Documentation

This is machine translation

Translated by Microsoft
Mouseover text to see original. Click the button below to return to the English verison of the page.

Note: This page has been translated by MathWorks. Please click here
To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

sylvester

Solve Sylvester equation AX + XB = C for X

Syntax

X = sylvester(A,B,C)

Description

example

X = sylvester(A,B,C) returns the solution, X, to the Sylvester equation.

Input A is an m-by-m matrix, input B is an n-by-n matrix, and both C and X are m-by-n matrices.

Examples

collapse all

Create the coefficient matrices A and B.

A = [1 -1 1; 1 1 -1; 1 1 1];
B = magic(3);

Define C as the 3-by-3 identity matrix.

C = eye(3);

Use the sylvester function to solve the Sylvester equation for these values of A, B, and C.

X = sylvester(A,B,C)
X = 

    0.1223   -0.0725    0.0131
   -0.0806   -0.0161    0.1587
   -0.0164    0.1784   -0.1072

The result is a 3-by-3 matrix.

Create a 4-by-4 coefficient matrix, A, and 2-by-2 coefficient matrix, B.

A = [1 0 2 3; 4 1 0 2; 0 5 5 6; 1 7 9 0];
B = [0 -1; 1 0];

Define C as a 4-by-2 matrix to match the corresponding sizes of A and B.

C = [1 0; 2 0; 0 3; 1 1]
C = 

     1     0
     2     0
     0     3
     1     1

Use the sylvester function to solve the Sylvester equation for these values of A, B, and C.

X = sylvester(A,B,C)
X = 

    0.4732   -0.3664
   -0.4006    0.3531
    0.3305   -0.1142
    0.0774    0.3560

The result is a 4-by-2 matrix.

Input Arguments

collapse all

Input matrices, specified as matrices. Input A is an m-by-m square matrix, input B is an n-by-n square matrix, and input C is an m-by-n rectangular matrix. The function returns an error if any input matrix is sparse.

Data Types: single | double
Complex Number Support: Yes

Output Arguments

collapse all

Solution, returned as a matrix of the same size as C. The function returns an error if the eigenvalues of A and -B are not distinct (in this case, the solution, X, is singular or not unique).

More About

collapse all

Sylvester Equation

The Sylvester equation is

AX+XB=C.

The equation has a unique solution when the eigenvalues of A and -B are distinct. In terms of the Kronecker tensor product, , the equation is

[IA+BTI]X(:)=C(:),

where I is the identity matrix, and X(:) and C(:) denote the matrices X and C as single column vectors.

Introduced in R2014a

Was this topic helpful?