Etymology does not help much. To say that two things are "adjugate" means that they are "yoked or joined together" from the Latin word for a yoke (such as you would use if your ox cart was being pulled by two oxen and they needed to pull together). And to say that they are "adjoint" means the same thing, from the word to join.
It is a good point that the word adjoint is properly associated to a linear operator rather than a matrix. On C^n with standard Hermitian inner product and the standard basis to represent operators, this leads to the conjugate transpose. But if you use a random basis (not an orthonormal one) then even in this situation the adjoint operator is not represented by the conjugate transpose matrix). [This caught me out eons ago when I was computing Hecke operators on spaces of modular forms. [Actually I still am...] I knew they were self-adjoint (for the experts out there, these were Hecke operators for Gamma_0(N)) but the matrices were real but not symmetric, even though their eigenvalues were all real, as they do not, of course, depend on the basis used.] The wonderful website "Earliest Known Uses of Some of the Words of Mathematics" (see http://jeff560.tripod.com/a.html for the A page) does not mention adjugate unfortunately, but does have this entry: ADJOINT MATRIX. The OED quotes from M. Bôcher's Introduction to Higher Algebra of 1907: "By the adjoint A of a matrix a is understood another matrix of the same order in which the element in the ith row and jth column is the cofactor of the element in the jth row and ith column of a." (p. 77) That site also has an entry for "adjoint linear form", Cayley 1856, which I just spent some time looking up, but it does not seem relevant (despite coming just before the definitions of resultant and discriminant). If you like long words which never quite caught on, Cayley is the mathematician for you. (I am looking forward to the next time I have a polynomial of degree 12 since I now know to call it a dodecadic!) John On Tue, Dec 21, 2010 at 5:17 AM, Rob Beezer <goo...@beezer.cotse.net> wrote: > 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 > > -- > 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
