A matrix pseudo-inversion lemma for positive semidefinite hermitian matrices and its application to adaptive blind deconvolution of MIMO systems

In the simplest case, the matrix inversion lemma gives an explicit formula of the inverse of a positive-definite matrix A added to a rank-one matrix bb(H) as follows: (A + bb(H))(-1) = A(-1) - A(-1) b(1 + b(H) A(-1)b)(-1)b(H) A(-1). It is well known in the literature that this formula is very useful to develop a recursive least-squares algorithm for the recursive identification of linear systems or the design of adaptive filters. We extend this result to the case when the matrix A is singular and present a matrix pseudo-inversion lemma along with some illustrative examples. Such a singular case may occur in a situation where a given problem is overdetermined in the sense that it has more equations than unknowns. This lemma is important in its own right, but in order to show the usefulness of the lemma, we apply it to develop an adaptive superexponential algorithm for the blind deconvolution of multi-input multi-output systems.