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2 % Standard operations from linear algebra.
5 % Needed for \lvert, \rVert, etc. and \operatorname.
10 % Absolute value (modulus) 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]{\left\langle{#1},
{#2}\right\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 ``transpose'' of a linear operator; namely, the adjoint, but
31 % specialized to real matrices.
32 \newcommand*
{\transpose}[1]{ #1^
{T
} }
34 % The trace of an operator.
35 \newcommand*
{\trace}[1]{ \operatorname{trace
}\of{{#1}} }
37 % The ``span of'' operator. The name \span is already taken.
38 \newcommand*
{\spanof}[1]{ \operatorname{span
}\of{{#1}} }
40 % The ``co-dimension of'' operator.
41 \newcommand*
{\codim}{ \operatorname{codim
} }
43 % The orthogonal projection of its second argument onto the first.
44 \newcommand*
{\proj}[2] { \operatorname{proj
}\of{#1,
#2} }
46 % The ``Automorphism group of'' operator.
47 \newcommand*
{\Aut}[1]{ \operatorname{Aut
}\of{{#1}} }
49 % The ``Lie algebra of'' operator.
50 \newcommand*
{\Lie}[1]{ \operatorname{Lie
}\of{{#1}} }
52 % The ``write a matrix as a big vector'' operator.
53 \newcommand*
{\vectorize}[1]{ \operatorname{vec
}\of{{#1}} }
55 % The ``write a big vector as a matrix'' operator.
56 \newcommand*
{\matricize}[1]{ \operatorname{mat
}\of{{#1}} }
58 % An inline column vector, with parentheses and a transpose operator.
59 \newcommand*
{\colvec}[1]{ \left(
{#1}\right)^
{T
} }
61 % Bounded linear operators on some space. The required argument is the
62 % domain of those operators, and the optional argument is the
63 % codomain. If the optional argument is omitted, the required argument
65 \newcommand*
{\boundedops}[2][]{
67 \if\relax\detokenize{#1}\relax