Perturbation theorems for the joint spectrum of commuting matrices: A conservative approach
A joint eigenvalue of an \(m\)-tuple \(A = (A_ 1, A_ 2, \dots, A_ m)\) of complex \(n\)-by-\(n\) matrices \(A_ i\) is a vector \(\lambda \in \mathbb{C}^ m\), \(\lambda = (\lambda_ 1, \lambda_ 2, \dots, \lambda_ m)^ T\), such that there exists a nonzero vector \(x \in \mathbb{C}^ n\) with \(A_ jx = \lambda_ j x\), \(j = 1,2, \dots, m\). If \(A_ i\) are commuting, then there is at least one joint eigenvalue, \(x\) is called a joint eigenvector, and the collection of the joint eigenvalues is called the joint spectrum and is denoted by \(Sp(A)\). The effect of a perturbation of a matrix on its spectrum has been investigated for a long time and now well understood. In contrast, study of the joint spectrum \(Sp(A)\) has started only recently by using Clifford-algebra tools. After introducing the relevant definitions of the Clifford operator \(\text{Cliff} (A)\) of \(A\), the underlying spaces, and the connection between the joint spectrum of \(A\) and the spectrum of \(\text{Cliff} (A)\), the author outlines the recent perturbation theorems for the joint spectrum of commuting matrices, which have been proved using Clifford- algebra tools. After establishing bounds on \(\| \text{Cliff} (A) \|\), he proves and improves these theorems by classical means as used in the case of the standard eigenvalue problem.
scientific article; zbMATH DE number 665779
scientific article