cossin(X, p=None, q=None, separate=False, swap_sign=False, compute_u=True, compute_vh=True)

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

X : array_like, iterable

complex unitary or real orthogonal matrix to be decomposed, or iterable of subblocks X11 , X12 , X21 , X22 , when p , q are omitted.

p : int, optional

Number of rows of the upper left block X11 , used only when X is given as an array.

q : int, optional

Number of columns of the upper left block X11 , used only when X is given as an array.

separate : bool, 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_sign : bool, optional

if True , the -S , -I block will be the bottom left, otherwise (by default) they will be in the upper right block.

compute_u : bool, optional

if False , u won't be computed and an empty array is returned.

compute_vh : bool, optional

if False , vh won't be computed and an empty array is returned.

Returns

u : ndarray

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) .

cs : ndarray

The cosine-sine factor with the structure described above.

If separate=True , this contains the theta array containing the angles in radians.

vh : ndarray

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) .

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

Examples

>>> 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

See :

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GitHub : /scipy/linalg/_decomp_cossin.py#12
type: <class 'function'>
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