This research introduces a row compression and nested product decomposition of an n × n hierarchical representation of a rank structured matrix A which extends the compression and nested product decomposition of quasiseparable matrices. The row compression is comprised of a sequence of small Householder transformations which are formed from the low-rank off-diagonal blocks of the hierarchical representation of a rank structured matrix. The row compression forms a factorization of matrix A = QC, where Q is the product of the Householder transformations, and C preserves the low-rank structure in both the lower and upper triangular parts of the matrix A. The nested product decomposition is formed after applying a sequence of unitary transformations to the compressed matrix C. By combining a fast matrix-vector product and system solver, linear systems involving the decomposition are directly solved with linear complexity. An application of hierarchically rank structured matrices in image processing will be shown.

Committee
Dr. Michael Stewart (co-chair)
Dr. Saeid Belkasim (co-chair)
Dr. Yi Pan
Dr. Raj Sunderraman
Dr. Jon Preston