Spectral Decomposition

Bruce C. Dieffenbach ()
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Bruce C. Dieffenbach: Independent author

Chapter 26 in Conjugate Duality in Economic Analysis, 2026, pp 195-205 from Springer

Abstract: Abstract A Euclidean Jordan algebra admits a spectral decomposition that generalizes the spectral decomposition of the Euclidean space of self-adjoint linear transformations. Each vector has unique real eigenvalues, and the spectral decomposition expresses the vector as the sum of the eigenvalues times idempotents. The cone of squares is the vectors having nonnegative eigenvalues. The trace is the sum of the eigenvalues. The trace inner product makes the algebra into a Euclidean space: the inner product of two vectors is the trace of their Jordan product. The linear representation is the square linear transformation $${\textit {{\sf { {L}}}}}_{\boldsymbol {x}}\left ( \boldsymbol {y}\right ) =\boldsymbol {x}\bullet \boldsymbol {y}$$ , which is self-adjoint under the trace inner product and is nonnegative if and only if x is nonnegative. The polar of the nonnegative cone is its negative.

Date: 2026
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DOI: 10.1007/978-3-032-21396-9_26

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