# Matrix Theory problem

1. Mar 2, 2012

### arpitm08

1. The problem statement, all variables and given/known data
If B≈A with (P^-1)*A*P = B and also (Q^-1)*A*Q = B, show that Q=RP where R is a nonsingular matrix that commutes with A.

2. Relevant equations
AR = RA

3. The attempt at a solution
Our professor told us that it is easier to just play around with these equations and get the answer. That breaking them down to their elements would be a ton of work. I tried multiplying by a bunch of various ways. First I changed the equivalence formulas to, AP=PB and AQ=QB, and I tried multiplying by R, because of the commuting factor, but I couldn't get anywhere doing that. If someone could give me a hint as to if I'm missing a property or something that could help me out, that'd be great. Thanks.

2. Mar 2, 2012

### tiny-tim

hi arpitm08!

hint: what happens if you multiply either equation by Q ?

3. Mar 2, 2012

### Deveno

i have an alternate approach:

note that B = P-1AP is the same as: A = PBP-1.

also, note that if R-1AR = A, then AR = RA.

what happens if you evaluate (QP-1)-1A(QP-1)?

4. Mar 2, 2012

### tiny-tim

isn't that the same as mine, except using P instead of Q ?

5. Mar 2, 2012