A covariance matrix with a special pattern (e.g., sparsity or block structure) is essential for conducting multivariate analysis on high-dimensional data. Recently, a block covariance or correlation pattern has been observed in various biological and biomedical studies, such as gene expression, proteomics, neuroimaging, exposome, and seed quality, among others. Specifically, this pattern partitions the population covariance matrix into uniform (i.e., equal variances and covariances) blocks. However, the unknown mathematical properties of matrices with this pattern limit the incorporation of this pre-determined covariance information into research. To address this gap, we propose a block Hadamard product representation that utilizes two lower-dimensional "coordinate" matrices and a pre-specific vector. This representation enables the explicit expressions of the square or power, determinant, inverse, eigendecomposition, canonical form, and the other matrix functions of the original larger-dimensional matrix on the basis of these "coordinate" matrices. By utilizing this representation, we construct null distributions of information test statistics for the population mean(s) in both single and multiple sample cases, which are extensions of Hotelling's $T^2$ and $T_0^2$, respectively.

翻译：具有特殊模式（例如稀疏性或分块结构）的协方差矩阵对于高维数据的多元分析至关重要。近年来，在基因表达、蛋白质组学、神经影像学、暴露组学及种子品质等多种生物学和生物医学研究中，观察到一种分块协方差或相关模式。具体而言，该模式将总体协方差矩阵划分为均匀（即等方差等协方差）的块。然而，具有该模式的矩阵其未知的数学性质限制了将这种预先确定的协方差信息纳入研究。为解决这一空白，我们提出一种利用两个较低维度的“坐标”矩阵与一个预设向量的块哈达玛积表示。该表示能够基于这些“坐标”矩阵，显式表达原始高维矩阵的平方或幂、行列式、逆、特征分解、规范型及其他矩阵函数。利用该表示，我们构建了单样本和多样本情形下总体均值信息检验统计量的零分布，这些统计量分别是Hotelling $T^2$ 和 $T_0^2$ 的推广。