# Eigenvectors of recursive set of similarity transformations

• Oct 27th 2009, 01:57 AM
Jodles
Eigenvectors of recursive set of similarity transformations
Hi!

I need some help with the following:
Consider the sequence
$\displaystyle A_{1} = A \\ A_{2} = P ^{-1}_{1} A_{1} P_{1} \\ A_{3} = P ^{-1}_{2} A_{2} P_{2} \\ . \\ . \\ . ... \\ A_{k+1} = P ^{-1}_{k} A_{k} P_{k} .$

1) Write an expression for the eigenvectors of $\displaystyle A_{k+1}$ using the eigenvectors of $\displaystyle A_{k}$. Then use this to write an expression for the eigenvectors of $\displaystyle A_{k}$ using the matrices $\displaystyle P_{1},..., P_{k-1}$, and the eigenvectors of $\displaystyle A_{1} = A$.

I am completely lost and have no idea where to start. Any pointing me in the right direction would be greatly appreciated!

j

edit: I can't get line breaks to work in the above code...(?)
• Oct 27th 2009, 03:37 AM
tonio
Quote:

Originally Posted by Jodles
Hi!

I need some help with the following:
Consider the sequence
$\displaystyle A_{1} = A \\ A_{2} = P ^{-1}_{1} A_{1} P_{1} \\ A_{3} = P ^{-1}_{2} A_{2} P_{2} \\ . \\ . \\ . ... \\ A_{k+1} = P ^{-1}_{k} A_{k} P_{k} .$

1) Write an expression for the eigenvectors of $\displaystyle A_{k+1}$ using the eigenvectors of $\displaystyle A_{k}$. Then use this to write an expression for the eigenvectors of $\displaystyle A_{k}$ using the matrices $\displaystyle P_{1},..., P_{k-1}$, and the eigenvectors of $\displaystyle A_{1} = A$.

I am completely lost and have no idea where to start. Any pointing me in the right direction would be greatly appreciated!

j

edit: I can't get line breaks to work in the above code...(?)

$\displaystyle \mbox {If v is an eigenvector of } A_k \mbox{ with eigenvalue }\lambda \,,\,\, then\; A_kv = \lambda v$. $\displaystyle \mbox { As P is invertible } \exists u\in V\; s.t.\; Pu=v$, and so:

$\displaystyle A_{k+1}(u)=P_k^{-1}A_kP_k(u)=P_k^{-1}A_k(v)=P_k^{-1}(\lambda v)=\lambda P_k^{-1}v=\lambda u$

Try to take it from here now.

Tonio
• Oct 29th 2009, 04:47 AM
Jodles
Tonio, thank you! You saved my day!

Here's my take on the second part ("Write an expression for the eigenvectors of $\displaystyle A_{k}$ using the matrices $\displaystyle P_{1},..., P_{k-1}$, and the eigenvectors of $\displaystyle A_{1} = A$).

(I'm using $\displaystyle v_k$ and $\displaystyle v_{k+1}$ instead of $\displaystyle v$ and $\displaystyle u$, respectively.)

From the first part we now have: $\displaystyle v_{k+1} = P_k^{-1} v_k$. Obviously $\displaystyle v_k$ would be $\displaystyle v_k = P_{k-1}^{-1} v_{k-1}$ (1) ,
and further:
$\displaystyle v_{k-1} = P_{k-2}^{-1} v_{k-2}$
$\displaystyle v_{k-2} = P_{k-3}^{-1} v_{k-3}$
... and so on...

Inserted into (1), we get:
$\displaystyle v_k = P_{k-1}^{-1} P_{k-2}^{-1}P_{k-3}^{-1} v_{k-3}$

Thus a pattern evolves:
$\displaystyle v_k = P_{k-1}^{-1} \cdot P_{k-2}^{-1} \cdot ... \cdot P_{1}^{-1} v_{1}$ (2)
where $\displaystyle v_1$ is the eigenvector of $\displaystyle A_1 = A$, i.e. $\displaystyle Av_1 = \lambda v_1$

Does that look somewhat alright? Something I've missed? Or better ways to write it? (especially (2))...

J
• Oct 29th 2009, 05:08 AM
tonio
Quote:

Originally Posted by Jodles
Tonio, thank you! You saved my day!

Here's my take on the second part ("Write an expression for the eigenvectors of $\displaystyle A_{k}$ using the matrices $\displaystyle P_{1},..., P_{k-1}$, and the eigenvectors of $\displaystyle A_{1} = A$).

(I'm using $\displaystyle v_k$ and $\displaystyle v_{k+1}$ instead of $\displaystyle v$ and $\displaystyle u$, respectively.)

From the first part we now have: $\displaystyle v_{k+1} = P_k^{-1} v_k$. Obviously $\displaystyle v_k$ would be $\displaystyle v_k = P_{k-1}^{-1} v_{k-1}$ (1) ,
and further:
$\displaystyle v_{k-1} = P_{k-2}^{-1} v_{k-2}$
$\displaystyle v_{k-2} = P_{k-3}^{-1} v_{k-3}$
... and so on...

Inserted into (1), we get:
$\displaystyle v_k = P_{k-1}^{-1} P_{k-2}^{-1}P_{k-3}^{-1} v_{k-3}$

Thus a pattern evolves:
$\displaystyle v_k = P_{k-1}^{-1} \cdot P_{k-2}^{-1} \cdot ... \cdot P_{1}^{-1} v_{1}$ (2)
where $\displaystyle v_1$ is the eigenvector of $\displaystyle A_1 = A$, i.e. $\displaystyle Av_1 = \lambda v_1$

Does that look somewhat alright? Something I've missed? Or better ways to write it? (especially (2))...

J

Looks fine to me though all those indexes get me dizzy. I'd do a specific example with k= 2 or 3 and see if we get the correct pattern.

Tonio
• Oct 29th 2009, 05:11 AM
Jodles
Hehe, alright! Thanks again!

All the best,
Joachim