Monday, February 1, 2010

Hessenberg Factorization

An [Graphics:Images/HessenbergMod_gr_2.gif] matrix with [Graphics:Images/HessenbergMod_gr_3.gif] for [Graphics:Images/HessenbergMod_gr_4.gif] is called a Hessenberg matrix. The form of a Hessenberg matrix is

[Graphics:Images/HessenbergMod_gr_5.gif]

Unitary Matrix

(i) For real matrices, a unitary matrix is a matrix [Graphics:Images/HessenbergMod_gr_6.gif] for which [Graphics:Images/HessenbergMod_gr_7.gif].

(ii) For complex matrices, a unitary matrix is a matrix [Graphics:Images/HessenbergMod_gr_8.gif] for which [Graphics:Images/HessenbergMod_gr_9.gif].

Hessenberg Factorization of a Matrix:

There are two cases to consider.

(iii) Given a real matrix [Graphics:Images/HessenbergMod_gr_10.gif], there exists a unitary matrix [Graphics:Images/HessenbergMod_gr_11.gif] and Hessenberg matrix [Graphics:Images/HessenbergMod_gr_12.gif] so that

[Graphics:Images/HessenbergMod_gr_13.gif].

(iv) Given a complex matrix [Graphics:Images/HessenbergMod_gr_14.gif], there exists a unitary matrix [Graphics:Images/HessenbergMod_gr_15.gif] and Hessenberg matrix [Graphics:Images/HessenbergMod_gr_16.gif] so that

[Graphics:Images/HessenbergMod_gr_17.gif].


Hessenberg Factorization of a Symmetric Matrix:

Given a real symmetric matrix [Graphics:Images/HessenbergMod_gr_18.gif], there exists a unitary matrix [Graphics:Images/HessenbergMod_gr_19.gif] and tri-diagonal symmetric matrix [Graphics:Images/HessenbergMod_gr_20.gif] so that

[Graphics:Images/HessenbergMod_gr_21.gif].

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

No comments:

Post a Comment

Introductory Methods of Numerical Analysis