QR Decomposition

Given a Matrix A, its QR-decomposition is of the form

$${\hbox{\sf A}}= {\hbox{\sf Q}}{\hbox{\sf R}},$$

where ${\hbox{\sf R}}$ is an upper Triangular Matrix and ${\hbox{\sf Q}}$ is an Orthogonal Matrix, i.e., one satisfying

$${\hbox{\sf Q}}^{\rm T}{\hbox{\sf Q}}={\hbox{\sf I}},$$

where I is the Identity Matrix. This matrix decomposition can be used to solve linear systems of equations.

See also Cholesky Decomposition, LU Decomposition, Singular Value Decomposition

References

Householder, A. S. The Numerical Treatment of a Single Non-Linear Equations. New York: McGraw-Hill, 1970.

Nash, J. C. Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 26-28, 1990.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``QR Decomposition.'' §2.10 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 91-95, 1992.

Stewart, G. W. ``A Parallel Implementation of the QR Algorithm.'' Parallel Comput. 5, 187-196, 1987. ftp://thales.cs.umd.edu/pub/reports/piqra.ps.