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2 % Standard operations from linear algebra.
5 % Needed for \lvert, \rVert, etc. and \operatorname.
10 % Absolute value (modulis) of a scalar.
11 \newcommand*
{\abs}[1]{\left\lvert{#1}\right\rvert}
14 \newcommand*
{\norm}[1]{\left\lVert{#1}\right\rVert}
16 % The inner product between its two arguments.
17 \newcommand*
{\ip}[2]{\langle{#1},
{#2}\rangle}
19 % The tensor product of its two arguments.
20 \newcommand*
{\tp}[2]{ {#1}\otimes{#2} }
22 % The Kronecker product of its two arguments. The usual notation for
23 % this is the same as the tensor product notation used for \tp, but
24 % that leads to confusion because the two definitions may not agree.
25 \newcommand*
{\kp}[2]{ {#1}\odot{#2} }
27 % The adjoint of a linear operator.
28 \newcommand*
{\adjoint}[1]{ #1^
{*
} }
30 % The ``span of'' operator. The name \span is already taken.
31 \newcommand*
{\spanof}[1]{ \operatorname{span
}\of{{#1}} }
33 % The ``co-dimension of'' operator.
34 \newcommand*
{\codim}{ \operatorname{codim
} }
36 % The trace of an operator.
37 \newcommand*
{\trace}[1]{ \operatorname{trace
}\of{{#1}} }
39 % The orthogonal projection of its second argument onto the first.
40 \newcommand*
{\proj}[2] { \operatorname{proj
}\of{#1,
#2} }
42 % The ``Automorphism group of'' operator.
43 \newcommand*
{\Aut}[1]{ \operatorname{Aut
}\of{{#1}} }
45 % The ``Lie algebra of'' operator.
46 \newcommand*
{\Lie}[1]{ \operatorname{Lie
}\of{{#1}} }
48 % The ``write a matrix as a big vector'' operator.
49 \newcommand*
{\vectorize}[1]{ \operatorname{vec
}\of{{#1}} }
51 % The ``write a big vector as a matrix'' operator.
52 \newcommand*
{\matricize}[1]{ \operatorname{mat
}\of{{#1}} }
54 % An inline column vector, with parentheses and a transpose operator.
55 \newcommand*
{\colvec}[1]{ \left(
{#1}\right)^
{T
} }
57 % Bounded linear operators on some space. The required argument is the
58 % domain of those operators, and the optional argument is the
59 % codomain. If the optional argument is omitted, the required argument
61 \newcommand*
{\boundedops}[2][]{
63 \if\relax\detokenize{#1}\relax