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% Copyright (C) 2018 - 2021 by ChairX
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% version.  The latest version of this license is in:
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%    http://www.latex-project.org/lppl.txt
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% 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:
%\[
%\IP[\big]{}{B}{z, w}{\perp}{R}
%\quad
%\textrm{and}
%\quad
%\IP[\Big]{\perp}{\algebra{B}}{\prod x_i, y}{\prime}{\algebra{A}}
%\]
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