We study a matrix that arises from a singular form of the Woodbury matrix identity. We present generalized inverse and pseudo-determinant identities for this matrix, which have direct applications for Gaussian process regression, specifically its likelihood representation and precision matrix. We extend the definition of the precision matrix to the Bott-Duffin inverse of the covariance matrix, preserving properties related to conditional independence, conditional precision, and marginal precision. We also provide an efficient algorithm and numerical analysis for the presented determinant identities and demonstrate their advantages under specific conditions relevant to computing log-determinant terms in likelihood functions of Gaussian process regression.

翻译：我们研究一个来自单一形式的Woodbury矩阵特征的矩阵,我们为该矩阵提供了普遍反向和伪决定性特征,这些特征直接适用于高斯进程回归,特别是其可能性的表示和精确矩阵;我们将精确矩阵的定义扩大到Bott-Duffin,与共变矩阵相反,保留与有条件独立、有条件精确和边际精确性有关的属性;我们还为所提出的决定因素身份提供有效的算法和数字分析,并在与计算高斯进程回归的可能功能的日志定义有关的特定条件下展示其优势。