Question

can someone explain the geometric implications of taking the exponential of a matrix please?

Answer 1

it was introduced to me in a problem a few days ago

Answer 2

Suppose your matrix translates to some sort of linear action on R^2 (or similar). The exponential is the same thing as applying the transformation twice.

Answer 3

or thrice. Or four times.

Answer 4

sorry i mean exp(A) not A^n

Answer 5

nevermind :(

Answer 6

\[\exp(A) =\sum{\frac{1}{r!} A^r}\]

Answer 7

how do i type infinity in the equation thing?

Answer 8

\infty

Answer 9

.... how odd

if i were to take a blind stab at this:

e^A = B

A = ln(B)

.... and thats as far as I get on that idea lol

Answer 10

just using the trylor expansion for e

Answer 11

*\(e^x\)

Answer 12

i'll find the question, i did it fine, i just want extra explanation

Answer 13

A^r = P-1 D P right? if its diag.able

Answer 14

D^r that is

Answer 15

sorry, im answering a question i wont be long :)

Answer 16

i solved it, i was just interested in getting more info on exponentials of matrices as this is the first i have encountered

Answer 17

Square matrices?

Answer 18

It seems like you might want to consider your matrix \(M\) as an equivalent to \(i=\sqrt{-1}\) and from there draw some parallels to the complex plane.

Answer 19

yes. i am also interested in what happens if its a general matrix M instead of that specific matrix

Answer 20

I honestly don't know how you could visualize exp(A) with a generalized matrix matrix of size nxn. Unless \(n=1\). Then it's just plain old \(e^a\) for some \(a\).

Answer 21

The matrix exponential for square matrices has a lot of relevance in rotation theory..

Answer 22

http://en.wikipedia.org/wiki/Matrix_exponential

Answer 23

this maths is so much more exciting than school work!