In this paper, we highlight the role played by orthogonal and symplectic Harish-Chandra integrals in the study of real-valued matrix product ensembles. By making use of these integrals and the matrix-valued Fourier-Laplace transform, we find the explicit eigenvalue distributions for particular Hermitian anti-symmetric matrices and Hermitian anti-self dual matrices, involving both sums and products. As a consequence of these results, the eigenvalue probability density function of the random product structure [Formula: see text], where each [Formula: see text] is a standard real Gaussian matrix, and [Formula: see text] is a real anti-symmetric matrix can be determined. For [Formula: see text] and [Formula: see text] the bidiagonal anti-symmetric matrix with 1’s above the diagonal, this reclaims results of Defosseux. For general [Formula: see text], and this choice of [Formula: see text], or [Formula: see text] itself a standard Gaussian anti-symmetric matrix, the eigenvalue distribution is shown to coincide with that of the squared singular values for the product of certain complex Gaussian matrices first studied by Akemann et al. As a point of independent interest, we also include a self-contained diffusion equation derivation of the orthogonal and symplectic Harish-Chandra integrals.

中文翻译：

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正交和辛 Harish-Chandra 积分和矩阵积集成

在本文中，我们强调了正交和辛 Harish-Chandra 积分在实值矩阵积集成研究中所起的作用。通过利用这些积分和矩阵值傅里叶-拉普拉斯变换，我们找到了特定 Hermitian 反对称矩阵和 Hermitian 反对自对偶矩阵的显式特征值分布，包括和和乘积。作为这些结果的结果，随机积结构[公式：见文]的特征值概率密度函数，其中每个[公式：见文]是一个标准的实高斯矩阵，[公式：见文]是一个真正的反-对称矩阵可以确定。对于 [Formula: see text] 和 [Formula: see text] 对角线上方为 1 的双对角反对称矩阵，这将回收 Defosseux 的结果。对于一般[公式：见正文]，并且[公式：参见文本]或[公式：参见文本]的这种选择本身是标准的高斯反对称矩阵，特征值分布首先与某些复高斯矩阵乘积的平方奇异值的分布一致由 Akemann 等人研究。作为一个独立的兴趣点，我们还包括正交和辛 Harish-Chandra 积分的独立扩散方程推导。