[vox-tech] linear algebra: equivalent matrices

[Richard Harke]

On Wed December 7 2005 12:56, Peter Jay Salzman wrote:
> On Wed 07 Dec 05,  3:05 PM, Aaron A. King <aaron.king at umich.edu> said:
> > Not sure if I'm understanding your question aright, Peter, but I think
> > you're asking if the equivalence relation (1) is isomorphic to the
> > equivalence relation (2).  That is not the case.  If you view a matrix as
> > defining a parallelopiped in n-space, then the determinant measures the
> > volume thereof. Now it is easy to have two parallelopipeds which are not
> > congruent but which have the same volume.
> >
> > The relation (2) defines a set of equivalence classes which is much finer
> > than the set defined by (1).  To put it another way, the determinant is
> > only one of many matrix invariants.  The full set of invariants under the
> > relation (2) can be summarized in the Jordan form of the matrix.
> >
> > Apologies for pontificating, especially if I've answered a question you
> > never asked.
> >
> > Cheers (and congrats on the Ph.D.!),
> >
> > Aaron
>
> Don't apologize -- this is exactly what I wanted to know!
>
> OK, so then it's not true that all matrices with the same determinant are
> related by a rotation.  I was going to ask about trace too,
>
>    Tr[M_b]
>
>       = Tr[S^{-1} M_a S ]
>
>       = Tr[S] Tr[M_a^{-1}] Tr[S^{-1}]
>
>       = Tr[M_a^{-1}]
>
>       != Tr[M_a]
>
> but trace isn't invariant under inverse, so the trace relation can't be
> equivalent to relation (2) either.
>
> Thanks for chiming in!
> Pete

You might want to look at how Clifford algebras express linear algebra.
Normally one works with frame free expressions where invariance
under change of basis is inherent. Both the trace and determinat
are easily expressed this way. Also, one see that there are additional
invariants for a total of n in n dimensional space.
Also, eigen values are handled in a more natural way. One can have,
for example, an eigenbivector which expresses the invariance of a plane
while vectors within the plane are not eigenvectors. For a rotation
in three space the axis is invariant but also the plane of the rotation is
invariant even though no vector in the plane is.

I reccomend "Geometric Algebra for Physicists" by Chris Doran and
Anthony Lasenby (Cambridge) As the book is fairly expensive I also
mention the web site http://www.mrao.cam.ac.uk/~clifford
There are a large number of papers posted there for free download.

Richard Harke