Advertisement

Matrix Algebra pp 227-263 | Cite as

Matrix Transformations and Factorizations

  • James E. Gentle
Chapter
  • 5.9k Downloads
Part of the Springer Texts in Statistics book series (STS)

Abstract

In most applications of linear algebra, problems are solved by transformations of matrices. A given matrix (which represents some transformation of a vector) is itself transformed. The simplest example of this is in solving the linear system Ax = b, where the matrix A represents a transformation of the vector x to the vector b. The matrix A is transformed through a succession of linear operations until x is determined easily by the transformed A and the transformed b.

References

  1. Bindel, David, James Demmel, William Kahan, and Osni Marques. 2002. On computing Givens rotations reliably and efficiently. ACM Transactions on Mathematical Software 28:206–238.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bischof, Christian H., and Gregorio Quintana-Ortí. 1998a. Computing rank-revealing QR factorizations. ACM Transactions on Mathematical Software 24:226–253.Google Scholar
  3. Bischof, Christian H., and Gregorio Quintana-Ortí. 1998b. Algorithm 782: Codes for rank-revealing QR factorizations of dense matrices. ACM Transactions on Mathematical Software 24:254–257.Google Scholar
  4. Björck, Åke. 1996. Numerical Methods for Least Squares Problems. Philadelphia: Society for Industrial and Applied Mathematics.CrossRefzbMATHGoogle Scholar
  5. Harville, David A. 1997. Matrix Algebra from a Statistician’s Point of View. New York: Springer-Verlag.CrossRefzbMATHGoogle Scholar
  6. Hill, Francis S., Jr., and Stephen M Kelley. 2006. Computer Graphics Using OpenGL, 3rd ed. New York: Pearson Education.Google Scholar
  7. Hong, H. P., and C. T. Pan. 1992. Rank-revealing QR factorization and SVD. Mathematics of Computation 58:213–232.MathSciNetzbMATHGoogle Scholar
  8. Kim, Hyunsoo, and Haesun Park. 2008. Nonnegative matrix factorization based on alternating non-negativity-constrained least squares and the active set method. SIAM Journal on Matrix Analysis and Applications 30:713–730.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Lee, Daniel D., and H. Sebastian Seung. 2001. Algorithms for non-negative matrix factorization. Advances in Neural Information Processing Systems, 556–562. Cambridge, Massachusetts: The MIT Press.Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • James E. Gentle
    • 1
  1. 1.FairfaxUSA

Personalised recommendations