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Householder Matrix

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Householder (1953) first considered the matrix that now bears his name in the first couple of pages of his book. A Householder matrix for a real vector v can be implemented in the Wolfram Language as: 

  HouseholderMatrix[v_?VectorQ] :=
    IdentityMatrix[Length[v]]
      - 2 Transpose[{v}] . {v} / (v.v)

Trefethen and Bau (1997) gave an incorrect version of the formula for complex v. D. Laurie gave a correct version by interpreting reflection along a given direction v not as

Hx=x-2av,
(1)

where

Px=x-av
(2)

is the projection onto the hyperplane orthogonal to v (since this is in general not a unitary transformation), but as

Hx=x-(a+a^(H))v.
(3)

Lehoucq (1996) independently gave an interpretation that still uses the formula Hx=x-2av, but choosing v to be unitary.

See also

QR Decomposition

Portions of this entry contributed by Dirk Laurie

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References

Bock, R. K. and Krischer, W. "Householder Transformation." In The Data Analysis Briefbook. http://rkb.home.cern.ch/rkb/AN16pp/node123.html.Householder, A. S. Principles of Numerical Analysis. New York: McGraw-Hill, pp. 135-138, 1953.Lehoucq, R. B. "The Computation of Elementary Unitary Matrices." ACM Trans. Math. Software 22, 393-400, 1996.Lepikult, T. "Householder Reflection." http://www.cs.ut.ee/~toomas_l/linalg/lin2/node6.html.Trefethen, L. N. and Bau, D. III. Numerical Linear Algebra. Philadelphia, PA: SIAM, 1997.

Referenced on Wolfram|Alpha

Householder Matrix

Cite this as:

Laurie, Dirk and Weisstein, Eric W. "Householder Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HouseholderMatrix.html

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