Find of : . I have found but I get stuck trying to find e-vectors and so on. Thanks.

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- April 20th 2009, 01:19 PMChief65Exponential Matrix
Find of : . I have found but I get stuck trying to find e-vectors and so on. Thanks.

- April 20th 2009, 01:38 PMMoo
- April 21st 2009, 06:57 PMChief65
- April 21st 2009, 11:53 PMMoo
Okay

Then, you have to find the vectors

Such that

Let's do it for (for , it's common techniques !)

That is :

From (2), you get

Substitute in (3) :

Now let for example a=1 (that's what's good with eigenvectors, you can choose arbitrarily one component)

So

And now compute

that's all (Nod)

Similar one for

For , it should be easy, because all the coefficients are real.

Once you get the 3 eigenvectors, place them in columns to form matrix such that

and where

(the order of the eigenvalues has to be the same as the order of the eigenvectors in P) - May 2nd 2009, 09:19 PMseadog
Just want to confirm my thought process on backsolving for A given an exponential matrix .

A matrix of sums is also a sum of matrices, and vice versa. Right? For example,

would it go like this?

Once I find my , just plug in n=1 to get A. Right? - May 2nd 2009, 10:56 PMMoo
Nope.

The n-th power of a matrix is not the matrix formed by the elements to the n-th power ~

Try

Do you have ?

You will see that it's not the case.

You don't need to find A, you need to find an expression for

And by finding its eigenvalues and the transition matrix, P, you'll have :

And hence

But as I said above, the n-th power of a diagonal matrix is a matrix formed by the n-th power of its diagonal elements.

So finally, you have :

Does it look clear ?