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Recursive eigen extrusion: Expanding eigenbasis conjecture

M Hariprasad

TL;DR

The paper proposes and studies a recursive eigen-direction map, where each step solves $A_iX_i=X_i\Lambda_i$ and then forms $A_{i+1}=\phi(X_i)$ with unit-norm eigenvectors, and conjectures that for almost all $A_0$ and $n\le7$ this process yields $X_i^{\dagger}X_i\to I$ and a limiting unitary $A_i$. It connects this dynamics to a distance-maximizing map on the unit sphere and provides extensive numerical evidence across random matrices, plus proofs for several special-structure cases (notably upper-triangular and a special 2×2 real matrix). The results identify a dimension threshold near $n=7$, with convergence observed below this bound, instability at $n=7$, and no convergence at $n=8$ in the tested scenarios, while also noting possible loops and discontinuities in the recursion. The work advances understanding of the geometric and spectral properties of iterative eigen-directions and suggests a deep link between unitary limits and sphere-packings-like optimization in low to moderate dimensions.

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 $A_iX_i=X_iΛ_i$ with $i=0,1,2 \dots$; and here $Λ_i$ is the diagonal matrix of eigenvalues and columns of $X_i$ are the eigenvectors. Note that $A_{i+1}=φ(X_i)$ where $φ$ normalizes all eigenvectors to unit $\mathcal{L}_2$ norm such that all diagonal elements $[φ(X)^\daggerφ(X)]_{jj}=1$. It is to be proven that for any matrix $A_o$ and $n \leq 7$, the limiting set of matrices $A_i$ with $i \to \infty$ is the set of unitary matrices $U(n)$ with $X_i^\dagger X_i \to I$. 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.

Recursive eigen extrusion: Expanding eigenbasis conjecture

TL;DR

The paper proposes and studies a recursive eigen-direction map, where each step solves and then forms with unit-norm eigenvectors, and conjectures that for almost all and this process yields and a limiting unitary . It connects this dynamics to a distance-maximizing map on the unit sphere and provides extensive numerical evidence across random matrices, plus proofs for several special-structure cases (notably upper-triangular and a special 2×2 real matrix). The results identify a dimension threshold near , with convergence observed below this bound, instability at , and no convergence at in the tested scenarios, while also noting possible loops and discontinuities in the recursion. The work advances understanding of the geometric and spectral properties of iterative eigen-directions and suggests a deep link between unitary limits and sphere-packings-like optimization in low to moderate dimensions.

Abstract

Consider linearly independent vectors in which form columns of a matrix . The recursive evaluation of eigen directions (normalized eigenvectors) of is the solution of an eigenvalue problem of the form with ; and here is the diagonal matrix of eigenvalues and columns of are the eigenvectors. Note that where normalizes all eigenvectors to unit norm such that all diagonal elements . It is to be proven that for any matrix and , the limiting set of matrices with is the set of unitary matrices with . Interestingly, this problem also represents a recursive map that maximizes some average distance among a set of points on the unit -sphere. We first formally pose this conjecture, present extensive numerical results highlighting it, and prove it for special cases.

Paper Structure

This paper contains 7 sections, 5 theorems, 30 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Numerical observations and a conjecture
  3. Proofs: Special cases
  4. Upper triangular matrices
  5. Loops and discontinuities
  6. $3 \times 3$ upper triangular matrices
  7. A special $2 \times 2$ matrix

Key Result

Theorem 3.1

For $2 \times 2$ upper triangular matrix of the form if $\theta_1 < 0$ (restricting to fourth quadrant) maintained for all the eigenvector matrices $X_i$, then

Figures (3)

  • Figure 1: Uniform $\mathcal{U}(0,1)$ random matrices used for verification of the conjecture. Convergence of $\det(X_i^{\dagger}X_i)$ is observed for dimension less than 7, instability is observed at $n$ = 7, and no convergence is observed for a dimension $n$ = 8.
  • Figure 2: Convergence: Gaussian $\mathcal{N}(0,1)$ are used random matrices for verification of the conjecture. Convergence is observed for dimensions less than 7, instability is observed at $n$ = 7, and no convergence is observed for a dimension $n$ = 8.
  • Figure 3: Average of $\log(-\log(\det(u^{\dagger}u))$ over 1000 Gaussian $\mathcal{N}(0,1)$ random matrices plotted with iteration number. Dotted line represent $y = -\frac{x}{2^t}$ curves for $t = 0,1,2, \cdots 7$.

Theorems & Definitions (12)

  • conjecture 1
  • conjecture 2
  • conjecture 3
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Corollary 3.3.1
  • ...and 2 more