On Thu, Dec 2, 2010 at 7:16 AM, John Cremona <john.crem...@gmail.com> wrote:
> 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.
Do you have a reference for this convention? I had never seen the word
"adjugate" before.
--
Something interesting:
if T : V --> V is an operator, it naturally induces a graded operator
T* in the exterior algebra of V. If V is of finite dimension n, there
are two pieces of the exterior algebra which are isomorphic to V,
namely
1. Lambda_1(V) = the dual space of V
2. Lambda_{n-1}(V) = the space of (n-1)-multivectors of V
Then the restriction of T* to Lambda_1(V) is the adjoint (or dual)
operator, but the dual space is not canonically isomorphic to V. Given
a duality (inner product) then you get the adjoint operator acting in
V. The matrix depends on the choice of basis.
On the other hand, Lambda_{n-1}(V) is naturally isomorphic to V (the
isomorphism is given by the determinant), and
so the restriction of T* to Lambda_{n-1}(V) naturally induces an
operator on V, which is precisely what I would call "adj T". Since
this is natural, an adjoint operation for matrices can be defined in
terms of this.
In summary, both notions of adjointness are quite related. The simple
one is just duality, and makes more sense for operators in inner
product spaces. The seemingly not-so-simple one is natural, makes
sense for operators or matrices, is usually expressed in terms of
matrices, and appears often in relation to quadratic forms.
> 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.
+1 to Gauss :-)
[he reduces ternary quadratic forms by alternating a binary reduction
step on the form itself and a binary reduction step on the adjoint, so
he may have invented adjoints for this purpose]
Gonzalo
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