I am trying to understand if there is some intuition to the idea of eigen vectors of a normalized eigen vector matrix of a matrix itself. To elaborate, let A be a n x n square matrix, we know that A can be decomposed to:
$ A = S \Lambda S^{-1} $ where $\Lambda$ is the diagonal matrix with eigenvalues as its entries and S is the normalized eigenvector matrix, with eigen vectors as its columns. Although these eigen vectors are not unique, but they define a direction, eigenspace for each eigenvalue.
Moreover, is there some pattern/intuition I can expect when I recursively start looking at eigenvectors of the eigenvector matrix itself. How does the eigenvector matrix (or eigen space) traverse the space $\mathbb{R}^{n}$ as we recursively setup such a matrices.
I tried looking at a simple case below:
Let A = $\begin{bmatrix}0 & 1 \\1 & 0\end{bmatrix}$ and look at the eigen vector decomposition $ A = S \Lambda S^{-1} $below:
Here S = $\begin{bmatrix}1 & 1 \\1 & -1\end{bmatrix} \text{ and } S^{-1} = \begin{bmatrix}\frac{1}2 & \frac{1}2 \\\frac{1}2 & \frac{-1}2\end{bmatrix} \text{ and }\Lambda = \begin{bmatrix}1 & 0 \\0 & -1\end{bmatrix}$
Now if I look at eigen decomposition (without normalization) of eigenvector matrix $ S = P \Omega P^{-1}$, we get:
P = $\begin{bmatrix}1+\sqrt{2} & 1-\sqrt{2} \\1 & 1\end{bmatrix} \text{ and } \Omega = \begin{bmatrix}\sqrt{2} & 0 \\0 & -\sqrt{2}\end{bmatrix}$ and so on...
Is there any connection between eigenvectors of A and eigenvectors of S or any interesting behavior as we repeatedly look at such eigenvectors recursively? (Although both seem to be orthogonal to each other, but when I do one level deep, that orthogonality disappears).
Is there any such thing or I am just looking at something random?
Clearly Identity matrix has all vectors as eigenvectors and hence we can create any random matrix as eigenvector matrix, but that might be some special case. What kind of path is traversed by EigenVector matrices(think as vectors). Do they converge?, may be in some cases? etc.
I hope the question is not vague. I tried searching online but could not find anything.
Adding some modifications based on the feedback from @Arthur in the comment section:
Since the Eigenvectors are not unique, especially when the multiplicity of eigenvalues is more than 1, because we could chose infinite set of eigen vectors for an eigen matrix. Although there might be some patterns here, but let us restrict ourselves to matrix A with real and distinct eigen values, which ensures that we get unique unit eigen vectors (directions). Now can we claim to have some patterns in the set of eigen values and eigen vectors as we look recursively at the normalized eigen vector matrices.