David Roundy and Henning Thielemann have been arguing about the nature
of vectors and matrices, in particular saying:
On Thu, Jun 30, 2005 at 02:20:16PM +0200, Henning Thielemann wrote:
On Thu, 30 Jun 2005, David Roundy wrote:
Matrices _and_ vectors! Because matrices represent operators on vectors
and it is certainly not sensible to support only the operators but not
the objects they act on ... Adding a vector type by a library that is
build on top of a matrix library seems to me like making the first step
after the second one.
No, matrices operate on matrices and return matrices. This is the
wonderful thing about matrix arithmetic, why it's unique, and why I'd like
to have a library that supports matrix arithemetic.
The really funny thing about that exchange is that you are *both right*
! You're just using different interpretations of the same objects.
1) Matrices represent linear operators which naturally act (via
application) on vectors
2) Matrices of compatible sizes, almost form a non-commutative graded
ring. It does not matter what a matrix represents here, this is true
purely algebraically. [to be a proper ring, the ``compatible sizes''
condition would need to be dropped]
There is a problem that the _types_ associated with both interpretations
are quite different. But if you want 2 different products on vectors
(inner and outer) represented as matrices, you have no choice -- the
``inner product'' will return a 1x1 matrix, not a real.
The same thing is true for differential operators, BTW. They can either
represent actions on spaces of smooth-enough functions, or represent
elements of a Weyl Algebra (or Ore Algebra if you really want to be
algebraic). You end up having the dichotomy of algebraic D-modules
versus analytic D-modules, where they share a number of theorems, but
the ``corner'' cases behave quite differently.
Jacques
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