On a formula of Thompson and McEnteggert for the adjugate matrix

Abstract

For an eigenvalue λ0 of a Hermitian matrix A, the formula of Thompson and McEnteggert gives an explicit expression of the adjugate of λ0I−A, Adj(λ0I−A), in terms of eigenvectors of A for λ0 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 Adj(A) is provided in terms of those of A. Finally, a generalization of the eigenvalue-eigenvector identity and three applications of the Thompson-McEnteggert's formula are presented.