Eigenvectors of recursive set of similarity transformations

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October 27th 2009, 01:57 AM
Jodles

Eigenvectors of recursive set of similarity transformations

Hi!

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

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

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...(?)
October 27th 2009, 03:37 AM
tonio

Quote:

Originally Posted by Jodles

Hi!

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

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

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...(?)

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

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

Try to take it from here now.

Tonio
October 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 $A_{k}$ using the matrices $P_{1},..., P_{k-1}$ , and the eigenvectors of $A_{1} = A$ ).

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

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

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

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

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

J
October 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 $A_{k}$ using the matrices $P_{1},..., P_{k-1}$ , and the eigenvectors of $A_{1} = A$ ).

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

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

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

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

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
October 29th 2009, 05:11 AM
Jodles

Hehe, alright! Thanks again!

All the best,
Joachim