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Supertrace
In the theory of superalgebras, if ''A'' is a commutative superalgebra, ''V'' is a free right ''A''-supermodule and ''T'' is an endomorphism from ''V'' to itself, then the supertrace of ''T'', str(''T'') is defined by the following trace diagram: : More concretely, if we write out ''T'' in block matrix form after the decomposition into even and odd subspaces as follows, :T=\beginT_&T_\\T_&T_\end then the supertrace :str(''T'') = the ordinary trace of ''T''00 − the ordinary trace of ''T''11. Let us show that the supertrace does not depend on a basis. Suppose e1, ..., ep are the even basis vectors and e''p''+1, ..., e''p''+''q'' are the odd basis vectors. Then, the components of ''T'', which are elements of ''A'', are defined as :T(\mathbf_j)=\mathbf_i T^i_j.\, The grading of ''T''''i''''j'' is the sum of the gradings of ''T'', e''i'', e''j'' mod 2. A change of basis to e1', ..., ep', e(''p''+1)', ..., e(''p''+''q'')' is given by the supermatrix :\mathbf_=\mathbf_i A^i_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supermatrix
In mathematics and theoretical physics, a supermatrix is a Z2-graded analog of an ordinary matrix (mathematics), matrix. Specifically, a supermatrix is a 2×2 block matrix with entries in a superalgebra (or superring). The most important examples are those with entries in a commutative superalgebra (such as a Grassmann algebra) or an ordinary field (mathematics), field (thought of as a purely even commutative superalgebra). Supermatrices arise in the study of super linear algebra where they appear as the coordinate representations of a linear transformations between finite-dimensional super vector spaces or free supermodules. They have important applications in the field of supersymmetry. Definitions and notation Let ''R'' be a fixed superalgebra (assumed to be unital algebra, unital and associative). Often one requires ''R'' be supercommutative as well (for essentially the same reasons as in the ungraded case). Let ''p'', ''q'', ''r'', and ''s'' be nonnegative integers. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace (matrix)
In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities). It can also be proved that for any two matrices and . This implies that similar matrices have the same trace. As a consequence one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). Definition The trace of an square matrix is defined as \operatorname(\mathbf) = \sum_^n a_ = a_ + a_ + \dots + a_ where denotes the entry on the th row and th column of . The entries of can be real numbers or (more generally) complex numbers. The trace is not def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berezinian
In mathematics and theoretical physics, the Berezinian or superdeterminant is a generalization of the determinant to the case of supermatrices. The name is for Felix Berezin. The Berezinian plays a role analogous to the determinant when considering coordinate changes for integration on a supermanifold. Definition The Berezinian is uniquely determined by two defining properties: *\operatorname(XY) = \operatorname(X)\operatorname(Y) *\operatorname(e^X) = e^\, where str(''X'') denotes the supertrace of ''X''. Unlike the classical determinant, the Berezinian is defined only for invertible supermatrices. The simplest case to consider is the Berezinian of a supermatrix with entries in a field ''K''. Such supermatrices represent linear transformations of a super vector space over ''K''. A particular even supermatrix is a block matrix of the form :X = \beginA & 0 \\ 0 & D\end Such a matrix is invertible if and only if both ''A'' and ''D'' are invertible matrices over ''K''. The Berezin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superalgebra
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. The prefix ''super-'' comes from the theory of supersymmetry in theoretical physics. Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry where they enter into the definitions of graded manifolds, supermanifolds and superschemes. Formal definition Let ''K'' be a commutative ring. In most applications, ''K'' is a field of characteristic 0, such as R or C. A superalgebra over ''K'' is a ''K''-module ''A'' with a direct sum decomposition :A = A_0\oplus A_1 together with a bilinear multiplication ''A'' × ''A'' → ''A'' such t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Superalgebra
In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements ''x'', ''y'' we have :yx = (-1)^xy , where , ''x'', denotes the grade of the element and is 0 or 1 (in Z) according to whether the grade is even or odd, respectively. Equivalently, it is a superalgebra where the supercommutator : ,y= xy - (-1)^yx always vanishes. Algebraic structures which supercommute in the above sense are sometimes referred to as skew-commutative associative algebras to emphasize the anti-commutation, or, to emphasize the grading, graded-commutative or, if the supercommutativity is understood, simply commutative. Any commutative algebra is a supercommutative algebra if given the trivial gradation (i.e. all elements are even). Grassmann algebras (also known as exterior algebras) are the most common examples of nontrivial supercommutative algebras. The supercenter of any superalgebra is the set of elements that supe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supermodule
In mathematics, a supermodule is a Z2-graded module over a superring or superalgebra. Supermodules arise in super linear algebra which is a mathematical framework for studying the concept supersymmetry in theoretical physics. Supermodules over a commutative superalgebra can be viewed as generalizations of super vector spaces over a (purely even) field ''K''. Supermodules often play a more prominent role in super linear algebra than do super vector spaces. These reason is that it is often necessary or useful to extend the field of scalars to include odd variables. In doing so one moves from fields to commutative superalgebras and from vector spaces to modules. :''In this article, all superalgebras are assumed be associative and unital unless stated otherwise.'' Formal definition Let ''A'' be a fixed superalgebra. A right supermodule over ''A'' is a right module ''E'' over ''A'' with a direct sum decomposition (as an abelian group) :E = E_0 \oplus E_1 such that multiplication by el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group is a group homomorphism . In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set ''S'' to itself. In any category, the composition of any two endomorphisms of is again an endomorphism of . It follows that the set of all endomorphisms of forms a monoid, the full transformation monoid, and denoted (or to emphasize the category ). Automorphisms An invertible endomorphism of is called an automorphism. The set of all automorphisms is a subset of with a group structure, called the automorphism group of and denoted . In the following diagram, the arrows denote implication: Endomorphism rings Any two endomorphisms of an abelian group, , can be added toge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Diagram
In mathematics, trace diagrams are a graphical means of performing computations in linear and multilinear algebra. They can be represented as (slightly modified) graphs in which some edges are labeled by matrices. The simplest trace diagrams represent the trace and determinant of a matrix. Several results in linear algebra, such as Cramer's Rule and the Cayley–Hamilton theorem, have simple diagrammatic proofs. They are closely related to Penrose's graphical notation. Formal definition Let ''V'' be a vector space of dimension ''n'' over a field ''F'' (with ''n''≥2), and let Hom(''V'',''V'') denote the linear transformations on ''V''. An ''n''-trace diagram is a graph \mathcal=(V_1\sqcup V_2\sqcup V_n, E), where the sets ''V''''i'' (''i'' = 1, 2, ''n'') are composed of vertices of degree ''i'', together with the following additional structures: * a ''ciliation'' at each vertex in the graph, which is an explicit ordering of the adjacent edges at that ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace
Trace may refer to: Arts and entertainment Music * Trace (Son Volt album), ''Trace'' (Son Volt album), 1995 * Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * The Trace (album), ''The Trace'' (album) Other uses in arts and entertainment * Trace (magazine), ''Trace'' (magazine), British hip-hop magazine * Trace (manhwa), ''Trace'' (manhwa), a Korean internet cartoon * Trace (novel), ''Trace'' (novel), a novel by Patricia Cornwell * The Trace (film), ''The Trace'' (film), a 1994 Turkish film * The Trace (video game), ''The Trace'' (video game), 2015 video game * Sama (film), ''Sama'' (film), alternate title ''The Trace'', a 1988 Tunisian film * Trace, a fictional character in the game ''Metroid Prime Hunters'' * Trace, the protagonist of ''Axiom Verge'' * Trace, another name for List of One Piece characters#Portgas D. Ace, Portgas D. Ace, a fictional character in the manga ''One Piece'' * TRACE, the main brand for a numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices. Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned. This notion can be made more precise for an n by m matrix M by partitioning n into a collection \text, and then partitioning m into a collection \text. The original matrix is then considered as the "total" of these groups, in the sense that the (i, j) entry of the original matrix corresponds in a 1-to-1 way with some (s, t) offset entry of some (x,y), where x \in \text and y \in \text. Block matrix algebra arises in general from biproducts in categories of matrices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis Independent
Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of the purchase of a security and the sale of a similar security **Basis of futures, the value differential between a future and the spot price **Basis (options), the value differential between a call option and a put option **Basis swap, an interest rate swap *Cost basis, in income tax law, the original cost of property adjusted for factors such as depreciation *Tax basis, cost of an asset and technology *Basis function *Basis (linear algebra) **Dual basis **Orthonormal basis **Schauder basis *Basis (universal algebra) *Basis of a matroid *Generating set of an ideal: **Gröbner basis ** Hilbert's basis theorem *Generating set of a group *Base (topology) *Change of basis *Greedoid *Normal basis *Polynomial basis *Radial basis function *Sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
