On a formula of Thompson and McEnteggert for the adjugate matrix

2021/03/25 by Castillo, Kenier, Zaballa, Ion
#15A15 #15A18 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2103.13739

Abstract

For an eigenvalue λ₀ of a Hermitian matrix A, the formula of Thompson and McEnteggert gives an explicit expression of the adjoint of λ₀ I-A, adj(λ₀ I-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 two applications of the Thompson-McEnteggert's formula are presented.

Related

• Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra
• Note on eigenvectors from eigenvalues
• Singular Vectors From Singular Values
• Eigenvalues and Diagonal Elements
• Thompson-type formulae