Schur Decomposition VI

Owning Palette: Linear Algebra VIs

Requires: Full Development System

Performs the Schur decomposition of a square matrix. Wire data to the Input Matrix input to determine the polymorphic instance to use or manually select the instance.

Details  

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Select an instance Real Schur DecompositionComplex Schur Decomposition

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Real Schur Decomposition

	Input Matrix must be a square real matrix.
	compute Schur vectors? specifies whether the VI calculates Schur Vectors. The default is FALSE.
	order specifies how to order the Eigenvalues and the corresponding Schur Form and Schur Vectors. 0 No Reorder (default)—Does not change the order of the Eigenvalues. 1 Real Ascending—Lists the Eigenvalues in ascending order according to their real parts. 2 Real Descending—Lists the Eigenvalues in descending order according to their real parts. 3 Magnitude Ascending—Lists the Eigenvalues in ascending order according to their magnitudes. 4 Magnitude Descending—Lists the Eigenvalues in descending order according to their magnitudes.
	Schur Form returns the block upper triangular matrix in real Schur form.
	Schur Vectors returns the orthogonal matrix.
	Eigenvalues returns a complex vector that contains all the computed eigenvalues of Input 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 Schur Decomposition

	Input Matrix must be a square complex matrix.
	compute Schur vectors? specifies whether the VI calculates Schur Vectors. The default is FALSE.
	order specifies how to order the Eigenvalues and the corresponding Schur Form and Schur Vectors. 0 No Reorder (default)—Does not change the order of the Eigenvalues. 1 Real Ascending—Lists the Eigenvalues in ascending order according to their real parts. 2 Real Descending—Lists the Eigenvalues in descending order according to their real parts. 3 Magnitude Ascending—Lists the Eigenvalues in ascending order according to their magnitudes. 4 Magnitude Descending—Lists the Eigenvalues in descending order according to their magnitudes.
	Schur Form returns the upper triangular matrix.
	Schur Vectors returns the unitary matrix.
	Eigenvalues returns a complex vector that contains all the computed eigenvalues of Input 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.

Schur Decomposition Details

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

A = QSQ^H

where S is in Schur form, and Q^H is the conjugate transpose of matrix Q.

Real Matrix

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.

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

Complex Matrix

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