奇异值与Eigenvalue统计量的精确关系
Exact Relation between Singular Value and Eigenvalue Statistics
摘要：
利用矩阵空间上调和分析的经典结果，研究了双酉不变（也称为各向同性或酉旋转不变）复随机矩阵的奇异值和特征值的联合密度之间的关系.我们证明，这些联合密度之一，决定了其他。此外，我们构造了一个显式公式，这两个联合密度在有限矩阵维数。这个关系涵盖了概率密度和符号密度。借助这种关系，我们得到了一般的分析关系相应的核和双正交函数的一类特定的多项式系综。此外，我们展示了如何推广的特征值和奇异值统计之间的关系，某些情况下，当合奏是由一个长期的变形，打破双酉不变性。
英文摘要：
We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint density of the singular values and of the eigenvalues of complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that one of these joint densities determines the other one. Moreover we construct an explicit formula relating both joint densities at finite matrix dimension. This relation covers probability densities as well as signed densities. With the help of this relation we derive general analytical relations among the corresponding kernels and biorthogonal functions for a specific class of polynomial ensembles. Furthermore we show how to generalize the relation between the eigenvalue and singular value statistics to certain situations when the ensemble is deformed by a term which breaks the bi-unitary invariance.
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