Transforming Matrix Addition to Multiplication without Exponentiation

[aliya]

Suppose two matrices A and B . i want to transform matrix addition to matrix multiplication. e.g A+B into A.B.
Can anybody please tell me of any way i can do it ,except exponentiation? exponentiation gives an infinite answer.

 

[HallsofIvy]

Suppose two matrices A and B . i want to transform matrix addition to matrix multiplication. e.g A+B into A.B.
Can anybody please tell me of any way i can do it ,except exponentiation? exponentiation gives an infinite answer.

It is certainly true that $e^{A+ B}= e^Ae^B$ but what do you mean by "exponentiation gives an infinite answer"?

 

[trambolin]

It is certainly true that $e^{A+ B}= e^Ae^B$ but what do you mean by "exponentiation gives an infinite answer"?

$e^{A+ B}= e^Ae^B$ if and only if $AB = BA$

 

[adaptation]

$e^{A+ B}= e^Ae^B$ if and only if $AB = BA$

Don't A and B also need to be n x n matices?

 

[aliya]

Thankyou all for your replies.
"By exponentiation gives an infinite answer" i meant that suppose two matrices A and B, their multiplication A.B gives a finite answer i.e another finite matrix. but e^A.e^B give an infinite answer. doesn't it? Also can you please tell if there exists any X such that A.B=X(e^A.e^B)?

 

[HallsofIvy]

Thankyou all for your replies.
"By exponentiation gives an infinite answer" i meant that suppose two matrices A and B, their multiplication A.B gives a finite answer i.e another finite matrix. but e^A.e^B give an infinite answer. doesn't it?

No, it doesn't. If A and B are n by n matrices, then so are $e^A$, $e^B$, and $e^Ae^B$.

Also can you please tell if there exists any X such that A.B=X(e^A.e^B)?

 

[aliya]

ok thanks but " can you please tell if there exists any X such that A.B=X(e^A.e^B)?"

 

[HallsofIvy]

?? Of course their is:
$$X= ABe^{-(A+ B)}$$

 

[trambolin]

I think he is trying to come up with a logarithm function. But that would be quite restrictive since the function wouldn't be distributive in general.