# A QLP Decomposition via Randomization

This paper is concerned with full matrix decomposition of matrices, primarily low-rank matrices. It develops a QLP-like decomposition algorithm such that when operating on a matrix A, gives A = QLP^T , where Q and P are orthonormal, and L is lower-triangular. The proposed algorithm, termed Rand-QLP, utilizes randomization and the unpivoted QR decomposition. This in turn enables Rand-QLP to leverage modern computational architectures, thus addressing a serious bottleneck associated with classical and most recent matrix decomposition algorithms. We derive several error bounds for Rand- QLP: bounds for the first k approximate singular values as well as the trailing block of the middle factor, which show that Rand-QLP is rank-revealing; and bounds for the distance between approximate subspaces and the exact ones for all four fundamental subspaces of a given matrix. We assess the speed and approximation quality of Rand-QLP on synthetic and real matrices with different dimensions and characteristics, and compare our results with those of multiple existing algorithms.

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