scipy.linalg.cossin#

scipy.linalg.cossin(X, p=None, q=None, separate=False, swap_sign=False, compute_u=True, compute_vh=True)[source]#

Compute the cosine-sine (CS) decomposition of an orthogonal/unitary matrix.

X is an (m, m) orthogonal/unitary matrix, partitioned as the following where upper left block has the shape of (p, q):

                           ┌                   ┐
                           │ I  0  0 │ 0  0  0 │
┌           ┐   ┌         ┐│ 0  C  0 │ 0 -S  0 │┌         ┐*
│ X11 │ X12 │   │ U1 │    ││ 0  0  0 │ 0  0 -I ││ V1 │    │
│ ────┼──── │ = │────┼────││─────────┼─────────││────┼────│
│ X21 │ X22 │   │    │ U2 ││ 0  0  0 │ I  0  0 ││    │ V2 │
└           ┘   └         ┘│ 0  S  0 │ 0  C  0 │└         ┘
                           │ 0  0  I │ 0  0  0 │
                           └                   ┘

U1, U2, V1, V2 are square orthogonal/unitary matrices of dimensions (p,p), (m-p,m-p), (q,q), and (m-q,m-q) respectively, and C and S are (r, r) nonnegative diagonal matrices satisfying C^2 + S^2 = I where r = min(p, m-p, q, m-q).

Moreover, the rank of the identity matrices are min(p, q) - r, min(p, m - q) - r, min(m - p, q) - r, and min(m - p, m - q) - r respectively.

X can be supplied either by itself and block specifications p, q or its subblocks in an iterable from which the shapes would be derived. See the examples below.

Parameters:

Xarray_like, iterable: complex unitary or real orthogonal matrix to be decomposed, or iterable of subblocks X11, X12, X21, X22, when p, q are omitted.
pint, optional: Number of rows of the upper left block X11, used only when X is given as an array.
qint, optional: Number of columns of the upper left block X11, used only when X is given as an array.
separatebool, optional: if True, the low level components are returned instead of the matrix factors, i.e. (u1,u2), theta, (v1h,v2h) instead of u, cs, vh.
swap_signbool, optional: if True, the -S, -I block will be the bottom left, otherwise (by default) they will be in the upper right block.
compute_ubool, optional: if False, u won’t be computed and an empty array is returned.
compute_vhbool, optional: if False, vh won’t be computed and an empty array is returned.

Returns:

undarray

When compute_u=True, contains the block diagonal orthogonal/unitary matrix consisting of the blocks U1 (p x p) and U2 (m-p x m-p) orthogonal/unitary matrices. If separate=True, this contains the tuple of (U1, U2).

csndarray

The cosine-sine factor with the structure described above.: If separate=True, this contains the theta array containing the angles in radians.

vhndarray

When compute_vh=True`, contains the block diagonal orthogonal/unitary matrix consisting of the blocks ``V1H (q x q) and V2H (m-q x m-q) orthogonal/unitary matrices. If separate=True, this contains the tuple of (V1H, V2H).

References

[1]

Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Examples

>>> import numpy as np
>>> from scipy.linalg import cossin
>>> from scipy.stats import unitary_group
>>> x = unitary_group.rvs(4)
>>> u, cs, vdh = cossin(x, p=2, q=2)
>>> np.allclose(x, u @ cs @ vdh)
True

Same can be entered via subblocks without the need of p and q. Also let’s skip the computation of u

>>> ue, cs, vdh = cossin((x[:2, :2], x[:2, 2:], x[2:, :2], x[2:, 2:]),
...                      compute_u=False)
>>> print(ue)
[]
>>> np.allclose(x, u @ cs @ vdh)
True