r"""
The symmetric positive definite cone `$S^{n}_{++}$` is the cone
consisting of all symmetric positive-definite matrices (as a subset of
$\mathbb{R}^{n \times n}$`. It is the interior of the symmetric positive
SEMI-definite cone.
"""

from sage.all import *
from mjo.cone.symmetric_psd import random_symmetric_psd

def random_symmetric_pd(V):
    r"""
    Generate a random symmetric positive-definite matrix over the
    vector space ``V``. That is, the returned matrix will be a linear
    transformation on ``V``, with the same base ring as ``V``.

    (We take a very loose interpretation of "random," here.)

    INPUT:

    - ``V`` -- The vector space on which the returned matrix will act.

    OUTPUT:

    A random symmetric positive-definite matrix, i.e. a linear
    transformation from ``V`` to itself.

    ALGORITHM:

    We request a full-rank positive-semidefinite matrix from the
    :func:`mjo.cone.symmetric_psd.random_symmetric_psd` function.

    SETUP::

        sage: from mjo.cone.symmetric_pd import random_symmetric_pd

    EXAMPLES:

    Well, it doesn't crash at least::

        sage: set_random_seed()
        sage: V = VectorSpace(QQ, 2)
        sage: A = random_symmetric_pd(V)
        sage: A.matrix_space()
        Full MatrixSpace of 2 by 2 dense matrices over Rational Field

    """
    # We accept_zero because the trivial matrix is (trivially)
    # symmetric and positive-definite, but it's also "zero." The rank
    # condition ensures that we don't get the zero matrix in a
    # nontrivial space.
    return random_symmetric_psd(V, rank=V.dimension())