Re: When do matrices B and C commute?
- From: Mariano Suárez-Alvarez <mariano.suarezalvarez@xxxxxxxxx>
- Date: Wed, 19 Nov 2008 12:43:50 -0800 (PST)
On Nov 19, 5:55 pm, amy666 <tommy1...@xxxxxxxxxxx> wrote:
alain wrote :
I think about a needed parenthood between both the
matrices.
Ex1: if B = A^ n , C = A^ p , n, p integer
numbers
B*C = C*B = A ^ (n+p)
Ex2: more generally B = p1(A) ,C = p2(A) p1 and
p2 polynomials
then B*C = C*B = p3(A) ; p3(u) = p2(u)*p1(u) =
p1(u)*p2(u) ,
I've built a case B =
{[18,27,36],[48,66,84],[28,40,52]}
C = {[5,10,15], [20,25,30]
= {[5,10,15], [20,25,30] ,[10,15,20]}
There is probably an extension to some kinds of
function f1,f2
B = f1(A) , C = f2(A)
Your comments will be welcomed,
Alain
i believe matrix A and B commute if
- assuming they have abs different from 1 or 0 -
What is "abs"?
- assuming both are square matrices -
There is no possible way two non-square matrices
can commute, for in that case the two products
have different sizes...
if A^C = B for some square matrix C
where C satisfies C D = D C for any square matrix D.
The only matrices C which commute with all other matrices
are the scalar matrices, that is, the scalar multiples
of the identity. In that case, and assuming that
by A^C you mean the conjugation of A by C, one
trivially has A^C = A.
- assuming all matrices are of the same size of course -
If they are not, the question does not make sense.
-- m
.
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