Documentation

# 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

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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 = 3×3 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 = 4×2 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 = 4×2 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

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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

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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).

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### Sylvester Equation

The Sylvester equation is

`$AX\text{​}+XB=C.$`

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

`$\left[I\otimes A+{B}^{T}\otimes I\right]X\left(:\right)=C\left(:\right),$`

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