Recursive eigen extrusion: Expanding eigenbasis conjecture

Abstract

Consider $n$ linearly independent vectors in ${\mathbb{C}}^n$ which form columns of a matrix $A$. The recursive evaluation of eigen directions (normalized eigenvectors) of $A$ is the solution of an eigenvalue problem of the form $A_i{X}_i=X_i{\Lambda }_i$ with $i=0,1,2\dots$; and here ${\Lambda }_i$ is the diagonal matrix of eigenvalues and columns of $X_i$ are the eigenvectors. Note that $A_{i+1}=\varphi (X_i)$ where $\varphi$ normalizes all eigenvectors to unit ${\mathcal{L}}_2$ norm such that all diagonal elements $[\varphi (X{)}^{†}\varphi (X){]}_{jj}=1$. It is to be proven that for any matrix $A_o$ and $n\le 7$, the limiting set of matrices $A_i$ with $i\to \infty$ is the set of unitary matrices $U(n)$ with $X_i^{†}{X}_i\to I$. Interestingly, this problem also represents a recursive map that maximizes some average distance among a set of $n$ points on the unit $n$-sphere. We first formally pose this conjecture, present extensive numerical results highlighting it, and prove it for special cases.