# Hessenberg Decomposition (G Dataflow)

Performs the Hessenberg decomposition of a matrix.

## matrix A

An n x n real matrix.

This input accepts a 2D array of double-precision, floating point numbers or 2D array of complex double-precision, floating point numbers.

## index low

The form of the balanced matrix.

You can obtain this input from the Matrix Balance node. If you do not wire this input, this node sets index low to 0.

Default: -1

## index high

The form of the balanced matrix.

You can obtain this input from the Matrix Balance node. If you do not wire this input, this node sets index low to n - 1.

Default: -1

## error in

Error conditions that occur before this node runs.

The node responds to this input according to standard error behavior.

Standard Error Behavior

Many nodes provide an error in input and an error out output so that the node can respond to and communicate errors that occur while code is running. The value of error in specifies whether an error occurred before the node runs. Most nodes respond to values of error in in a standard, predictable way.

error in does not contain an error error in contains an error
If no error occurred before the node runs, the node begins execution normally.

If no error occurs while the node runs, it returns no error. If an error does occur while the node runs, it returns that error information as error out.

If an error occurred before the node runs, the node does not execute. Instead, it returns the error in value as error out.

Default: No error

## Hessenberg form H

An n x n matrix in Hessenberg form.

## orthogonal matrix Q

An n x n orthogonal matrix.

## error out

Error information.

The node produces this output according to standard error behavior.

Standard Error Behavior

Many nodes provide an error in input and an error out output so that the node can respond to and communicate errors that occur while code is running. The value of error in specifies whether an error occurred before the node runs. Most nodes respond to values of error in in a standard, predictable way.

error in does not contain an error error in contains an error
If no error occurred before the node runs, the node begins execution normally.

If no error occurs while the node runs, it returns no error. If an error does occur while the node runs, it returns that error information as error out.

If an error occurred before the node runs, the node does not execute. Instead, it returns the error in value as error out.

## Algorithm for Calculating Hessenberg Decomposition

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

A = Q H Q 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
• 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.

$H=\left[\begin{array}{ccccc}{h}_{11}& {h}_{12}& \dots & \dots & {h}_{1n}\\ {h}_{21}& {h}_{22}& \ddots & \ddots & ⋮\\ 0& {h}_{32}& \ddots & \ddots & ⋮\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ 0& \dots & 0& {h}_{n,\left(n-1\right)}& {h}_{n,n}\end{array}\right]$

Where This Node Can Run:

Desktop OS: Windows

FPGA: Not supported

Web Server: Not supported in VIs that run in a web application