Recursive eigen extrusion: Expanding eigenbasis conjecture

TL;DR

This paper investigates a recursive eigenvector normalization process, conjecturing that for matrices of size up to 7, the process converges to unitary matrices, with implications for eigenbasis expansion and point distribution on spheres.

Contribution

It formally states and numerically explores a conjecture about the convergence of a recursive eigenvector normalization process to unitary matrices, providing proofs for special cases.

Findings

Numerical results support the conjecture for matrices up to size 7.

The process converges to unitary matrices with orthonormal eigenvectors.

The problem relates to maximizing average distances among points on a sphere.

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 $A_iX_i=X_i\Lambda_i$ 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 $A_{i+1}=\phi(X_i)$ where $\phi$ normalizes all eigenvectors to unit $\mathcal{L}_2$ norm such that all diagonal elements $[\phi(X)^\dagger\phi(X)]_{jj}=1$. 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.

Equations

$$[u,v]=eig(A);$$

$$fori=1:N$$

$$[u,v]=eig(u);$$

$$A=\left[\begin{array}{cc}1& \cos {\theta }_1\\ 0& \sin {\theta }_1\end{array}\right],$$

$$\underset{i\to \infty }{\lim }{X}_i=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right].$$

$$Av=\sin ({\theta }_1)v$$

$$\left[\begin{array}{cc}1& \cos {\theta }_1\\ 0& \sin {\theta }_1\end{array}\right]\left[\begin{array}{c}\cos {\theta }_2\\ \sin {\theta }_2\end{array}\right]=\sin {\theta }_1\left[\begin{array}{c}\cos {\theta }_2\\ \sin {\theta }_2\end{array}\right].$$

$$\cos {\theta }_2+\cos {\theta }_1\sin {\theta }_2=\sin {\theta }_1\cos {\theta }_2$$

$$\cos {\theta }_2=\sin ({\theta }_1-{\theta }_2).$$

$$\sin \left(\frac{\pi }{2}+{\theta }_2\right)=\sin ({\theta }_1-{\theta }_2)$$

$${\theta }_2=\frac{-\pi }{4}+\frac{{\theta }_1}{2}.$$

$$\theta$$

$$\left[\begin{array}{ccccc}1& 0& 0& \cdots & x_1\\ 0& 1& 0& \cdots & x_2\\ 0& 0& \ddots & ⋮& ⋮\\ 0& 0& \cdots & 1& x_{n-1}\\ 0& 0& \cdots & 0& x_n\end{array}\right]\left[\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_{n-1}\\ y_n\end{array}\right]=x_n\left[\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_{n-1}\\ y_n\end{array}\right].$$

$$x_i{y}_n$$

$$\frac{x_i}{x_j}$$

$$s_{2n-1}{c}_{2n}+s_{n-1}{c}_n{s}_{2n}$$

$$c_{2n}$$

$$A=\left[\begin{array}{ccc}1& 0& a\\ 0& -1& b\\ 0& 0& ci\end{array}\right]\left[\begin{array}{c}x\\ y\\ zi\end{array}\right]=ci\left[\begin{array}{c}x\\ y\\ zi\end{array}\right].$$

$$x+azi$$

$$-y+bzi$$

$$azi$$

$$bzi$$

$$\frac{a}{b}$$

$$\frac{|a|}{|b|}$$

$${\cos }^2{\theta }_2{\sin }^2{\theta }_4$$

$${\tan }^2{\theta }_4$$

$$ta{n}^2\theta =1+2+4+8+16\cdots$$

$$A=\left[\begin{array}{cc}\cos {\theta }_1& \cos {\theta }_2\\ \sin {\theta }_1& \sin {\theta }_2\end{array}\right],$$

$$\underset{i\to \infty }{\lim }{X}_i=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right].$$

$$\cos {\theta }_1-\sin {\theta }_1$$

$$\sin {\theta }_2-\cos {\theta }_2$$

$$\left[\begin{array}{cc}\cos {\theta }_1& \cos {\theta }_2\\ \sin {\theta }_1& \sin {\theta }_2\end{array}\right]\left[\begin{array}{c}\cos {\theta }_3\\ \sin {\theta }_3\end{array}\right]={\lambda }_1\left[\begin{array}{c}\cos {\theta }_3\\ \sin {\theta }_3\end{array}\right].$$

$$\sin {\theta }_1\cos {\theta }_3+\sin {\theta }_2\sin {\theta }_3$$

Full text

Recursive eigen extrusion : Expanding eigenbasis conjecture

M. Hariprasad

Department of Computational and Data Sciences

Indian Institute of Science, Bangalore-560012, India

e-mail: mhariprasadkansur@gmail.com

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 $A_{i+1}=\phi(X_{i})$ where $\phi$ normalizes all eigenvectors to unit $\mathcal{L}_{2}$ norm such that all diagonal elements $[\phi(X)^{\dagger}\phi(X)]_{jj}=1$. 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.

Keywords: Eigenvalue problems; Unitary matrices; Fixed point theory; Limiting sets.

2010 Mathematics Subject Classification: 35P05, 15A18, 51F25, 58C30

1 Introduction

Geometrical properties of the unit $n$-sphere $\mathbb{U}^{n}$ ( [1],[4], [3] ) and distribution of points in its surface or volume ([6], [8]) are widely studied from theoretical and application perspective. Also, the structure of eigenvector matrices ([10],[7]), its condition number ([2], [5]) and correlation between its columns [9] are very well studied in the literature. In this article we study the limiting behavior of recursive evaluation of a linear operator and the resulting eigen directions, and more importantly its significant properties as a function of the dimension of space $n$. We conjecture that the limiting set of vectors are orthogonal only for $n\leq 7$, and this recursion is also equivalent to an iterative map maximizing the distance between $n$ points on a $n$-sphere. Along with symmetry and other interesting geometric properties, we note that the relative $n$-area of a sphere compared to a face of the larger circumscribing $n$-cube increases until $n$ increases to 7, and decreases thereafter. Here we present a conjecture supported by numerical evidence, where this significant dimension $n$ =7, reappears. Note that a set of $n$ unit vectors have $\binom{n}{2}$ degrees of freedom given by the angles between them. Given this procedure of extrusion of eigen directions, the eigenvalue problem includes only the real and imaginary parts of the eigenvalue (magnitude and phase), and the norm, as the three parameters for every eigenvector. It should be noted that for $n\leq 7$, $\binom{n}{2}\leq 3n$.

2 Numerical observations and a conjecture

In this section first we state the conjecture, and present the numerical observations. We say a property holds for almost all matrices, if for any matrix $A$ we have matrix $A^{\prime}$ such that $|A(i,j)-A^{\prime}(i,j)|<\epsilon$ for any $\epsilon>0$ satisfying that property.

conjecture 1**.**

(Expanding Eigen basis) : For almost all matrices up to dimension 7, If we repeatedly construct the eigenvector matrices $X_{i}$, each of them having columns of unit $\mathbb{L}_{2}$ norm, then in the limit of large $i$, we have $X_{i}^{\dagger}X_{i}\to I$.

conjecture 2**.**

(Expanding orthogonal similarity variant ) : For almost all matrices, up to dimension 7, If we repeatedly do the procedure of constructing the eigenvector matrix and similarity transformation with a random orthogonal (or unitary) matrix, $Y_{i}=Q_{i}Z_{i}Q_{i}^{\dagger}$, ($Z_{1}$ being eigenvector matrix of $X_{1}$, $Z_{i}$ being the eigenvector matrix of $Y_{i-1}$), then in the limit of large $i$, we have $Y_{i}^{\dagger}Y_{i}\to I$.

conjecture 3**.**

**(Expanding orthogonal product variant ) : For almost all matrices, up to dimension 7, If we repeatedly do the procedure of constructing the eigenvector matrix and multiply it with a random orthogonal (or unitary) matrix, $Y_{i}=Q_{i}Z_{i}$, ($Z_{1}$ being eigenvector matrix of $X_{1}$, $Z_{i}$ being the eigenvector matrix of $Y_{i-1}$), then in the limit of large $i$, we have $Y_{i}^{\dagger}Y_{i}\to I$.

**

Here, convergence of $X_{i}^{\dagger}X_{i}$ need not necessarily imply convergence of $X_{i}$ itself to a particular unitary matrix. Instead, $X_{i}$ tend to be more unitary with increase in $i$. Also the conjecture is stated for almost all matrices, because when algebraic and geometric multiplicities of an eigenvalue are different, we get a rank deficient eigenvector matrix, which can never be orthogonal. So we exclude all matrices which give rank deficient eigenvector matrix at any iteration. Also, there can be cases when the eigenvector matrices can get stuck into loops as shown in section 3.1.1.

In figure 1 and figure 2, uniform and Gaussian random matrices are considered. The following MATLAB commands are executed for verifying expanding eigenbasis conjecture,

[TABLE]

Determinant of $u^{\dagger}u$ is plotted over iteration from one to $N$. This is repeated for many matrices in a single plot.

In figure 3, we plot the average (over many matrices for a particular iteration) value of $\log(-\log(\det(u^{\dagger}u)))$ with respect to the iteration number for expanding eigenbasis and its variant conjectures.

• •

It can be seen that at the $k^{\text{th}}$ iteration the $\det(u^{\dagger}u)\approx e^{-e^{-\frac{k}{2^{d}}}}$ where $d+2$ is dimension of the matrix, for expanding eigen basis conjecture (figure 3(a)).

• •

Convergence for low dimensions (dimension two and three) have been slowed down in expanding eigenvector similarity variant (figure 3(b)).

• •

Convergence is faster (up to dimension six, we have $\det(u^{\dagger}u)\approx e^{-e^{-\frac{k}{2^{d-1}}}}$ , here $d+2$ representing dimension of the matrix,) for expanding orthogonal product variant where eigenvector matrix is multiplied by an orthogonal matrix before constructing next eigenvector matrix (figure 3(c)).

3 Proofs: Special cases

In this section, we consider a matrix of a particular type denoted by $A$. Eigenvector matrix of $A$ is denoted by $X_{1}$. $X_{2}$ denotes the eigenvector matrix of $X_{1}$, similarly $X_{i}$ denoting the eigenvector matrix of the matrix $X_{i-1}$. Let all $X_{i}$ have columns of unit $\mathcal{L}_{2}$ norm. We impose condition on choosing the eigenvectors, using the fact : if $x$ is an eigenvector of a matrix $A$ satisfying $Ax=\lambda x$, then $\alpha x$ is also an eigenvector for $\alpha\in\mathbb{C}$. By this conditioning, we force $X_{i}$ itself to converge to a unitary matrix.

3.1 Upper triangular matrices

First, we consider $2\times 2$ upper triangular matrix of the form $A=\begin{bmatrix}1&\cos\theta\\ 0&\sin\theta\\ \end{bmatrix},$ when $\theta<0$ (restricted to fourth quadrant). Note that the eigenvectors of the matrix $A$ can be chosen such that eigenvector matrix is also of the same form, that is, $X_{1}=\begin{bmatrix}1&\cos\beta\\ 0&\sin\beta\end{bmatrix}$ with $\beta<0$ (restricted to fourth quadrant).

Theorem 3.1**.**

For $2\times 2$ upper triangular matrix of the form

[TABLE]

if $\theta_{1}<0$ (restricting to fourth quadrant) maintained for all the eigenvector matrices $X_{i}$, then

[TABLE]

Proof.

We can see that $e1=\begin{bmatrix}1\\ 0\end{bmatrix}$ is an eigenvector. Let the second eigenvector be $v=\begin{bmatrix}\cos\theta_{2}\\ \sin\theta_{2}\end{bmatrix}$. Then we have

[TABLE]

From this relation we get,

[TABLE]

We have $\theta_{2}$ in the fourth quadrant, so we get

[TABLE]

When we repeat the procedure, let $\theta$ be the angle corresponding to limiting eigenvector, we have

[TABLE]

Therefore, in the limit of repeated extrusion of the eigenvectors, the upper triangular matrix converges to the diagonal matrix, which is unitary. ∎

Theorem 3.2**.**

*For a $n\times n$ real upper triangular matrix,

$A=\begin{bmatrix}1&0&0&\cdots&x_{1}\\ 0&1&0&\cdots&x_{2}\\ 0&0&\ddots&\vdots&\vdots\\ 0&0&\cdots&1&x_{n-1}\\ 0&0&\cdots&0&x_{n}\end{bmatrix}$ with $\sum_{i=1}^{n}x_{i}^{2}=1$. and $x_{i}\geq 0$ for $1\leq i<n$ and $x_{n}<0$. If we impose conditions $\sum_{i=1}^{n}y_{i}^{2}=1$. and $y_{i}\geq 0$ for $1\leq i<n$ and $y_{n}<0$ on the eigenvector satisfying,*

[TABLE]

*Then in the process of repeated extrusion of eigenvectors,

limiting eigenvector matrix is, $\begin{bmatrix}1&0&0&\cdots&0\\ 0&1&0&\cdots&0\\ 0&0&\ddots&\vdots&\vdots\\ 0&0&\cdots&1&0\\ 0&0&\cdots&0&-1\end{bmatrix}$.*

Proof.

Let $X_{1}=\begin{bmatrix}1&0&0&\cdots&y_{1}\\ 0&1&0&\cdots&y_{2}\\ 0&0&\ddots&\vdots&\vdots\\ 0&0&\cdots&1&y_{n-1}\\ 0&0&\cdots&0&y_{n}\end{bmatrix}$ be the eigenvector matrix of $A$.

So we have the relation,

[TABLE]

This implies ,

[TABLE]

Which means,

[TABLE]

Let us denote $\cos\theta_{i}$ by $c_{i}$ and $\sin\theta_{i}$ by $s_{i}$, then we have eigenvector of $A$ in the form, $\begin{bmatrix}c_{1}c_{2}\cdots c_{n}\\ s_{1}c_{2}c_{3}...c_{n}\\ s_{2}c_{3}\cdots c_{n}\\ \vdots\\ s_{n}\end{bmatrix}$. Because of the constraints we can represent last column of $A$ by $\begin{bmatrix}c_{1}c_{2}\cdots c_{n}\\ s_{1}c_{2}c_{3}...c_{n}\\ s_{2}c_{3}\cdots c_{n}\\ \vdots\\ s_{n}\end{bmatrix}$ and eigenvector by $\begin{bmatrix}c_{n+1}c_{n+2}\cdots c_{2n}\\ s_{n+1}c_{n+2}c_{n+3}\cdots c_{2n}\\ s_{n+2}c_{n+3}\cdots c_{2n}\\ \vdots\\ s_{2n}\end{bmatrix}$. Then from the relation (13) and the positivity of $x_{i}$ and $y_{i}$, we have $\theta_{i}=\theta_{i+n}$ for $1\leq i<n$.

From the equation (12) we get,

[TABLE]

Note that equation (15) is same as equation (5), so from the arguments as in theorem 3.1, we get the limiting eigenvector matrix, $\begin{bmatrix}1&0&0&\cdots&0\\ 0&1&0&\cdots&0\\ 0&0&\ddots&\vdots&\vdots\\ 0&0&\cdots&1&0\\ 0&0&\cdots&0&-1\end{bmatrix}$.

∎

3.1.1 Loops and discontinuities

While selecting the condition on the eigenvectors to make $X_{i}$ itself converge to an orthogonal matrix, we may get into two difficult scenarios.

• •

In general, for upper triangular matrices, if we don’t have the restriction on the eigenvectors, the procedure of repeated extrusion of eigenvectors may get into a loop. For example, the matrices $\begin{bmatrix}1&\frac{\sqrt{3}}{2}\\ 0&\frac{1}{2}\end{bmatrix}$ and the matrix $\begin{bmatrix}1&\frac{-\sqrt{3}}{2}\\ 0&\frac{1}{2}\end{bmatrix}$ are the eigenvector matrices of each other.

• •

Discontinuity: Consider a $2\times 2$ upper triangular matrix $\begin{bmatrix}1&\epsilon\\ 0&\sqrt{1-\epsilon^{2}}\end{bmatrix}$ for a small $\epsilon>0$ we can see that its eigenvector $\begin{bmatrix}x\\ y\end{bmatrix}$ corresponding to eigenvalue $\sqrt{1-\epsilon^{2}}$, then as $\epsilon\to 0$, we have $y\to 0$ and $x\to 1$. So we have the following

$\begin{bmatrix}1&\epsilon\\ 0&\sqrt{1-\epsilon^{2}}\end{bmatrix}$ matrix has eigenvector matrix $\begin{bmatrix}1&\pm\sqrt{1-\delta^{2}}\\ 0&\delta\end{bmatrix}$ for small $\epsilon,\delta>0$.

3.1.2 $3\times 3$ upper triangular matrices

Theorem 3.3**.**

For a $3\times 3$ upper triangular matrix of the form $A=\begin{bmatrix}1&0&a\\ 0&-1&b\\ 0&0&ci\end{bmatrix}$ where $|a|^{2}+|b|^{2}+|c|^{2}=1$ and $c\in\mathbb{R},$ and $a,b\in\mathbb{C}$ if we maintain the constraint to the eigenvector matrix $X_{1}=\begin{bmatrix}1&0&x\\ 0&-1&y\\ 0&0&zi\end{bmatrix},$ $|x|^{2}+|y|^{2}+|z|^{2}=1$ and $z\in\mathbb{R},$ and $x,y\in\mathbb{C}$ in the limit of repeatedly taking eigenvectors, we get the matrix $\begin{bmatrix}1&0&0\\ 0&-1&0\\ 0&0&i\end{bmatrix}$.

Proof.

We have a relation with eigenvectors,

[TABLE]

From this relation we get,

[TABLE]

Rearranging,

[TABLE]

Using the ratio of equations (19) and (20),

[TABLE]

Note $-1i\leq ci\leq 1i$ so $\frac{ci-1}{ci+1}$ is a point on the unit circle, so using the magnitude in (21), we get

[TABLE]

If we represent vector $\begin{bmatrix}a\\ b\\ ci\end{bmatrix}$ as $\begin{bmatrix}\cos\theta_{1}\cos\theta_{2}e^{i\alpha}\\ \sin\theta_{1}\cos\theta_{2}e^{i\beta}\\ \sin\theta_{2}i\end{bmatrix}$, and vector $\begin{bmatrix}x\\ y\\ zi\end{bmatrix}$ as $\begin{bmatrix}\cos\theta_{3}\cos\theta_{4}e^{i\gamma}\\ \sin\theta_{3}\cos\theta_{4}e^{i\delta}\\ \sin\theta_{4}i\end{bmatrix}$.

Then we have from the relation (22), $\tan\theta_{1}=\tan\theta_{3}$. Thus we have $\theta_{1}=\theta_{3}$.

Multiplying the equations (19) and (20), the angles are given by $\alpha+\beta=\gamma+\delta$. Also using the magnitude and the fact $\theta_{1}=\theta_{3}$, we get

[TABLE]

In the limit of repeated extrusion of eigenvectors (24) gives

[TABLE]

So the limiting eigenvector matrix is, $\begin{bmatrix}1&0&0\\ 0&-1&0\\ 0&0&i\end{bmatrix}$. ∎

Corollary 3.3.1**.**

For a real upper triangular matrix of dimension 3, while recursive evaluation eigenvector matrices, first constraining the second eigenvector as in theorem 3.1 we obtain $\begin{bmatrix}1&\epsilon&a\\ 0&-1+\delta&b\\ 0&0&c\end{bmatrix}$ for small $\epsilon,\delta>0$ , then constraining the third column as in theorem 3.3 along with the second column, gives $\begin{bmatrix}1&0&0\\ 0&-1&0\\ 0&0&i\end{bmatrix}$ as a limiting eigenvector matrix.

3.2 A special $2\times 2$ matrix

Consider a $2\times 2$ matrix with real entries such that $A=\begin{bmatrix}\cos\theta_{1}&\cos\theta_{2}\\ \sin\theta_{1}&\sin\theta_{2}\end{bmatrix}$ with $\theta_{1}$ restricted to the first quadrant and $\theta_{1}<\frac{\pi}{4}$ and $\theta_{2}$ restricted to the fourth quadrant and $|\theta_{2}|>\frac{\pi}{4}$. Then we can form an eigenvector matrix of the same form $X=\begin{bmatrix}\cos\theta_{3}&\cos\theta_{4}\\ \sin\theta_{3}&\sin\theta_{4}\end{bmatrix}$ with the restrictions $\theta_{3}$, $\theta_{4}$ respectively in the first and fourth quadrants. If $\theta_{1}-\frac{\pi}{2}=\theta_{2}$, then the matrix is orthogonal, and so we exclude that condition. We prove that in the limit of repeatedly taking eigenvectors, the eigenvector matrix converges to $\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}.$

Theorem 3.4**.**

For $2\times 2$ real matrix of the form

[TABLE]

starting with $\theta_{1}<\frac{\pi}{4},|\theta_{2}|>\frac{\pi}{4}$ and $\theta_{1}-\frac{\pi}{2}\neq\theta_{2}$, if $\theta_{1}$ restricted to the first quadrant and $\theta_{2}$ restricted to the fourth quadrant for all the eigenvector matrices $X_{i}$, then

[TABLE]

Proof.

Consider the first eigenvector matrix, $X=\begin{bmatrix}\cos\theta_{3}&\cos\theta_{4}\\ \sin\theta_{3}&\sin\theta_{4}\end{bmatrix}$ We have from the Gerschgorin circle theorem the eigenvalues $\lambda_{1}$ and $\lambda_{2}$ of matrix $A$ satisfying,

[TABLE]

We have

[TABLE]

Which gives

[TABLE]

Using the fact that $\sin\theta_{2}<0$, and $-\sin\theta_{1}-\sin\theta_{2}>0$,

[TABLE]

This implies $\theta_{1}>\theta_{3}$.

Similarly we have,

[TABLE]

Which gives,

[TABLE]

Rearranging (36) gives

[TABLE]

This implies $|\theta_{2}|<|\theta_{4}|$. So the matrices formed are in a increasing sequence of angles corresponding to $|\theta_{2}|$ and decreasing sequence of angles corresponding to $\theta_{1}$.

Let us denote $-\sin\theta_{2}-\sin\theta_{1}$ by $\delta>0$, from equation (32) we have,

[TABLE]

From equation (42) we can see that, for sufficiently large $\theta_{1}>0$, difference between $\tan\theta_{3}$ and $\tan\theta_{1}$ are not arbitrary small. So the sequence corresponding to $\theta_{1}$ should converge to zero. Similarly we get the relation from equation (38),

[TABLE]

Which says that consecutive difference between $\tan\theta$ increases and hence the sequence correspond to angles will converge to $\frac{-\pi}{2}$.

So in the limit of repeatedly taking eigenvectors, the eigenvector matrix converges to $\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}$.

∎

Conclusion

The proposed conjecture is on the limiting behavior of recursive evaluation of eigen directions of linear operators up to dimension seven. It is to be proven in general that, the limiting eigenvector matrix having columns as points on the unit $n$-ball, tend to be unitary. Numerical results clearly show the convergence behavior up-to dimension seven, supporting the conjecture. Some special cases of this recursive eigen extrusion procedure are proved affirmatively. Thus opening the further connection between the geometry of unit n-ball and eigen properties of a linear operator.

Acknowledgement

The author would like to thank Prof. Murugesan Venkatapathi for much helpful discussions and insights.