# Sylvester Equations (G Dataflow)

Version:

Solves the Sylvester matrix equation.

## matrix type

Type of matrix A and matrix B.

Specifying the matrix type allows this node to execute more quickly by avoiding unnecessary computations, which could introduce numerical inaccuracy.

Name Value Description
General 0 The input matrix is a matrix that you cannot describe with one of the other categories.
Upper Triangular 3 The input matrix is upper triangular.

Default: Upper Triangular

## matrix A

Matrix A in the Sylvester equation.

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

This input must be a square matrix or upper quasi-triangular matrix in canonical Schur form.

## matrix B

Matrix B in the Sylvester equation.

This input accepts the following data types:

• 2D array of double-precision, floating point numbers
• 2D array of complex double-precision, floating point numbers

This input must be a square matrix or upper quasi-triangular matrix in canonical Schur form.

## matrix C

Matrix C in the Sylvester equation.

## sign

Form of the Sylvester equation.

Name Value Description
Plus 0 op(A)X + X op(B) = aC, where A is the input matrix A, B is the input matrix B, and C is the input matrix C.
Minus 1 op(A)X - X op(B) = aC, where A is the input matrix A, B is the input matrix B, and C is the input matrix C.

Default: Plus

## error in

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

Default: No error

## operation A

Operation this node performs on matrix A in the Sylvester equation.

Name Value Description
Not Transposed 0 op(A) = A, where A is the input matrix A.
Transposed 1 op(A) = transpose of A, where A is the input matrix A.

Default: Not Transposed

## operation B

Operation this node performs on matrix B in the Sylvester equation.

Name Value Description
Not Transposed 0 op(B) = B, where B is the input matrix B.
Transposed 1 op(B) = transpose of B, where B is the input matrix B.

Default: Not Transposed

## matrix X

Solution of the Sylvester equation.

## scale

Scaling factor a of the Sylvester equation.

## perturbed

A Boolean that specifies whether this node uses perturbed values to solve the equation.

 True Uses perturbed values to solve the Sylvester equation. The eigenvalues of matrix A and matrix B are common or close and indicate the solution of the Sylvester equation is not unique. False Does not use perturbed values to solve the Sylvester equation.

## error out

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

## Solving the Sylvester Matrix Equation

The following equations define the Sylvester matrix equation:

op(A)X + Xop(B) = aC

or

op(A)X - Xop(B) = aC

where

• op(A) is A or the conjugate transpose of A
• op(B) is B or the conjugate transpose of B
• a is a scaling factor to avoid overflow in X

The Sylvester matrix equation has a unique solution if and only if λ ± β ≠ 0, where λ and β are the eigenvalues of A and B, respectively, and the sign (±) depends on the equation you want to solve. When the solution of the Sylvester matrix equation is not unique, this node sets perturbed to True and might not return the correct solution.

Where This Node Can Run:

Desktop OS: Windows

FPGA: Not supported