English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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.

Simultaneous spectral decomposition in Euclidean Jordan algebras and related systems