# Schur Decomposition (G Dataflow)

Version:
Last Modified: January 9, 2017

Performs the Schur decomposition of a square matrix.  ## matrix A

A square real matrix.

This input accepts a 2D array of double-precision, floating point numbers or 2D array of complex double-precision, floating point numbers. ## compute Schur vectors?

A Boolean that determines whether this node calculates the orthogonal matrix.

 True Calculates the orthogonal matrix and the block upper triangular matrix in real Schur form. False Calculates only the block upper triangular matrix in real Schur form.

Default: False ## order

Method to order the eigenvalues and the corresponding triangular matrix in real Schur form and the orthogonal matrix.

Name Value Description
No Reorder 0 Does not change the order of the eigenvalues.
Real Ascending 1 Lists the eigenvalues in ascending order according to their real parts.
Real Descending 2 Lists the eigenvalues in descending order according to their real parts.
Magnitude Ascending 3 Lists the eigenvalues in ascending order according to their magnitudes.
Magnitude Descending 4 Lists the eigenvalues in descending order according to their magnitudes.

Default: No Reorder ## error in

Error conditions that occur before this node runs. The node responds to this input according to standard error behavior.

Default: No error ## Schur form

The block upper triangular matrix in real Schur form. ## Schur vectors

The orthogonal matrix. ## eigenvalues

A complex vector that contains all the computed eigenvalues of the input matrix. ## error out

Error information. The node produces this output according to standard error behavior.

## Algorithm for Calculating Schur Decomposition

The following expression defines the Schur decomposition of a square n × n matrix A.

A = QSQH

where S is in Schur form and QH is the conjugate transpose of matrix Q.

For a real matrix A, Q is an n × n orthogonal matrix. S is a block upper triangular matrix in real Schur form, whose elements on the main diagonal are all 1 × 1 or 2 × 2 blocks, as shown in the following matrix.

$S=\left[\begin{array}{cccc}{S}_{11}& {S}_{12}& \dots & {S}_{1m}\\ 0& {S}_{22}& \ddots & {S}_{2m}\\ ⋮& \ddots & \ddots & ⋮\\ 0& 0& \dots & {S}_{mm}\end{array}\right]$

where Sii are square blocks of dimension 1 or 2 and i = 1, 2, ..., m.

For a complex matrix A, Q is an n × n unitary matrix. S is an upper triangular matrix in complex Schur form.

Where This Node Can Run:

Desktop OS: Windows

FPGA: Not supported