# 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

```>>> 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
```