Re: calculating an algebraic structure from matrices

On Jul 23, 6:02 pm, Jack Schmidt <Jack.Schmidt.SciM...@xxxxxxxxx>
wrote:
Hey,

Given any finite set of matrices, there will be
be some unique algebra
generated under the free product (matrix
multiplication) of these
matrices. If we are given the matrices exactly,
either with real or
complex coefficients, how can we develop the theory
of the algebra
which they generate.
I am really interested the special cases that the
the matrices are
invertible and both nxn, and thus the algebra is a
group.
Any references will be greatly appreciated.

Ben

When the matrices are nxn invertible matrices, the algebra
they generate is called a group, or a linear group, or a
matrix group.

Some very simple questions about these groups are known to
have no algorithmic solution. There is no algorithm which
takes as input a finite set of n+1 4x4 matrices with
rational entries and outputs whether the n+1st matrix is
contained in the group generated by the first n.

If the group they generate is finite, then you can in
principal determine most things algorithmically, but
again there are some simple questions which there do
not exist efficient algorithms to answer.

In other words, you may need to know some specific
properties of your matrices (both individually and how
they interact), before you have much hope of understanding
the group they generate.

Books on abstract algebra, group theory, linear groups,
and representation theory of groups may be your best bet
for references. In particular, if you have an engineering
background, you may find group theory/representation
theory books written for chemists more legible.


Hi,

I think I am satisfied. I wanted to keep the question general and
so no specific properties can be assumed, I guess. I think that a
straightforward undecidability proof would be easy given the
constraint , or promise, that the matrices generate an infinite
group. I take the problem of finding all equivalence classes of that
group, given just the values of the elements, to be undecidable.
Perhaps it is equivalent to doing a word problem for an arbitrary
infinite group, which is known to be undecidable.
I suppose the only thing to consider is other possible
presentations of the matrices themselves. Perhaps, if I was "given"
the matrices in some special way, then the problem might not be
undecidable, even though no explicit algebraic relations are given.
Thank you for your help.

Ben

.

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