This information is part of the Modelica Standard Library maintained by the Modelica Association.

Syntax

  As = Matrices.householderSimilarityTransformation(A,u);

Description

This function computes the Householder similarity transformation

As = S*A*S

with

S = I -2*u*u'/(u'*u).

This transformation is widely used for transforming non-symmetric matrices to a Hessenberg form.

Example

// First step of Hessenberg decomposition
  import   Modelica.Math.Vectors.Utilities;

  Real A[4,4] = [1,2,3,4;
                 3,4,5,6;
                 9,8,7,6;
                 1,2,0,0];
  Real Ar[4,4];
  Real u[4]={0,0,0,0};

  u[2:4]=Utilities.householderVector(A[2:4,1],{1,0,0});
  // u= = {0, 0.8107, 0.5819, 0.0647}

  Ar=householderSimilarityTransformation(A,u);
 //  Ar = [1.0,     -3.8787,    -1.2193,    3.531;
          -9.5394, 11.3407,      6.4336,   -5.9243;
           0.0,     3.1307,      0.7525,   -3.3670;
           0.0,     0.8021,     -1.1656,   -1.0932]

See also

Matrices.Utilities.householderReflection,
Vectors.Utilities.householderReflection,
Vectors.Utilities.householderVector

SAS = householderSimilarityTransformation(A, u)

A	Type: Real[:,size(A, 1)] Description: Square matrix A
u	Type: Real[size(A, 1)] Description: Householder vector

SAS	Type: Real[size(A, 1),size(A, 1)] Description: Transformation of matrix A