Abstract
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.
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Notes
- 1.
Show that we have equality ⇔ R is diagonal ⇔ A ∗A is diagonal.
- 2.
Consider the matrix Q TA −.
References
Å. Björck, Numerical Methods in Matrix Computations (Springer, 2015)
G.H. Golub, C.F. Van Loan, Matrix Computations, 4th Edition (The John Hopkins University Press, Baltimore, MD, 2013)
G.W. Stewart, Matrix Algorithms Volume I: Basic Decompositions (SIAM, Philadelphia, 1998)
J.H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon Press, Oxford, 1965)
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Lyche, T. (2020). Orthonormal and Unitary Transformations. In: Numerical Linear Algebra and Matrix Factorizations. Texts in Computational Science and Engineering, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-030-36468-7_5
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DOI: https://doi.org/10.1007/978-3-030-36468-7_5
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