# sylvester#

## Purpose#

Computes the solution to the Sylvester matrix equation, $$AX + XB = C$$.

## Format#

X = sylvester(A, B, C)#
Parameters:
• A (MxM real or complex matrix) – data

• B (NxN real or complex matrix) – data

• C (MxN real or complex matrix) – data

Returns:

X (MxN matrix) – solution to the equation $$AX + XB = C$$.

## Examples#

### Real input#

// Create a 3 x 3 real matrix
A = {  0.9069  -0.3150  -0.9732,
0.6023   0.6848   0.4925,
-0.8555  -0.7430   0.6521 };

// Create a 2 x 2 real matrix
B = { -0.9876   0.4503,
-0.3043   0.9807 };

// Create a 3 x 2 real matrix
C = { -0.8625   0.5247,
0.6331  -0.3334,
0.7912   0.0711 };

// Solve the Sylvester matrix equation
X = sylvester(A, B, C);


After the code above, X will equal the 3x2 matrix:

X =  -0.4279   0.3246
-1.0525  -0.0013
1.1609  -0.1071


### Complex input#

// Create a 3 x 3 complex matrix
A = { 7 + 7i     4 + 10i     2 + 8i,
10 - 3i    -7 -  5i   -10 - 7i,
3 + 5i   -10 -  2i     2 - 4i };

// Create a 2 x 2 complex matrix
B = { 5 +  1i    -5 - 8i,
8 - 10i    8 - 1i };

// Create a 3 x 2 complex matrix
C = { -9 - 3i   -1 - 1i,
9 - 8i   -5 + 8i,
-1 - 2i   -5 + 5i };

// Solve the Sylvester matrix equation
X = sylvester(A, B, C);


After the code above, X will equal the 3x2 complex matrix:

X =   0.1697 - 0.2242i    -0.5923 + 0.2221i
-0.5684 + 0.4562i     0.3670 - 0.7153i
-0.7502 + 0.2470i    -0.0636 - 0.4208i


## Remarks#

The equation $$AX + XB = C$$ will not have a unique solution if the eigenvalues of the matrices A and -B are equal. In this case an error will be returned.