[Paper Review] A Note On The Spectral Norms of The Matrices Connected Integer Numbers Sequence

Durmuş Bozkurt|arXiv (Cornell University)|May 9, 2011

Advanced Mathematical Theories and Applications5 references3 citations

TL;DR

This paper derives a closed-form expression for the spectral norm of matrices formed from the difference of integer sequences, specifically proving that the spectral norm equals the sum of squared differences between all pairs of sequence elements. The result is applied to Fibonacci and Lucas sequences, yielding explicit formulas involving their terms and sums of products of terms at combined indices.

ABSTRACT

In this paper, we compute the spectral norms of the matrices related with integer squences and we give two examples related with Fibonacci and Lucas numbers.

Motivation & Objective

• To establish a general formula for the spectral norm of matrices constructed from differences of integer sequences.
• To analyze the spectral properties of skew-symmetric matrices derived from such sequences.
• To apply the general result to specific sequences like Fibonacci and Lucas numbers.
• To compute explicit expressions for the spectral norms of Fibonacci and Lucas matrices.

Proposed method

• Define the matrix $ A_x = [x_i - x_j]_{i,j=1}^n $, which is skew-symmetric.
• Use row operations to transform $ A_x $ into a matrix of rank 2, preserving rank and spectral norm.
• Consider the matrix $ iA_x $, which becomes symmetric with real eigenvalues, enabling spectral analysis.
• Compute the characteristic polynomial of $ iA_x $, showing it has only two non-zero eigenvalues.
• Use the sum of principal 2-minors of $ iA_x $ to determine the coefficient of $ λ^{n-2} $ in the characteristic polynomial.
• Derive the spectral norm as the square root of the sum of squared differences $ \sum_{1\leq r<s\leq n} (x_r - x_s)^2 $.

Experimental results

Research questions

• RQ1 What is the spectral norm of a matrix formed by the pairwise differences of an integer sequence?
• RQ2 How does the spectral norm depend on the structure of the sequence, particularly for linear recurrences?
• RQ3 Can the spectral norm be expressed in closed form for Fibonacci and Lucas sequences?
• RQ4 What role do principal minors and eigenvalue structure play in computing the spectral norm for skew-symmetric matrices?
• RQ5 How do the spectral norms of Fibonacci and Lucas matrices differ based on parity of $ n $?

Key findings

• The spectral norm of the matrix $ A_x = [x_i - x_j] $ is $ \left\|A_x\right\|_2 = \sum_{1\leq r<s\leq n} (x_r - x_s)^2 $, valid for $ n \geq 4 $.
• For the Fibonacci matrix $ F $, the spectral norm is $ \left\|F\right\|_2 = (n-1)F_{n+1}F_n - \frac{2}{5}\left(L_n - 2 + \sum_{r=1}^{n-1}\sum_{s=r+1}^n L_{r+s}\right) $ when $ n $ is even.
• For the Fibonacci matrix $ F $, the spectral norm is $ \left\|F\right\|_2 = (n-1)F_{n+1}F_n - \frac{2}{5}\left(L_n - 1 + \sum_{r=1}^{n-1}\sum_{s=r+1}^n L_{r+s}\right) $ when $ n $ is odd.
• For the Lucas matrix $ L $, the spectral norm is $ \left\|L\right\|_2 = (n-1)(L_{n+1}L_n - 2) - 2\left(L_n - 2 + \sum_{r=1}^{n-1}\sum_{s=r+1}^n L_{r+s}\right) $ when $ n $ is even.
• For the Lucas matrix $ L $, the spectral norm is $ \left\|L\right\|_2 = (n-1)(L_{n+1}L_n - 2) - 2\left(L_n - 1 + \sum_{r=1}^{n-1}\sum_{s=r+1}^n L_{r+s}\right) $ when $ n $ is odd.
• The spectral norm of $ A_x $ is fully determined by the sum of squared pairwise differences of the sequence elements, independent of the sequence's specific values beyond these differences.