Every real symmetric positive-definite matrix has a unique factorization in which is a lower triangular matrix with positive diagonal entries.
Given a Cholesky decomposition , a rank-one Cholesky update computes a triangular factor of in operations without refactorizing the matrix from scratch.
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Cholesky decomposition is a mathematical technique used in linear algebra to decompose a symmetric, positive definite matrix into a product of a lower triangular matrix and its conjugate transpose. Specifically, if \( A \) is a symmetric positive definite matrix, the Cholesky decomposition states that: \[ A = L L^T \] where: - \( L \) is a lower triangular matrix with real and positive diagonal entries.