Determining the eigenvalues of a square matrix through known information of its submatrix

TL;DR

This paper introduces a novel property linking a matrix's eigenvalues to those of its submatrix, enabling more efficient eigenvalue computation and eigenvector sharing, especially for matrices with specific restrictions or degeneracies.

Contribution

It presents a new property of eigenvalues related to submatrices, facilitating eigenvalue determination and eigenvector sharing, which reduces computational complexity.

Findings

Eigenvalues of a matrix can be partially determined from its submatrix eigenvalues.

The method allows for algebraic calculation of eigenvalues and eigenvectors for certain matrices.

Eigenvectors of a submatrix can be shared with the original matrix while preserving key properties.

Abstract

In this paper we bring to light an unprecedented property of the eigenvalues of a matrix A with the eigenvalues and eigenvectors of a submatrix of A. This property can be used, through the technique developed here, to determine some of eigenvalues of A and, thus, reduce the degree of the characteristic polynomial associated with this matrix. This reduction can occur in two ways: when restrictions between some elements of A are checked (or imposed) and/or when some eigenvalues of the submatrix of A are degenerate. As an application, we show how to obtain matrices of arbitrary size, not trivial, that allow the algebraic calculation of eigenvalues and eigenvectors and how to share a set of eigenvectors of a submatrix of A with matrix A itself, preserving norm, direction and sense.