Decay Rates for Inverses of Band Matrices

Some Observations on Inverses of Band Matrices and Low Rank Perturbations of Triangular Matrices

The reverse order law for Moore-Penrose inverses of operators on Hilbert C*-modules

Suppose $T$ and $S$ are Moore-Penrose invertible operators betweenHilbert C*-module. Some necessary and sufficient conditions are given for thereverse order law $(TS)^{ dag} =S^{ dag} T^{ dag}$ to hold.In particular, we show that the equality holds if and only if $Ran(T^{*}TS) subseteq Ran(S)$ and $Ran(SS^{*}T^{*}) subseteq Ran(T^{*}),$ which was studied first by Greville [{it SIAM Rev. 8 (1966...

Perturbation bounds for $g$-inverses with respect to the unitarily invariant norm

Let complex matrices $A$ and $B$ have the same sizes. Using the singular value decomposition, we characterize the $g$-inverse $B^{(1)}$ of $B$ such that the distance between a given $g$-inverse of $A$ and the set of all $g$-inverses of the matrix $B$ reaches minimum under the unitarily invariant norm. With this result, we derive additive and multiplicative perturbation bounds of the nearest per...

Wavelet Sparse Approximate Inverse Preconditioners

There is an increasing interest in using sparse approximate inverses as preconditioners for Krylov subspace iterative methods. Recent studies of Grote and Huckle [21] and Chow and Saad [11] also show that sparse approximate inverse preconditioner can be effective for a variety of matrices, e.g. Harwell-Boeing collections. Nonetheless a drawback is that it requires rapid decay of the inverse ent...

Finding left inverses for operators on l(Z) with polynomial decay

We study the left-invertibility of infinite matrices indexed by metric spaces with polynomial growth. Under different conditions on the rows and the columns, we prove that being bounded-below in lp for some 1 ≤ p ≤ ∞, implies that there is a left-inverse which is bounded in lq, for all 1 ≤ q ≤ ∞. In particular, this applies to matrices with polynomial decay, indexed by discrete groups of polyno...