Linear and Multilinear Algebra ( IF 0.9 ) Pub Date : 2021-07-31 , DOI: 10.1080/03081087.2021.1960259 M. Seetharama Gowda 1
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.
中文翻译:
欧几里得约旦代数及相关系统中的同时谱分解
本文讨论欧几里得约旦代数中的元素族具有同时(阶)谱分解的必要条件和充分条件。受著名的矩阵理论结果的启发,即任何成对交换复数厄尔米特矩阵族都同时(单一地)对角化,我们表明,在一般欧几里得约旦代数的设置中,任何成对算子交换元素族都具有同时谱分解,即存在一个共同的 Jordan 框架 { e 1 , e 2 , …, en } 相对于给定族中的每个元素都具有λ 1 e 1 + λ形式的特征值分解2 e 2 + · · · + λ n e n。同时阶谱分解进一步要求对特征值进行排序λ 1 ≥ λ 2 ≥ · · · ≥ λ n。我们通过成对的强算子交换性条件 〈x , y〉 = 〈λ ( x ), λ ( y )〉 或等效地λ ( x + y ) = λ ( x ) + λ ( y ) 来表征这一点,其中λ ( x ) 表示按降序排列的x的特征值向量。超越欧几里得约旦代数,我们在所谓的 Fan-Theobald-von Neumann 系统的设置中制定了交换条件,该系统包括正常分解系统(伊顿三元组)和由双曲多项式诱导的某些系统。