Algebraic equivalence of matrix conjugate direction and matrix multistage filters for estimating random vectors

We consider matrix iterative subspace filters for solving minimum mean-squared error estimation problems in low-dimensional subspaces. Very general equivalences are established between matrix conjugate direction and matrix multistage filters, wherein the direction matrices of a matrix conjugate direction filter and the stagewise matrices of a matrix multistage filter are related through a one-term autoregressive recursion. By virtue of this recursion, the expanding subspaces of the two filters are identical, even though their bases for them are different. As a consequence, the subspace filters, gradient matrices, and error covariances in the respective filters are identical at each stage of the subspace iteration. If the matrix conjugate direction filter is a matrix conjugate gradient filter, then the equivalent stagewise filter is a matrix orthogonal multistage filter, and vice-versa. If either the matrix conjugate gradient filter or the matrix orthogonal multistage filter is initialized at the cross-covariance matrix between the signal and the measurement, then each of the matrix subspace filters iteratively turns out a basis for a Krylov subspace, which expands blockwise.