Representations and computation of the BT inverse and its randomized algorithms

This text explores characterizations of the BT inverse and proposes several methods for calculating the BT inverse. One of the representations examines the invertibility of an appropriate bordered matrix whose lower right block coincides with the BT inverse. Considerations of the BT inverse in relation to the group inverse and the standard matrix inverse are discussed. It is verified that the BT inverse solution is the unique solution to a specific class of constrained singular linear equations. Utilizing Cramer’s rule, we derive elementnwise solution of such systems. We also develop iterative methods for calculating the BT inverse and provide a criterion for the convergence of these methods. Modifications of the successive matrix square algorithm and the Newton iterative method are explored. The Singular Value Decomposition (SVD) approach for calculating the matrix BT inverse is developed. Additionally, the corresponding adopted Randomized SVD (RSVD) algorithm offers enhanced performance, making it a strong choice for large-scale data practical applications.