PLUS Factorization of Matrices


PLUS factorization was proposed as a new framework of matrix factorization, A=PLUS, where matrices P, L and U are almost the same as in LU factorization, permutation, unit lower and upper triangular matrices, respectively, and S is a very special matrix, which is unit, lower and triangular, but only with a small number of non-zeros. Different from LU factorization, all the diagonal elements of U in PLUS factorization are customizable, i.e., the elements can be assigned by users almost freely. With PLUS factorization, the matrix A is easily factorized further into a series of special matrices similar to S. The new computational mechanics with PLUS factorization has a few very elegant and promising properties that other factorizations do not have, such as in-place computation and simple inversion. PLUS factorization also allows of transforming integers reversibly and losslessly if the diagonal elements of U are all designated as 1, -1, i, or - i.


The theory was mainly published in the following two papers:


* Pengwei Hao, "Customizable Triangular Factorizations of Matrices", Linear Algebra and Its Applications, Vol. 382, pp. 135-154, May 2004.


* Pengwei Hao and Qingyun Shi, "Matrix Factorization for Reversible Integer Mapping", IEEE Transactions on Signal Processing, Vol. 49, No. 10, pp. 2314-2324, Oct. 2001.


It has applications in lossless source coding, fast image registration and fast volumetric data rendering.


All our publications are downloadable on

Our programs in C (its EXE version) and MATLAB are also available for research ONLY, and please cite the above papers in your publications.


For any commercial applications, a written permission MUST be obtained from the authors.