Hessenberg Factorization

matrix with

for

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 for which .

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

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

.

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.

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Numerical Analysis

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