Fast transforms: Banded matrices with banded inverses

Abstract

It is unusual for both A and A −1 to be banded-but this can be a valuable property in applications. Block-diagonal matrices F are the simplest examples; wavelet transforms are more subtle. We show that every example can be factored into A ¼ F 1 …F N where N is controlled by the bandwidths of A and A −1 (but not by their size, so this extends to infinite matrices and leads to new matrix groups).Bruhat | CMV matrix | factorization | wavelet | permutation

Groups of banded matrices with banded inverses

“…Suppose A ∞ and its inverse have bandwidth w = N , coming from N blocks centered on each pair of rows. We showed in [10] how to find N block diagonal factors in A ∞ = F 1 . .…”

“…At each step we move from left to right, exchanging pairs of neighbors that are in the wrong order: We conjectured in [10] that N ≤ 2w −1 in all cases. A beautiful proof is given by Greta Panova [6], using the "wiring diagram" to decide the sequence of exchanges in advance.…”

“…Those are block Toeplitz matrices in the wavelet case, and "CMV matrices" in other cases. Our earlier paper [10] applied an observation of Ilya Spitkovsky to separate these matrices into the minimum number N = w of factors F (each with bandwidth 1). We believe that these special (and short) factorizations can lead to useful algorithms.…”

“…Each zero is produced by a "Gauss matrix" that has a single nonzero next to its diagonal part I. The O w 3 estimate in [10] was over-generous.…”

“…Products of block diagonal matrices give many more examples (including useful ones). The main theorem in an earlier paper [10] is that all examples are produced this way, from multiplying invertible block diagonal matrices of bandwidth 1.…”