This paper introduces a generalized inverse for matrices, which is the unique solution to a set of equations. This generalized inverse exists for any matrix with complex elements, regardless of its dimensions. The paper discusses its applications in solving linear matrix equations and finding principal idempotent elements of a matrix. It also presents a new type of spectral decomposition. The generalized inverse, denoted as \( A^+ \), is defined by a set of equations and is shown to be unique. The paper provides a detailed proof of the existence and uniqueness of \( A^+ \) and explores its properties, including its relationship with the conjugate transpose and the trace of the matrix. Additionally, the paper discusses the application of the generalized inverse to solving linear matrix equations and finding hermitian idempotents. It also introduces the concept of principal idempotent elements and provides an explicit formula for them in terms of the generalized inverse. Finally, the paper presents a new spectral decomposition for any matrix, which is particularly useful for rectangular matrices.This paper introduces a generalized inverse for matrices, which is the unique solution to a set of equations. This generalized inverse exists for any matrix with complex elements, regardless of its dimensions. The paper discusses its applications in solving linear matrix equations and finding principal idempotent elements of a matrix. It also presents a new type of spectral decomposition. The generalized inverse, denoted as \( A^+ \), is defined by a set of equations and is shown to be unique. The paper provides a detailed proof of the existence and uniqueness of \( A^+ \) and explores its properties, including its relationship with the conjugate transpose and the trace of the matrix. Additionally, the paper discusses the application of the generalized inverse to solving linear matrix equations and finding hermitian idempotents. It also introduces the concept of principal idempotent elements and provides an explicit formula for them in terms of the generalized inverse. Finally, the paper presents a new spectral decomposition for any matrix, which is particularly useful for rectangular matrices.