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# scipy.linalg.cossin

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

  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:
    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)`.

  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