Orthonormal and Unitary Transformations

  • Tom Lyche
Part of the Texts in Computational Science and Engineering book series (TCSE, volume 22)


In Gaussian elimination and LU factorization we solve a linear system by transforming it to triangular form. These are not the only kind of transformations that can be used for such a task. Matrices with orthonormal columns, called unitary matrices can be used to reduce a square matrix to upper triangular form and more generally a rectangular matrix to upper triangular (also called upper trapezoidal) form. This lead to a decomposition of a rectangular matrix known as a QR decomposition and a reduced form which we refer to as a QR factorization. The QR decomposition and factorization will be used in later chapters to solve least squares- and eigenvalue problems.


  1. 2.
    Å. Björck, Numerical Methods in Matrix Computations (Springer, 2015)Google Scholar
  2. 3.
    G.H. Golub, C.F. Van Loan, Matrix Computations, 4th Edition (The John Hopkins University Press, Baltimore, MD, 2013)zbMATHGoogle Scholar
  3. 17.
    G.W. Stewart, Matrix Algorithms Volume I: Basic Decompositions (SIAM, Philadelphia, 1998)CrossRefGoogle Scholar
  4. 21.
    J.H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon Press, Oxford, 1965)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Tom Lyche
    • 1
  1. 1.Blindern, University of OsloOsloNorway

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