The discussion revolves around the transformation of matrix addition into matrix multiplication, specifically exploring whether it is possible to express the sum of two matrices A and B as a product A.B without resorting to exponentiation. Participants examine the implications of matrix exponentiation and the conditions under which certain identities hold.
Participants express differing views on the nature of matrix exponentiation and its implications, particularly regarding the finiteness of results. The discussion includes both agreement on certain mathematical identities and disagreement on the interpretation of those identities and their applications.
Participants highlight the need for matrices A and B to be of the same dimensions (n x n) for the discussed properties to hold, and there is an acknowledgment of the complexities involved in transforming addition into multiplication.
aliya said: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 said:It is certainly true that [itex]e^{A+ B}= e^Ae^B[/itex] but what do you mean by "exponentiation gives an infinite answer"?
trambolin said:[itex]e^{A+ B}= e^Ae^B[/itex] if and only if [itex]AB = BA[/itex]
No, it doesn't. If A and B are n by n matrices, then so are [itex]e^A[/itex], [itex]e^B[/itex], and [itex]e^Ae^B[/itex].aliya said: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)?