adjoint transforms / adjoint operators

NB: for complex, still have transpose,
  but adjoint (transpose + complex conj.) is better, usually...

adjoint transform: only thing to say is:
 row rank = column rank

adjoint operator:
do everything you can in terms of the natural pairing
V x V^* -> K

and only -then- specialize to when you have an iso V -> V^*
(clarifies a lot about self-adjoint/normal operators)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
adjoint, opposite, transpose

adjoint:
given $T \from V \to W$, the adjoint is the dual map
  $T^* \from W^* \to V^*$ (Russians and Algebraic Geometers denote
  this by $T^\vee \from W^\vee \to V^\vee$)

opposite:
Given a bilinear form $B\from V \times V \to K$,
you get the opposite form $B^\op$ by switching factors

transpose:
given a matrix (equivalently, a map $K^n \to K^m$), you get the
transpose map; transpose makes sense -in coordinates-

how these relate:
adjoint - transpose
  given coordinates on V, W;
  the matrix for $T^*$ is the transpose

opposite - transpose
  given coordinates on V, the matrix for $B^\op$ is the transpose

adjoint - opposite
  can think of $B$ as $B\from V \to V^*$:
  for fixed first variable, get functional in second variable;
  also get $B^* \from V^{**}=V \to V^*$, which is the opposite.
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$B+B^\op$ is symmetric; $B-B^\op$ is skew-sym;
away from 2 get splitting $(V\otimes V) = \Sym^2 V \oplus \Alt^2 V$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Frobenius defined "equivalence, similarity, etc.";

how stuff transforms:
given a map $V \to W$, writing matrix, changing basis on V changes on
right, changing basis on W changes on left
given a map $V \to V$, writing matrix, changing basis on V changes by
$A^{-1}MA$
given a map $V \to V^*$, writing matrix, changing basis on V changes by
$A^*MA$
...that's why these are natural;
and unitary transforms are the ones that respect both of these latter
