On a formula of Thompson and McEnteggert for the adjugate matrix

TL;DR

This paper generalizes Thompson and McEnteggert's formula for the adjugate matrix from Hermitian matrices to arbitrary fields, providing new formulas for elementary divisors and applications to eigenvalue identities.

Contribution

It extends the classical formula to matrices over any field and derives new results on elementary divisors and eigenvalue-eigenvector relations.

Findings

Generalized the Thompson-McEnteggert formula to arbitrary fields.

Derived formulas for elementary divisors of the adjugate matrix.

Presented applications to eigenvalue-eigenvector identities.

Abstract

For an eigenvalue $\lambda_0$ of a Hermitian matrix $A$, the formula of Thompson and McEnteggert gives an explicit expression of the adjoint of $\lambda_0 I-A$, $\mathrm{adj}(\lambda_0 I-A)$, in terms of eigenvectors of $A$ for $\lambda_0$ and all its eigenvalues. In this paper Thompson-McEnteggert's formula is generalized to include any matrix with entries in an arbitrary field. In addition, for any nonsingular matrix $A$, a formula for the elementary divisors of $\mathrm{adj}(A)$ is provided in terms of those of $A$. Finally, a generalization of the eigenvalue-eigenvector identity and two applications of the Thompson-McEnteggert's formula are presented.