Simultaneous spectral decomposition in Euclidean Jordan algebras and related systems
M. Seetharama Gowda · Linear and Multilinear Algebra · 2021
This article deals with necessary and sufficient conditions for a family of elements in a Euclidean Jordan algebra to have simultaneous (order) spectral decomposition. Motivated by a well-known matrix theory result that any family of pairwise commuting complex Hermitian matrices is simultaneously (unitarily) diagonalizable, we show that in the setting of a general Euclidean Jordan algebra, any family of pairwise operator commuting elements has a simultaneous spectral decomposition, i.e. there exists a common Jordan frame {e1,e2,…,en} relative to which every element in the given family has the eigenvalue decomposition of the form λ1e1+λ2e2+⋯+λnen. The simultaneous order spectral decomposition further demands the ordering of eigenvalues λ1≥λ2≥⋯≥λn. We characterize this by a pairwise strong operator commutativity condition ⟨x,y⟩=⟨λ(x),λ(y)⟩ or, equivalently, λ(x+y)=λ(x)+λ(y), where λ(x) denotes the vector of eigenvalues of x written in the decreasing order. Going beyond Euclidean Jordan algebras, we formulate commutativity conditions in the setting of the so-called Fan–Theobald–von Neumann system that includes normal decomposition systems (Eaton triples) and certain systems induced by hyperbolic polynomials.