Partition-Matrix Theory Applied to the Computation of Generalized-Inverses for MIMO Systems in Rayleigh Fading Channels

Partition-Matrix Theory and Generalized-Inverses are interesting topics explored in linear algebra and matrix computation. Partition-Matrix Theory is associated with the problem of properly partitioning a matrix into block matrices (i.e. an array of matrices), and is a matrix computation tool widely employed in several scientific-technological application areas. For instance, blockwise Toeplitz-based covariance matrices are used to model structural proper‐ ties for space-time multivariate adaptive processing in radar applications [1], Jacobian re‐ sponse matrices are partitioned into several block-matrix instances in order to enhance medical images for Electrical-Impedance-Tomography [2], design of state-regulators and partial-observers for non-controllable/non-observable linear continuous systems contem‐ plates matrix blocks for controllable/non-controllable and observable/non-observable eigen‐ values [3]. The Generalized-Inverse is a common and natural problem found in a vast of applications. In control robotics, non-collocated partial linearization is applied to underactu‐ ated mechanical systems through inertia-decoupling regulators which employ a pseudoin‐ verse as part of a modified input control law [4]. At sliding-mode control structures, a RightPseudoinverse is incorporated into a state-feedback control law in order to stabilize electromechanical non-linear systems [5]. Under the topic of system identification, definition of a Left-Pseudoinverse is present in auto-regressive moving-average models (ARMA) for matching dynamical properties of unknown systems [6]. An interesting approach arises whenever Partition-Matrix Theory and Generalized-Inverse are combined together yielding attractive solutions for solving the problem of block matrix inversion [7-10]. Nevertheless,

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