# Thread: Alternate Formula for Matrix Exponential

1. ## Alternate Formula for Matrix Exponential

Trying to solve a proof for Linear Algebra:

for a in the set of complex numbers, e^a = lim k-> ∞ (1 + a/k)^k

Establish the matrix analog for any A in the set of real nxn square matrices

e^A = lim k->∞ (I + A/k)^k

I started off by trying the following:

A = P D P^-1

A^k = P D^k P^-1

e^At = P e^D^kt P^-1

not sure where to go from here or if this is the right thought process, I would appreciate any help I can get!

Thanks

2. Here is a link which better describes what I am trying to prove:

The Matrix Exponential as a Limit of Powers

if anyone could help me get further I would really appreciate it!

3. Originally Posted by Demagodnewb
Trying to solve a proof for Linear Algebra:

for a in the set of complex numbers, e^a = lim k-> ∞ (1 + a/k)^k

Establish the matrix analog for any A in the set of real nxn square matrices

e^A = lim k->∞ (I + A/k)^k

I started off by trying the following:

A = P D P^-1
You appear to be assuming here that A is "diagonalizable" which you are not given.

A^k = P D^k P^-1

e^At = P e^D^kt P^-1

not sure where to go from here or if this is the right thought process, I would appreciate any help I can get!

Thanks

4. Shoot forgot to write that down. Sorry the problem does hint to solve by diagonalize, and that the matrix can be assumed to be diagonalizeable

Thanks