My opinion: What you call the classical adjoint is really the adjugate. That is abbreviated to adj, and since there is also an adjoint, it is a common error to call the adjugate the adjoint.
I would not be surprised if there plenty of elementary linear algebra texts out there who describe adj(A) as the adjoint of A. For a complex square matrix the genuine adjoint is denoted A^* and is the conjugate transpose. That is a special case of the adjoint of a linear operator on an inner product space (in the case of C^n with the standard inner product). However: the first ever occurrence of what I just said is really the adjugate is in Gauss's Disquisitiones Mathematicae in Art. 267 (page 293 of the modern Springer translation), where he calls it the adjoint! But this is in a very special case: symmetric 3x3 integer matrices, representing ternary quadratic forms, and Gauss does not use our standard matrix notation. So he is defining the adjoint of ternary quadratic forms rather than matrices. I support your proposal but it would have to be very well documented. We already have the problem, for those wanting to use Sage to teach linear algebra, that it uses the Magma convention where matrices act on the right on row vectors instead of the more common convention in textbooks of a left action on column vectors. This is potentially another situation like that... John On Thu, Dec 2, 2010 at 4:47 AM, Rob Beezer <goo...@beezer.cotse.net> wrote: > What does the "adjoint of a matrix" mean to you? > > I was brought up to understand it to mean the transpose of the matrix > of signed minors, a matrix close to being the inverse of the > original. Poking around (Wikipedia, Planet Math, Math World) would > imply this is known as the "classical adjoint." Hmmm, I'm not that > old. Anyway, it is also known now as the "adjugate matrix." > > It seems that the term "adjoint" is now more commonly used for the > conjugate-transpose of a matrix (and for vectors) and that's what I > find in most any relatively new textbook on matrix algebra. > > Presently, in Sage, "adjoint" gives the "classical" interpretation. I > would much prefer to define and implement the adjoint of a matrix to > be the conjugate transpose. So two questions: > > 1. Thoughts on what "adjoint" should be? > > 2. If adjoint were to be redefined to a more modern interpretation, > its current use could be aliased to "adjugate" and deprecated, but > that won't make it available for reuse until the deprecation period > runs its course. Any precedent, or techniques, for radically > redefining a method name? > > Rob > > -- > To post to this group, send an email to sage-devel@googlegroups.com > To unsubscribe from this group, send an email to > sage-devel+unsubscr...@googlegroups.com > For more options, visit this group at > http://groups.google.com/group/sage-devel > URL: http://www.sagemath.org > -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
