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).

Notes

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.

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