Motivated by statistical challenges arising in modern scientific fields, notably genomics, this paper seeks embeddings in which relevant covariance models are sparse. The work exploits a bijective mapping between a strictly positive definite matrix and its orthonormal eigen-decomposition, and between an orthonormal eigenvector matrix and its principle matrix logarithm. This leads to a representation of covariance matrices in terms of skew-symmetric matrices, for which there is a natural basis representation, and through which sparsity is conveniently explored. This theoretical work establishes the possibility of exploiting sparsity in the new parametrization and converting the conclusion back to the one of interest, a prospect of high relevance in statistics. The statistical aspects associated with this operation, while not a focus of the present work, are briefly discussed.

受现代科学领域（尤其是基因组学）出现的统计挑战的推动，本文寻求相关协方差模型稀疏的嵌入。这项工作利用了严格正定矩阵与其正交本征分解之间，正交本征向量矩阵与其主矩阵对数之间的双射映射。这导致以偏对称矩阵的形式表示协方差矩阵，对于该矩阵具有自然的基础表示形式，并且可以方便地探索稀疏性。这项理论工作建立了在新的参数化中利用稀疏性并将结论转换回有趣的可能性的可能性，这是统计学中高度相关的前景。与该操作相关的统计方面，