# scipy.linalg.solve_sylvester

scipy.linalg.solve_sylvester(a, b, q)

  Computes a solution (X) to the Sylvester equation \(AX + XB = Q\).

  Parameters:
    a(M, M) array_like

      Leading matrix of the Sylvester equation

    b(N, N) array_like

      Trailing matrix of the Sylvester equation

    q(M, N) array_like

      Right-hand side

  Returns:
    x(M, N) ndarray

      The solution to the Sylvester equation.

  Raises:
    LinAlgError

      If solution was not found

  Notes

  Computes a solution to the Sylvester matrix equation via the Bartels- Stewart algorithm. The A and B matrices first undergo Schur decompositions. The resulting matrices are used to construct an alternative Sylvester equation (`RY + YS^T = F`) where the R and S matrices are in quasi-triangular form (or, when R, S or F are complex, triangular form). The simplified equation is then solved using `*TRSYL` from LAPACK directly.

  New in version 0.11.0.

  Examples

  Given a, b, and q solve for x:

      >>> import numpy as np
      >>> from scipy import linalg
      >>> a = np.array([[-3, -2, 0], [-1, -1, 3], [3, -5, -1]])
      >>> b = np.array([[1]])
      >>> q = np.array([[1],[2],[3]])
      >>> x = linalg.solve_sylvester(a, b, q)
      >>> x
      array([[ 0.0625],
             [-0.5625],
             [ 0.6875]])
      >>> np.allclose(a.dot(x) + x.dot(b), q)
      True