Re: [boost] Re: Boost Mathematicians

From: Christoph Ludwig (cludwig_at_[hidden])
Date: 2004-07-07 04:33:04

On Tue, Jul 06, 2004 at 10:24:49AM -0500, David B. Held wrote:
> "David Abrahams" <dave_at_[hidden]> wrote in message
> news:uoemxdu50.fsf_at_boost-consulting.com...
> > Walter Landry <wlandry_at_[hidden]> writes:
> >
> > > David Abrahams <dave_at_[hidden]> wrote:
> > >> * prod should be named operator*; this is a linear algebra library
> > >>   after all.
> > >
> > > Actually, operator* can be thought of as either an inner product or
> > > an outer product.  I'm not familiar with the innards of uBLAS, but
> > > that is a problem common to most linear algebra libraries.
> >
> > IIUC in the domain of linear algebra when mathemeticians  write
> >
> >    AB
> >
> > for 2D matrices A and B, they mean ordinary matrix multiplication (I
> > think inner product) yielding another 2D matrix, and that's what
> > operator* should mean in a linear algebra library.  If you want an
> > outer product yielding a 4D matrix I think you're basically outside
> > the domain of linear algebra.
> 
> Actually, taking a look at mathworld.wolfram.com and a few other
> sites, I have reached the tentative conclusion that outer products
> can only be formed over vectors, not matrices.  Matrix multiplication
> is well-defined and unambiguous, whereas vector multiplication
> comes in at least four varieties: dot product, cross product (also
> called the "outer product"), perpendicular dot product, and vector
> direct product.

You forgot the "Kronecker product" which is the tensor product
explicitly written down for matrices. Let A = (a_{i,j}) be an m by n
matrix and B a p by q matrix. Then the Kronecker product A x B is the
following block matrix 

          / a_{1,1}B  ...  a_{1,n}B \
          |     .              .    |
  A x B = |     .              .    |
          |     .              .    |
          \ a_{m,1}B  ...  a_{m,n}B /

The Kronecker product has its place in linear algebra, but I agree
that in a C++ library operator* should be reserved for the "common"
matrix product. 

Christoph
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