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scipy.linalg.

# eig

scipy.linalg.eig(a, b=None, left=False, right=True, overwrite_a=False, overwrite_b=False, check_finite=True, homogeneous_eigvals=False)

  Solve an ordinary or generalized eigenvalue problem of a square matrix.

  Find eigenvalues w and right or left eigenvectors of a general matrix:

      a   vr[:,i] = w[i]        b   vr[:,i]
      a.H vl[:,i] = w[i].conj() b.H vl[:,i]

  where `.H` is the Hermitian conjugation.

  The documentation is written assuming array arguments are of specified “core” shapes. However, array argument(s) of this function may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details. Note that calls with zero-size batches are unsupported and will raise a `ValueError`.

  Parameters:
    a(M, M) array_like

      A complex or real matrix whose eigenvalues and eigenvectors will be computed.

    b(M, M) array_like, optional

      Right-hand side matrix in a generalized eigenvalue problem. Default is None, identity matrix is assumed.

    left bool, optional

      Whether to calculate and return left eigenvectors. Default is False.

    right bool, optional

      Whether to calculate and return right eigenvectors. Default is True.

    overwrite_a bool, optional

      Whether to overwrite a; may improve performance. Default is False.

    overwrite_b bool, optional

      Whether to overwrite b; may improve performance. Default is False.

    check_finite bool, optional

      Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

    homogeneous_eigvals bool, optional

      If True, return the eigenvalues in homogeneous coordinates. In this case `w` is a (2, M) array so that:

          w[1,i] a vr[:,i] = w[0,i] b vr[:,i]

      Default is False.

  Returns:
    w(M,) or (2, M) double or complex ndarray

      The eigenvalues, each repeated according to its multiplicity. The shape is (M,) unless `homogeneous_eigvals=True`.

    vl(M, M) double or complex ndarray

      The left eigenvector corresponding to the eigenvalue `w[i]` is the column `vl[:,i]`. Only returned if `left=True`. The left eigenvector is not normalized.

    vr(M, M) double or complex ndarray

      The normalized right eigenvector corresponding to the eigenvalue `w[i]` is the column `vr[:,i]`. Only returned if `right=True`.

  Raises:
    LinAlgError

      If eigenvalue computation does not converge.

  See also

  `eigvals`

    eigenvalues of general arrays

  `eigh`

    Eigenvalues and right eigenvectors for symmetric/Hermitian arrays.

  `eig_banded`

    eigenvalues and right eigenvectors for symmetric/Hermitian band matrices

  `eigh_tridiagonal`

    eigenvalues and right eigenvectors for symmetric/Hermitian tridiagonal matrices

  Examples

      >>> import numpy as np
      >>> from scipy import linalg
      >>> a = np.array([[0., -1.], [1., 0.]])
      >>> linalg.eigvals(a)
      array([0.+1.j, 0.-1.j])

      >>> b = np.array([[0., 1.], [1., 1.]])
      >>> linalg.eigvals(a, b)
      array([ 1.+0.j, -1.+0.j])

      >>> a = np.array([[3., 0., 0.], [0., 8., 0.], [0., 0., 7.]])
      >>> linalg.eigvals(a, homogeneous_eigvals=True)
      array([[3.+0.j, 8.+0.j, 7.+0.j],
             [1.+0.j, 1.+0.j, 1.+0.j]])

      >>> a = np.array([[0., -1.], [1., 0.]])
      >>> linalg.eigvals(a) == linalg.eig(a)[0]
      array([ True,  True])
      >>> linalg.eig(a, left=True, right=False)[1] # normalized left eigenvector
      array([[-0.70710678+0.j        , -0.70710678-0.j        ],
             [-0.        +0.70710678j, -0.        -0.70710678j]])
      >>> linalg.eig(a, left=False, right=True)[1] # normalized right eigenvector
      array([[0.70710678+0.j        , 0.70710678-0.j        ],
             [0.        -0.70710678j, 0.        +0.70710678j]])