Linear and Multilinear Algebra ( IF 0.9 ) Pub Date : 2020-10-05 , DOI: 10.1080/03081087.2020.1830020 J. Tao 1 , J. Jeong 2 , M. Seetharama Gowda 3
Motivated by Horn's log-majorization (singular value) inequality and the related weak-majorization inequality for square complex matrices, we consider their Hermitian analogs for positive semidefinite matrices and for general (Hermitian) matrices, where denotes the Jordan product of A and B and denotes the componentwise product in . In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form for and for all a and b, where and denote, respectively, the quadratic representation and the eigenvalue vector of an element u. We also describe inequalities of the form , where A is a real symmetric positive semidefinite matrix and is the Schur product of A and b. In the form of an application, we prove the generalized Hölder type inequality , where denotes the spectral p-norm of x and with . We also give precise values of the norms of the Lyapunov transformation and relative to two spectral p-norms.
中文翻译:
欧几里得乔丹代数中的一些对数和弱主要化不等式
受霍恩的对数多数(奇异值)不等式的启发以及相关的弱多数化不等式对于复方矩阵,我们考虑它们的 Hermitian 类似物对于正半定矩阵和对于一般(Hermitian)矩阵,其中表示A和B的 Jordan 积,表示中的组件乘积. 在本文中,我们将这些不等式扩展到欧几里德乔丹代数的设置,形式为为了和对于所有a和b,其中和分别表示元素u的二次表示和特征值向量。我们还描述了形式的不等式,其中A是实对称半正定矩阵,并且是A和b的舒尔积。以应用的形式,我们证明了广义的 Hölder 型不等式, 在哪里表示x的谱p范数和和. 我们还给出了 Lyapunov 变换范数的精确值和相对于两个光谱p范数。