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**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