MOORE-PENROSE PSEUDOINVERSE AND APPLICATIONS.
ABSTRACT
In this abstract, we focus on establishing algebraic analogs of a fundamental theorem attributed to Gauss and Legendre. This theorem states that for an over-determined system, there exist solutions that minimize the quantity kAx − bk^2, where A is a matrix and x, b are vectors. Notably, this result holds even when A is singular or rectangular, and the minimizing solutions can be obtained using the generalized inverse of A.
Our objective is to extend this theorem to arbitrary operators defined on complex Hilbert spaces, thereby generalizing its applicability. Moreover, we aim to explore the analog of this result in the context of the Moore-Penrose Inverse, a widely-used generalized inverse matrix.
To achieve our goals, we employ the generalized inverse matrix associated with the Moore-Penrose Inverse. By utilizing this mathematical framework, we investigate the existence and uniqueness of solutions for both over-determined and under-determined linear systems, while maintaining alignment with the principles of the least squares method.
Through our research, we establish algebraic counterparts of the Gauss-Legendre theorem in the realm of complex Hilbert spaces. By leveraging the power of the Moore-Penrose Inverse, we provide insights into the solutions that minimize the objective function, shedding light on diverse scenarios encompassing singularity, rectangularity, and under-determination. This investigation contributes to a deeper understanding of the fundamental principles governing linear systems and paves the way for broader applications in various fields of science and engineering.
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