Exponent of polynomials (of matrices)

Question

$A$ is a matrix over $\mathbb R$ (reals).

Prove that for every $f,g\in \mathbb R[x]$, $\displaystyle e^{f(A)}\times e^{g(A)} = e^{f(A)+g(A)}$

I tried using the sigma writing but got stuck (I wrote $f(x)=\sum a_ix^i$, $g(x)=\sum bix^i$ and then started to develop the left part of the equation and then I got stuck ..)

Any suggestions ? thanks guys

score 1 · Answer 1 · answered Jun 14 '14 at 14:35

1

You should know that for square matrices $A,B$ the following holds:

$AB=BA \implies e^Ae^B=e^{A+B}$.

Can you prove that if $P,Q$ are polynomials and $A$ a square matrix, $P(A)Q(A)=Q(A)P(A)$ ?

answered Jun 14 '14 at 14:35

Gabriel Romon

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I was in the process of saying pretty much the same in a comment and I had this link ready for the first part, so here it goes: $AB=BA \implies e^Ae^B=e^{A+B}$ (random text to open in new tab). – Git Gud Jun 14 '14 at 14:36

Exponent of polynomials (of matrices)

1 Answers1