A singular Woodbury and pseudo-determinant matrix identities and application to Gaussian process regression

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