colour.algebra.eigen_decomposition

colour.algebra.eigen_decomposition(a: ArrayLike, eigen_w_v_count: int | None = None, descending_order: bool = True, covariance_matrix: bool = False) → Tuple[NDArrayFloat, NDArrayFloat]

Compute the eigenvalues \(w\) and eigenvectors \(v\) of the specified array \(a\) in the specified order.

Parameters:

• a (ArrayLike) – Array to compute the eigenvalues \(w\) and eigenvectors \(v\) for.

• eigen_w_v_count (int | None) – Number of eigenvalues \(w\) and eigenvectors \(v\) to return.

• descending_order (bool) – Whether to return the eigenvalues \(w\) and eigenvectors \(v\) in descending order.

• covariance_matrix (bool) – Whether to compute the eigenvalues \(w\) and eigenvectors \(v\) of the array \(a\) covariance matrix \(A = a^T \cdot a\).

Returns:

Tuple of eigenvalues \(w\) and eigenvectors \(v\). The eigenvalues are in the specified order, each repeated according to its multiplicity. The column v[:, i] is the normalized eigenvector corresponding to the eigenvalue w[i].

Return type:

tuple

Examples

>>> a = np.diag([1, 2, 3])
>>> w, v = eigen_decomposition(a)
>>> w
array([ 3.,  2.,  1.])
>>> v
array([[ 0.,  0.,  1.],
       [ 0.,  1.,  0.],
       [ 1.,  0.,  0.]])
>>> w, v = eigen_decomposition(a, 1)
>>> w
array([ 3.])
>>> v
array([[ 0.],
       [ 0.],
       [ 1.]])
>>> w, v = eigen_decomposition(a, descending_order=False)
>>> w
array([ 1.,  2.,  3.])
>>> v
array([[ 1.,  0.,  0.],
       [ 0.,  1.,  0.],
       [ 0.,  0.,  1.]])
>>> w, v = eigen_decomposition(a, covariance_matrix=True)
>>> w
array([ 9.,  4.,  1.])
>>> v
array([[ 0.,  0.,  1.],
       [ 0.,  1.,  0.],
       [ 1.,  0.,  0.]])