So, maybe some background I should have included somewhere along the
way.  I'm adding significant amounts of Sage code to my open source
linear algebra textbook and am working hard to make the linear algebra
code more welcoming to beginners (and thanks to those who have been
helping).

The confusion (obvious above) is one thing I wanted to tackle.  The
patch at

http://trac.sagemath.org/sage_trac/ticket/10501

deprecates adjoint in favor of adjugate.  It requires one change in
the crypto code (inverting an integer matrix) and two or three changes
to code within quadratic forms routines.  But it does introduce new
confusion, such as getting a quadratic form adjoint, via the adjugate,
which is implemented with PARI's matadjoint method.

I have no good source for the term adjugate (but don't have my books
handy lately) and don't like the word much myself.  Having taught a
"matrix analysis" course twice now, it seems that adjoint is what gets
used regularly for the conjugate-transpose, so that is my motivation.
I'll admit the distinction between matrices and operators is less
interesting to me.

I didn't think properties were going to fly, but they are close to
ready at #8094.  I could probably live with a conjugate_transpose
method (#10471) and the new H property as a less cumbersome version.
Then I would document Sage's use of the adjoint carefully in the
textbook (and maybe in some docstrings) rather than waiting for it to
come out of a deprecation waiting period.  My one reservation is that
beginners get very confused about why, for example, A.tranpose() is
correct, yet A.transpose is not.  Having A.H work seems to just add to
that misunderstanding.  (I'm not against properties, I just would not
use them with beginners.)

Strong opinions welcome, otherwise Gonzalo and I will come to some
agreement and move on.

Rob

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