"Penrose广义逆矩阵：定义、性质及存在性"

Penrose's generalized inverse matrix is a concept that extends the idea of the inverse matrix from singular (or full-rank) square matrices to non-square matrices. Penrose's generalized inverse matrix covers both scenarios. For a full-rank matrix A, A can be inverted to A^(-1), satisfying the properties A*A^(-1) = A^(-1)*A = I. This concept can be extended to singular or non-square matrices. Penrose's definition states that for a matrix A ∈ C^(m*n), if there exists a matrix Z∈C^(n*m) that satisfies the following four equations: AZA = A, ZAZ = Z, (AZ)^H = AZ, (ZA)^H = ZA, then Z is called the Moore-Penrose (generalized) inverse of A, denoted as A^†. These four equations are known as the Penrose equations. In conclusion, Penrose's generalized inverse matrix extends the concept of the inverse matrix to encompass singular and non-square matrices. It provides a way to find a generalized inverse for matrices that may not have a traditional inverse, allowing for further applications and analysis in linear algebra.