Simultaneous spectral decomposition in Euclidean Jordan algebras and related systems
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Date
2021-07-31
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Citation of Original Publication
M. Seetharama Gowda;
Simultaneous spectral decomposition in Euclidean Jordan algebras and related systems;
Linear and Multilinear Algebra , 31 July, 2021;
https://doi.org/10.1080/03081087.2021.1960259
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Abstract
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 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.