% \iffalse meta-comment % % Copyright (C) 2018 - 2021 by ChairX % % This file may be distributed and/or modified under the % conditions of the LaTeX Project Public License, either % version 1.3 of this license or (at your option) any later % version. The latest version of this license is in: % % http://www.latex-project.org/lppl.txt % % and version 1.3 or later is part of all distributions of % LaTeX version 2005/12/01 or later. % % This file contains the documentation of all linear algebra related macros . % % Macros have to be described by (delete the first %) % \DescribeMacro{\macro} % Description and usage of the macro. % % The description will appear in the usage % part of the documentation. Use \subsubsection{} etc. for structuring. % % The implementation of the macros defined here has to be written in % chairxmathLinalg.dtx %\fi % %\subsubsection{General Linear Algebra} % %\DescribeMacro{\tr} % Trace of a linear map |\tr(A)|: $\tr(A)$ \\ % Uses |operatorfont|. % %\DescribeMacro{\rank} % Rank of a linear map |\rank(A)|: $\rank(A)$ \\ % Uses |operatorfont|. % %\DescribeMacro{\codim} % Codimension |\codim U|: $\codim U$ \\ % Uses |operatorfont|. % %\DescribeMacro{\diag} % Diagonal (for filling matrices etc.) |\diag(1,-1, -1)|: $\diag(1,-1, -1)$ \\ % Uses |operatorfont|. % %\DescribeMacro{\Trans} % Transposition of matrices |A^\Trans|: $A^\Trans$ \\ % Uses |scriptfont|. % %\DescribeMacro{\Mat} % Matrices |\Mat_n(\mathbb{R})|: $\Mat_n(\mathbb{R})$ \\ % Uses |operatorfont|. % %\DescribeMacro{\SymMat} % Symmetric matrices |\SymMat_n(\mathbb{R})|: $\SymMat_n(\mathbb{R})$ \\ % Uses |operatorfont|. % %\DescribeMacro{\ann} % Annihilator of a subspace |U^\ann|: $U^\ann$ \\ % Uses |scriptfont|. % %\DescribeMacro{\Span} % Span of something |\Span\{v, u\}|: $\Span\{v, u\}$ % and with optional argument % |\Span[\mathbb{C}]\{v,u\}|: $\Span[\mathbb{C}]\{v,u\}$ \\ % Uses |operatorfont|. % %\DescribeMacro{\basis} % Font for basis vectors |\basis{e}_i|: $\basis{e}_i$ \\ % Uses |basisfont|. % %\subsubsection{Tensors} % %\DescribeMacro{\tensor} % Generic tensor product over some ring |a \tensor b|: $a \tensor b$.\\ % With optional subscript |V \tensor[\algebra{A}] U|: $V \tensor[\algebra{A}] U$ % %\DescribeMacro{\Tensor} % Tensor powers, tensor algebra |\Tensor^\bullet(V)|: $\Tensor^\bullet(V)$ \\ % Uses |operatorfont|. % %\DescribeMacro{\Anti} % Antisymmetric tensor powers, Grassmann algebra |\Anti(V)|: $\Anti(V)$ % %\DescribeMacro{\Sym} % Symmetric tensor powers, symmetric algebra |\Sym^\bullet(V)|: $\Sym^\bullet(V)$ \\ % Uses |operatorfont|. % %\DescribeMacro{\Symmetrizer} % Symmetrizer |\Symmetrizer_n|: $\Symmetrizer_n$ % %\DescribeMacro{\AntiSymmetrizer} % Anti-symmetrizer |\AntiSymmetrizer|: $\AntiSymmetrizer$ % %\DescribeMacro{\ins} % Generic insertion map |\ins_X|: $\ins_X$ \\ % Uses |operatorfont|. % %\DescribeMacro{\jns} % Generic right insertion map |\jns_X|: $\jns_X$ \\ % Uses |operatorfont|. % %\DescribeMacro{\insa} % Antisymmetric insertion map |\insa(X)|: $\insa(X)$ \\ % Uses |operatorfont|, |scriptfont|. % %\DescribeMacro{\inss} % Symmetric insertion map |\inss(v)|: $\inss(v)$ \\ % Uses |operatorfont|, |scriptfont|. % %\DescribeMacro{\dega} % Antisymmetric degree |\dega(a) = ka|: $\dega(a) = ka$ \\ % Uses |operatorfont|, |scriptfont|. % %\DescribeMacro{\degs} % Symmetric degree |\degs(X) = \ell X|: $\degs(X) = \ell X$ \\ % Uses |operatorfont|, |scriptfont|. % %\subsubsection{Inner Products} % %\DescribeMacro{\SP} % Simple scalar product |\SP{x, y}|: $\SP{x, y}$. % %\DescribeMacro{\littlepara} % Small parallel to be used as a subscript |v_\littlepara|: $v_\littlepara$ % %\DescribeMacro{\IP} % Generic inner product with five arguments to decorate it |\IP[]{}{}{}{}{}| and an optional argument to adjust the size: %\[ %\IP[\big]{}{B}{z, w}{\perp}{R} %\quad %\textrm{and} %\quad %\IP[\Big]{\perp}{\algebra{B}}{\prod x_i, y}{\prime}{\algebra{A}} %\]