed category V and a suitable monad T on the category of sets, we introduce the notion of reflexive and transitive (T,V)-algebra and show that various old and new structures are instances of such algebras. As a further example, we introduce a new structure called ultracategory which simultaneously generalizes the notions of topological space and of category. www.mat.uc.pt /~cmuc/pubdetails.php?pub=661&lid=1   (139 words)

[ref] 59 Vector Spaces Note that this is meaningful only if the mechanism of computing nice and ugly vectors (see Vector Spaces Handled By Nice Bases) is invariant under closures of the basis; this is the case for example if the vectors are matrices, Lie objects, or elements of structure constants algebras. Examples of such vector spaces are vector spaces of field elements (but not the fields themselves) and non-Gaussian row and matrix spaces (see IsGaussianSpace). The canonical basis of a Gaussian matrix space is defined as the unique semi-echelonized (see IsSemiEchelonized) basis for which the list of concatenations of the basis vectors forms the canonical basis of the corresponding Gaussian row space. mad.epfl.ch /gap/ref/CHAP059.htm   (139 words)

Atlas: Matrix calculus for metric, topological and approach spaces by Walter Tholen This leads to the definition and study of so-called (T, V)-algebras where V is any symmetric closed-monoidal category and T is a monad on Set which allows for a suitable extension to the (bi)category Mat(V) of V- matrices. We take Lawvere's presentation of a metric space as a V-category (where V is the closed interval from 0 to infinity) as the starting point to describe topological spaces and approach spaces in a similar fashion. Matrix calculus for metric, topological and approach spaces atlas-conferences.com /c/a/g/x/50.htm   (212 words)

15: Linear and multilinear algebra; matrix theory In this category we might include 15A06: Linear equations, 15A09: Matrix inversion and generalized inverses, 15A18: Eigenvalues and singular values, 15A23: Factorization of matrices (SVD, LU, QR, etc.), 15A12: Conditioning of matrices, as well as applications to physics (15A90), Control Theory (93) and Statistics (62) such as what is there known as Principal Component Analysis. Also crowded near the center of the diagram are several fields concerned with linear spaces and linear transformations, and in some cases the reflection of those ideas in the corresponding matrices. In the upper right are the topics appropriate for 60: Probability and 62: Statistics, including 15A51: Stochastic matrices and 15A52: Random matrices, and applications to statistical mechanics (82) and the sciences (92). www.math.niu.edu /~rusin/known-math/index/15-XX.html   (212 words)

15: Linear and multilinear algebra; matrix theory In this category we might include 15A06: Linear equations, 15A09: Matrix inversion and generalized inverses, 15A18: Eigenvalues and singular values, 15A23: Factorization of matrices (SVD, LU, QR, etc.), 15A12: Conditioning of matrices, as well as applications to physics (15A90), Control Theory (93) and Statistics (62) such as what is there known as Principal Component Analysis. There are several connections with ring theory (16: Noncommutative Rings, 17: Nonassociative Rings, 19: Algebraic K-Theory); indeed many of the key examples of such rings involve collections of matrices, including the full matrix rings and Lie rings, and rings of matrices are used for representing groups and general rings. At the far right are several large areas of activity in numerical linear algebra and related topics, typically, the study of individual matrices or transformations between (large-dimensional) real vector spaces. www.math.niu.edu /~rusin/known-math/index/15-XX.html   (212 words)

Fernando Muro research Moreover, we see that for a 2-dimensional space the proper L-S category is 2 if and only if the fundamental pro-group belong to that class, if and only if it is a proper co-H-space. Here we compute the suspension functor in the homotopy category of qadratic complexes (Baues' Peiffer-nilpotent version of 2-crossed modules) as well as the co-H-structure on a suspended quadratic complex, which turns out to be a strict cogroup structure. We also prove that this class is closed under retracts in the category of finitely presented towers and construct proper Eilenberg-MacLane spaces for these towers. www.aloj.us.es /fmuro/research.htm   (212 words)

morestuff.html Much of what we learned in class today (7/5) may be rephrased as the simple statement "nxn invertible matrices form a group with matrix multiplication as the operation." Note that the set of all mxn matrices is also a group with respect to addition for any m and n. So here the forgetful functor sends the field (T,*,+) to the ring (T,*,+), but when a field arrives in the category of rings, the only difference is that in its new home, nobody notices that its group of units is abelian and contains all nonzero elements. The group of nxn invertible matrices with entries in R (the set of real numbers) has a name: it is called GL(n,R). www.math.columbia.edu /~hundley/morestuff.html   (212 words)

Additive category - Wikipedia, the free encyclopedia However, this concept makes sense — such matrices have 0 entries are determined uniquely by their size alone — and while they are rather degenerate, they do need to be included to get an additive category, since an additive category must have a zero object 0. If we define morphism composition to be multiplication of matrices, then Mat ( R) becomes an additive category, and A Thus additive categories can be seen as the most general context in which the algebra of matrices makes sense. en.wikipedia.org /wiki/Additive_category   (212 words)

Morita.txt The bimodules which induce the equivalences of module categories ca* *n both be taken to be Rn, but viewed as `row vectors' (or 1 x n matrices) and `column vec* *tors' (or n x 1 matrices) respectively. Since the category of right Rop-modules is isomorphic to the cate* *gory of left R-modules, we can view M as an Sop-Rop-bimodule and N as an Rop-Sop-bimodule, a* *nd then they provide the equivalence of categories between Mod-Ropand Mod-Sop. The endomorphism ring End R(Rn) ~= Mn(R) is the ring of n x n matrices with entries in R. So R and the matrix ring Mn(R)* * are Morita equivalent. hopf.math.purdue.edu /Schwede/Morita.txt   (212 words)

15: Linear and multilinear algebra; matrix theory In this category we might include 15A06: Linear equations, 15A09: Matrix inversion and generalized inverses, 15A18: Eigenvalues and singular values, 15A23: Factorization of matrices (SVD, LU, QR, etc.), 15A12: Conditioning of matrices, as well as applications to physics (15A90), Control Theory (93) and Statistics (62) such as what is there known as Principal Component Analysis. At the far right are several large areas of activity in numerical linear algebra and related topics, typically, the study of individual matrices or transformations between (large-dimensional) real vector spaces. Eigenvalues of a symmetric matrix and the symmetric part of a general square matrix. www.math.niu.edu /~rusin/known-math/index/15-XX.html   (212 words)

A Generalization of Redfield's Master Theorem (ResearchIndex) The main result of this theory is the eqivalence between the category of finite-dimensional linear representations of the symmetric group S d and the category of polynomial homogeneous degree d functors on the category of finite-dimensional linear spaces.... Abstract: Introduction In our paper [3], it is shown that the natural environment for P'olya's fundamental enumeration theorem and for one of its possible generalizations, is Schur-Macdonald's theory of invariant matrices (cf. 0.5: A Generalization of Pólya's Enumeration Theorem or the.. citeseer.ist.psu.edu /558744.html   (259 words)

Arlettaz-survey.txt Thus, K* *2(-) is a covariant functor from the category of rings to the category of abelian groups. The category of finitely generated projective modules over a ring R was actually in the center of the pr* *eoccupations of the first K-theorists because of its relationships with linear groups which play a crucia* *l role in almost all subjects in mathematics. The functors K1 and K2 One of the main objects of interest in linear algebra over a ring R is the gene* *ral linear group GLn(R) consisting of the multiplicative group of n x n invertible matrices with coeffi* *cients in R. hopf.math.purdue.edu /Arlettaz/Arlettaz-survey.txt   (8942 words)

Working Seminar on sl_2 Categorification The basic idea of sl_2 categorification is to find representations of the Lie algebra sl_2 of trace zero matrices over C as functors on a category (as opposed to linear operators). sl2 categorifications appear in the work of Bernstein, Frenkel and Khovanov [1], who use various blocks of the so-called category O of representations of simple Lie algebras of type A to construct categorifications of sl2. The aim of the seminar will be to study the Chuang-Rouquier paper [2], in particular looking at affine Hecke algebras, central characters, adjunctions, sl_2 categorification, and the examples concerning symmetric groups and the category O (and, if there is interest/time, other examples). www.math.le.ac.uk /RESEARCH/PURE/SEMINARS/REP_SEMINAR/sl2.html   (8942 words)

categories.html This category contains subroutines which map and/or transform scalars, vectors, or matrices from one representation into another. Here also are routines which generate mapping and/or rotation matrices that can be used in conjunction with matrix multiply routines for transforming both vectors and matrices associated with physical entities. Vector routines manipulate vectors to produce scalars or other vectors. roger.ecn.purdue.edu /~masl/documents/vector/vector.current/categories.html   (8942 words)

Group representation In abstract algebra, a representation of a finite group G is a group homomorphism from G to the general linear group GL( n'','''C') of invertible complex n -by- n matrices. Rather generally, a representation of a group G in a category 'C' is a functor from G (as one-object category) to 'C' A set-theoretic representation is a representation in the category of sets, i.e. In the study of mathematical groups, a group representation is a "description" of a group as a concrete group of transformations (or automorphism group) of some mathematical object. www.serebella.com /encyclopedia/article-Group_representation.html   (8942 words)

Professor Fred DePiero Subgraph isomorphism is proven to be in a combinatorically-wicked category of problems (NP-Complete). (This operation requires N^2 effort to visit each location in the adjacency matrices.) Verification using the reordered adjacency matrices can be used to eliminate the possibility of a false-positive result being reported for an isomorphism. Subgraph isomorphism is a condition of isomorphism that exists between two subgraphs of G and H. www.ee.calpoly.edu /~fdepiero/fdepiero_research/subgraph.html   (8942 words)

Goldblatt. Topoi: The Categorial Analysis of Logic Matrices over a commutative ring K form the arrows of a category Matr(K) where the objects are natural numbers and an nxm matrix is an arrow from m to n. The natural numbers N gives an example of a category (with one object and natural numbers as arrows and addition as composition) which is not a preorder. These categories are examples of preorders: categories in which there is at most one arrow between any two objects (identity corresponds to reflexivity and composition corresponds to transitivity). www.andrew.cmu.edu /~cebrown/notes/goldblatt.html   (8942 words)

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