
Matrix Transformations
Hello everyone
can anybody tell me if there exists any transformation between matrix addition to matrix multiplication (in any domain)e.g an addition in one domain for fourier is multiplication in the other. ??? i actually want to multiply a random matrix instead of adding it in my equation. what changes/effects do i have to make?
regards
aliya

Well, one option would be exponentiation. You can exponentiate diagonalizable matrices in a rather straightforward manner: if $\displaystyle A=PDP^{1}$, where $\displaystyle D$ is diagonal, then $\displaystyle e^{A}=Pe^{D}P^{1}$, and $\displaystyle e^{D}$ you can compute by simply exponentiating each number on the main diagonal. Because matrix multiplication is not, in general, commutative, you might also need to have some condition on the commutator in order actually to set $\displaystyle e^{A+B}=e^{A}e^{B}.$

thanx
Thankyou ackbeet for your reply. It indeed was useful to some extent. But it can make my solution complex as this method requires three assumptions:
1. matrix A and B be commutive
2. matrix A be diagonizable
3. matrix B be nilpotent
is there any solution which is more general than this? (i.e., without assumptions)

Well, taking a step back, what's preventing you from simply multiplying in your equation? Multiplication is not commutative, it is true. So you do have to be careful. But if you have control over the way your equations look, then it seems to me you can just change to multiplication by fiat. I guess what I'm getting at is that a little more context of the problem would be helpful.

Thankyou ackbeet . I think i am much clear on this now.

You're welcome. Good luck!

Hello,
i need a bit more help please. with what factor is e^A.e^B is different from A.B? does any close approximation exist?

I'm not sure what you're asking. Can you provide more context?

suppose two matrices A and B. Now simple matrix multiplication (i.e., A.B) is not equal to multiplication of exponent of matrices (i.e., e^A.e^B). right?
i was just curious if these two multiplications have any relationship. ? for example does there any quantity 'X' exist for which i can say A.B=X(e^A.e^B)? or vice versa.

I'm not aware of any X that will do that for you. On the RHS of that equation, you have two infinite series being multiplied together. On the LHS, you have simple matrix multiplication. There might conceivably be some sort of operation you could do, something like Fourier analysis, on the RHS in order to get you the LHS. But Fourier analysis on matrices is not something I've even seen. There is such a thing as the Fourier matrix (you can google it), but I've never studied it. I think you've definitely reached the end of my knowledge here, I'm afraid.