Groups and orthogonal matrices question

[vr88]

Let A and B be nxn matrices which generate a group under matrix multiplication. Assume A and B are not orthogonal. How can I determine an nxn matrix X such that X^-1AX and X^-1BX are both orthogonal matrices? Is it possible to do this without any special knowledge of the group in question?

 

[Bacle]

If I understand you well, matrices M,M' are orthogonal if MM'=0 , where 0 is the

multiplicative 0 matrix. Then you need to check that the product of your matrices

is the zero matrix in your group. So at least to that extent (knowing the

0 element in your group, and the multiplication table to verify the product

equals the zero element) , you need to know your group.

Is that what you meant?

 

[vr88]

I mean that a matrix M is orthogonal if M^-1=M^t.

 

[Stephen Tashi]

I don't know the answer to your question. My interpretation of what you are asking is: "Can I set up a bunch of simultaneous equations and solve for a matrix [itex] X [/tex] so that conjugation by $X$ will map two given matrices to orthogonal matrices? - or must I resort to group theory?".

I suspect that if it can be done by group theory, there would be a way to do it with simultaneous equations of some sort. However, it might be simpler to take some hints from group theory.

The mapping defined by $$f(Y) = X^{-1} Y X$$ is an isomorphism from the group $$G$$ generated by $$A$$ and $$B$$ to the group W generated by $$P = X^{-1} A X$$ and $$Q = X^{-1} B X$$, which is a group of orthogonal matrices.

The groups of n by n orthogonal matrices have been much studied and so have their subgroups. W must be one of these subgroups. It must be finitely generated. That narrows down the list of possible "targets" for the mapping $$f$$. If the group G happened to be finite (in addition to being finitely generated) that would greatly simplify matters. Finite groups are isomorphic to permuation groups. Are the matrices for permuation groups orthogonal matrices?

 

[chiro]

I mean that a matrix M is orthogonal if M^-1=M^t.

Yes it is:

http://en.wikipedia.org/wiki/Orthogonal_matrix

 

[vr88]

Stephen- That's essentially what I'm looking for. I am more interested in the case of finite groups. I don't know how much permutations help, though the permutation matrices are orthogonal. For example, if G=A5 is the alternating group on 5 points, and I have two 3x3 matrices which generate this group. I'm not sure how to use permutations to find two orthogonal 3x3 matrices which generate an isomorphic copy of this group. Though I know it is possible, since the symmetry group of the icosahedron is A5.

 

[Stephen Tashi]

It isn't clear to me yet whether you are particularly interested in the simultaneous equations approach vs the group theory approach. It also isn't clear whether you want something like a general purpose computer algebra program to solve this problem for any given A and B or whether you only care about one particular pair of matrices.

The name for the study of representing groups as matrices (or more generally as transformations on perhaps infinite dimensional vectors spaces) is the "theory of group representations". I found one paper dealing with the representation theory of A5. http://www.math.toronto.edu/murnaghan/courses/mat445/mwesslen.pdf (See page 14)

A nice Wiki devoted to groups is: http://groupprops.subwiki.org/wiki/Main_Page

 

[vr88]

I'm more interested in a general solution with a linear algebra or a simultaneous equations approach in the hopes that such an approach will work for whatever initial matrices I start with.