A new proof of an equality associated with submatrices and eigenvector elements

TL;DR

This paper presents a new, shorter proof of a mathematical equality involving eigenvectors, eigenvalues, and submatrix eigenvalues, using matrix block methods to simplify previous approaches.

Contribution

It introduces a novel proof technique employing matrix blocks to establish a known eigenvector-related equality more efficiently.

Findings

Provides a shorter, more elegant proof of the eigenvector equality

Utilizes matrix block methods to simplify the proof process

Confirms the validity of the previously established equality

Abstract

By using the methods of Cauchy-Binet type formula and adjugate matrix respectively, a wonderful equality relating to the elements of eigenvectors, the eigenvalues and the submatrix eigenvalues is proved in arXiv:1908.03795. In the note, we use matrix block to provide a new and shorter proof for the equality.