1. ## Eigenvectors and eigenvalues.

Okay there's a question I've been working on all day.

First it said: If v is an eigenvector of A with corresponding eigenvalue λ, show that v is an eigenvectors of A^-1. What is its corresponding eigenvalue?

Then the second part of the question: Suppose a matrix A has real eigenvalues λ1 > λ2 > ..... > λn. Then was is lim k -> ∞A^kx0 for most initial vectors x0?

(The x0 is supposed to be "x knot" )

If anyone can help me with that second part I will appreciate it SO MUCH! Thanks

2. Originally Posted by nicolem1051
Okay there's a question I've been working on all day.

First it said: If v is an eigenvector of A with corresponding eigenvalue λ, show that v is an eigenvectors of A^-1. What is its corresponding eigenvalue?

Then the second part of the question: Suppose a matrix A has real eigenvalues λ1 > λ2 > ..... > λn. Then was is lim k -> ∞A^kx0 for most initial vectors x0?

(The x0 is supposed to be "x knot" )

If anyone can help me with that second part I will appreciate it SO MUCH! Thanks
By any chance, does $\displaystyle \lambda_1=1$?

3. No it doesn't say that it does. This question is just confusing to me!

4. This problem is dealing with Markov Chains.

$\displaystyle A^k=XD^{k}X^{-1}\mathbf{x}_0$

If $\displaystyle \lambda_1=1$, then $\displaystyle D^k=\begin{bmatrix} 1^k & \hdots & & & 0\\ \vdots & & & & \\ & & \ddots & & \\ & & & & \\ 0 & & & & \lambda_n^k \end{bmatrix}\rightarrow\begin{bmatrix} 1^{\infty} & \hdots & & & 0\\ \vdots & & & & \\ & & \ddots & & \\ & & & & \\ 0 & & & & \lambda_n^{\infty} \end{bmatrix}\rightarrow\begin{bmatrix} 1 & \hdots & & & 0\\ \vdots & & & & \\ & & \ddots & & \\ & & & & \\ 0 & & & & 0 \end{bmatrix}$.

Everything then would zero out except for the first time of $\displaystyle D^K$

Therefore, $\displaystyle A^k=\begin{bmatrix} \mathbf{x}_{1,1}*X^{-1}_{1,1}*D^k_{1,1}*X_{1,1}\\ \vdots\\ \\ \\ \mathbf{x}_{1,1}*X^{-1}_{1,1}*D^k_{1,1}*X_{n,1} \end{bmatrix}$ You would then have a nx1 matrix where the first term is the only surviving term.