Finding Limit of Matrix A^n with n→∞

[bamboula100]

Limit of a Matrix??

Hello,

I have a 3x3 Matrix A.

I was asked to find A^n.

I found it using eigenvalues/Eigenvectors
and the fact that

A^n=P.B^n.(P^-1) with B the diagonal matrix and P the basis of the eigenvectors.

Then The question is

Find the Limit of

1/c^n. A^n for n goes to +inf for all c>0 (if it exists)

Basicallt how do we generally find this limit? I have

1/c^n. A^n => 3x3 Matrix

Sometimes Row1Col1 goes to +inf
Row1Col2 goes to - Inf

:/

Help Please :)

 

[Tedjn]

There is no way we can help if we don't know what A is, and what you found A^n to be.

 

[arkajad]

Can you reduce this problem to finding the answer for your B-matrix?

If so, order its eigenvalues according to their values.

When lambda^n/c^n has a finite limit? And what this limit can be? When x^n has a limit? And what this limit can only be?

 

[Tedjn]

Take arkajad's advice and see what happens when c is less than all eigenvalues, equal to some eigenvalue, or in between two eigenvalues. See what your center matrix tends to. See if you can work from there.

 

[stevekho]

Basically I should only work my my diagonal matrix B?
But how can we estimate the limit of a matrix?

For instance, if c<3,
3^n/c^n will go to infinity when n goes to inf (like 6 and 9)

For 6>c>3
(3/c)^n goes to 0
(6/c)^n goes to inf
(9/c)^n goes to inf

How from that I can find a limit for all the matrix? :/ I just found here a limit for each space.

Moreover, P and P^-1 should influence the sign no?

 

[Tedjn]

Then maybe the limit will exist only for some c. And maybe the limit is different for different c. This is all very possible. P and P^{-1} could influence the result; you may want to multiply out.

 

[stevekho]

So you think I should multiply and then if I find for instance
That row1col1 goes to + inf and row2col1 goes to -inf then there is no limit?

 

[Tedjn]

I don't have the time to do this question out myself, but if you find that some entries tend to infinity for certain values of c, then for those values of c, there is no matrix limit (since we are likely only considering matrices with finite entries).

 

[arkajad]

Notice that c_n.P.B^n.P^-1 = P.c_n.B^n.P^-1. Therefore c_nA^n has a limit if and only if c_n.B^n has a limit.

A sequence of matrices is said to have a limit if for each of the matrix elements of this sequence has a limit.

 

[bamboula100]

Greaaaaat :)
I got it
thanks guys

 

[adricu]

Hi Guys,

I have to solve the same problem but I don't get which of your answers is the correct one. If someone has the solution can you please post it and explain as it would be incredibly helpful. Thanks