Hessenberg Decomposition VI

Owning Palette: Linear Algebra VIs

Requires: Full Development System

Performs the Hessenberg decomposition of Input Matrix. Wire data to the Input Matrix input to determine the polymorphic instance to use or manually select the instance.

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Select an instance Real Hessenberg DecompositionComplex Hessenberg Decomposition

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Real Hessenberg Decomposition

	Input Matrix is an n × n real matrix.
	index low is the index low value from the Matrix Balance VI. If you balance Input Matrix with the Matrix Balance VI, wire the index low output of the Matrix Balance VI to this input. If the index low input of this VI equals –1 (default), the VI uses 0 for index low.
	index high is the index high value from the Matrix Balance VI. If you balance Input Matrix with the Matrix Balance VI, wire the index high output of the Matrix Balance VI to this input. If the index high input of this VI equals –1 (default), the VI uses n – 1 for index high.
	Hessenberg Form H returns an n × n matrix in Hessenberg form.
	Orthogonal Matrix Q returns the n × n orthogonal matrix.
	error returns any error or warning from the VI. You can wire error to the Error Cluster From Error Code VI to convert the error code or warning into an error cluster.

Complex Hessenberg Decomposition

	Input Matrix is an n × n complex matrix.
	index low is the index low value from the Matrix Balance VI. If you balance Input Matrix with the Matrix Balance VI, wire the index low output of the Matrix Balance VI to this input. If the index low input of this VI equals –1 (default), the VI uses 0 for index low.
	index high is the index high value from the Matrix Balance VI. If you balance Input Matrix with the Matrix Balance VI, wire the index high output of the Matrix Balance VI to this input. If the index high input of this VI equals –1 (default), the VI uses n – 1 for index high.
	Hessenberg Form H returns an n × n matrix in Hessenberg form.
	Orthogonal Matrix Q returns the n × n unitary matrix.
	error returns any error or warning from the VI. You can wire error to the Error Cluster From Error Code VI to convert the error code or warning into an error cluster.

Hessenberg Decomposition Details

The following expression defines the Hessenberg decomposition of an n × n matrix A:

A = QHQ^H

where Q is an orthogonal matrix when matrix A is a real matrix and a unitary matrix when matrix A is a complex matrix, Q^H is the conjugate transpose of matrix Q, and H is a Hessenberg matrix.

By definition, a Hessenberg matrix is a matrix with zeros under the main subdiagonal, as shown by the following matrix.