Raising a bunch of matrices to a power

[sjeddie]

Let A, B, and C be nxn matrices,
I'm wondering
1. is it possible to simplify (ABC)^5 to expand it ( maybe into something like (A^5)(B^5)(C^5) )
2. what's the fastest way of solving (ABC)^5? I'm thinking actually multiply ABC out, then diagonalize it. Is there a faster way?

 

[CompuChip]

Let A, B, and C be nxn matrices,
I'm wondering
1. is it possible to simplify (ABC)^5 to expand it ( maybe into something like (A^5)(B^5)(C^5) )

Under certain conditions. If A, B and C commute among themselves (that is, AB = BA, AC = CA and BC = CB) then this is possible.

2. what's the fastest way of solving (ABC)^5? I'm thinking actually multiply ABC out, then diagonalize it. Is there a faster way?

I'm thinking the same thing. Alternatively you could try writing down an expression in terms of A^5, B^5, C^5 and the commutators [A, B] = AB - BA, [A, C] = AC - CA and [B, C] = BC - CB, e.g.
(A B C)^2 = A B C A B C
... = A B A C B C + A B [C, A] B C
... = A^2 B C B C + A [B, A] C B C + A B [C, A] B C
... = A^2 B^2 C^2 - A^2 B [B, C] C - A [A, B] C B C - A B [A, C] B C

But that is in general ugly and not very helpful.

 

[sjeddie]

Thank you CompuChip