Hessenberg Matrix and Factorization

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Hessenberg Matrix and Factorization

xample 1.  Find the upper-Hessenberg form for the real symmetric matrix  .   Use the subroutine Hessenberg.

Solution 1.
Make the first Mathematica subroutine  Hessenberg1  active, then obtain the answer.

Module for
Hessenberg Matrix and Factorization
Background

   
    If A is symmetric then Householder's method can be used to reduce it to a similar symmetric tridiagonal matrix.  If A is nonsymmetric then Householder's method can be used to reduce it to a similar Hessenberg matrix.  These are the preliminary steps that are made before applying the QR method for finding eigenvalues of .

Definition  (Hessenberg Matrix) 

    An   matrix with    for is called a Hessenberg matrix.  The form of a Hessenberg matrix is

          

Definition  (Unitary Matrix) 

(i)     For real matrices, a unitary matrix is a matrix    for which  . 
   
(ii)    For complex matrices, a unitary matrix is a matrix    for which  . 

Theorem (Hessenberg Factorization of a Matrix)  There are two cases to consider. 

(iii)    Given a real matrix , there exists a unitary matrix and Hessenberg matrix so that 

        . 

(iv)     Given a complex matrix , there exists a unitary matrix and Hessenberg matrix so that 

        . 
Proof  Hessenberg Factorization

Theorem (Hessenberg Factorization of a Symmetric Matrix)  Given a real symmetric matrix , there exists a unitary matrix and tri-diagonal symmetric matrix so that

        . 

Remark.  This is the case that is easiest to illustrate in a first course in numerical methods.
Proof  Hessenberg Factorization

Theorem  If    then is similar to and the eigenvalues of are the same as the eigenvectors of .  

Remark.  The eigenvectors of are in general different from the eigenvectors of . 
Proof  Hessenberg Factorization

Computer Programs  Hessenberg Factorization
Program (Householder Reduction to Upper-Hessenberg Form).  To reduce the    real matrix    to a Hessenberg matrix form by using   Householder transformations.  The following version of the program uses "loops" extensively and is more traditional in programming structure.  It also contains a print statement so that you can watch the Householder transformations perform their "magic." 
Disclaimer.  The following subroutine is for pedagogical purposes only, it may not work for all cases. 

Program (Householder Reduction to Upper-Hessenberg Form).  To reduce the    real matrix    to a Hessenberg form form by using   Householder transformations.  The following version of the program uses "objects" extensively and is more like "object oriented programming."  
Disclaimer.  The following subroutine is for pedagogical purposes only, it may not work for all cases. 

Caveat.  The above subroutines will not reduce an    complex matrix to Hessenberg form.  You should use Mathematica's built in procedure  HessenbergDecomposition  if you use complex matrices.  If you do use Mathematica's built in procedure you will need to learn how software developers are permitted to dig into Mathematica's kernel and use its internal mathematical subroutines (i.e. picking Mathematica's brain).
Example 1.  Find the upper-Hessenberg form for the real symmetric matrix  .   Use the subroutine Hessenberg.
Solution 1.

Example 2.  Find the upper-Hessenberg form for the real symmetric matrix  . 
Use Mathematica's built in subroutine HessenbergDecomposition. 
Solution 2.

Example 3.  Find the upper-Hessenberg form for the real symmetric matrix  .   Explore the eigenvalues and eigenvectors. 
Solution 3.

Example 4.  Find the upper-Hessenberg form for the real matrix  .   Explore the eigenvalues.
Solution 4.

Example 5.  Find the upper-Hessenberg form for the real matrix  .   Explore the eigenvalues.
Solution 5.

Example 6.  Find the upper-Hessenberg form for the real matrix  .   Explore the eigenvalues.
Solution 6.

Example 7.  Find the upper-Hessenberg form for the real matrix  .   Explore the eigenvalues.
Solution 7.

Example 8.  Find the upper-Hessenberg form for the real matrix  .   Explore the eigenvalues.
Solution 8.

Exercise 9.   Find the upper-Hessenberg form for the real matrix    .   Explore the eigenvalues.
Solution 9.

Research Experience for Undergraduates
Hessenberg Factorization  Internet hyperlinks to web sites and a bibliography of articles. 

Download this Mathematica Notebook Hessenberg Factorization