exp(At)

[Maddox]

Evaluate e^At
Where A is an upper triangular matrix given by
1 2 3
0 1 2
0 0 1

What is the answer?  Determinent is obviously 1...

I just had that question on my diff eqns test I had no idea how to do it. I'm not sure if he talked about it in class since I don't go. 

I took linear algebra about a year ago so I've forgotten a lot of stuff.

[Yoni]

http://mathworld.wolfram.com/MatrixExponential.html

[Maddox]

http://mathworld.wolfram.com/MatrixExponential.html

ok... I think this is how you do it.

Code Select Expand


A is equal to
[ 0 2 3 ]    [ 1 0 0 ]
[ 0 0 2 ]  + [ 0 1 0 ]
[ 0 0 0 ]    [ 0 0 1 ]

Let B be the matrix on the left.
Then exp(At) is equal to exp(t(B+I)) where I is the identity matrix.
This then gives us exp(tB)exp(tI)


B is a nilpotent matrix, but I don't know how to get the matrix polynomial.  I think someone said it has to do with taylor polynomials.

Oh well...

[Yoni]

Your evaluation is not correct. exp(A+B) != exp(A) + exp(B).

[Maddox]

Your evaluation is not correct. exp(A+B) != exp(A) + exp(B).

err, stupid mistake.   

fixed.

[Maddox]

Argh, I studied this for the final and he didn't put it on there.