"""
from sage.all import *
-from mjo.cone.symmetric_psd import factor_psd, is_symmetric_psd
+from mjo.cone.symmetric_psd import factor_psd
from mjo.cone.doubly_nonnegative import (is_doubly_nonnegative,
is_extreme_doubly_nonnegative)
msg = 'The matrix ``A`` cannot be symbolic.'
raise ValueError.new(msg)
- if not is_symmetric_psd(A):
+ if not A.is_positive_semidefinite():
return False
n = A.nrows() # Makes sense since ``A`` is symmetric.
msg = 'The matrix ``A`` cannot be symbolic.'
raise ValueError(msg)
- if not is_symmetric_psd(A):
+ if not A.is_positive_semidefinite():
return False
n = A.nrows() # Makes sense since ``A`` is symmetric.
from sage.all import *
from mjo.cone.symmetric_psd import (factor_psd,
- is_symmetric_psd,
random_symmetric_psd)
from mjo.basis_repr import basis_repr
# It's nonnegative, so all we need to do is check that it's
# symmetric positive-semidefinite.
- return is_symmetric_psd(A)
+ return A.is_positive_semidefinite()
# Short circuit, we know the zero matrix is extreme.
return True
- if not is_symmetric_psd(A):
+ if not A.is_positive_semidefinite():
return False
# Step 1.5, appeal to Theorem 3.1 in reference #1 to short
from sage.all import *
from mjo.cone.symmetric_psd import random_symmetric_psd
-def is_symmetric_pd(A):
- """
- Determine whether or not the matrix ``A`` is symmetric
- positive-definite.
-
- INPUT:
-
- - ``A`` - The matrix in question.
-
- OUTPUT:
-
- Either ``True`` if ``A`` is symmetric positive-definite, or
- ``False`` otherwise.
-
- SETUP::
-
- sage: from mjo.cone.symmetric_pd import (is_symmetric_pd,
- ....: random_symmetric_pd)
-
- EXAMPLES:
-
- The identity matrix is obviously symmetric and positive-definite::
-
- sage: set_random_seed()
- sage: A = identity_matrix(ZZ, ZZ.random_element(10))
- sage: is_symmetric_pd(A)
- True
-
- The following matrix is symmetric but not positive definite::
-
- sage: A = matrix(ZZ, [[1, 2], [2, 1]])
- sage: is_symmetric_pd(A)
- False
-
- This matrix isn't even symmetric::
-
- sage: A = matrix(ZZ, [[1, 2], [3, 4]])
- sage: is_symmetric_pd(A)
- False
-
- The trivial matrix in a trivial space is trivially symmetric and
- positive-definite::
-
- sage: A = matrix(QQ, 0,0)
- sage: is_symmetric_pd(A)
- True
-
- The implementation of "is_positive_definite" in Sage leaves a bit to
- be desired. This matrix is technically positive definite over the
- real numbers, but isn't symmetric::
-
- sage: A = matrix(RR,[[1,1],[-1,1]])
- sage: A.is_positive_definite()
- Traceback (most recent call last):
- ...
- ValueError: Could not see Real Field with 53 bits of precision as a
- subring of the real or complex numbers
- sage: is_symmetric_pd(A)
- False
-
- TESTS:
-
- Every gram matrix is positive-definite, and we can sum two
- positive-definite matrices (``A`` and its transpose) to get a new
- positive-definite matrix that happens to be symmetric::
-
- sage: set_random_seed()
- sage: V = VectorSpace(QQ, ZZ.random_element(5))
- sage: A = random_symmetric_pd(V)
- sage: is_symmetric_pd(A)
- True
-
- """
-
- if A.base_ring() == SR:
- msg = 'The matrix ``A`` cannot be symbolic.'
- raise ValueError.new(msg)
-
- # First make sure that ``A`` is symmetric.
- if not A.is_symmetric():
- return False
-
- # If ``A`` is symmetric, we only need to check that it is positive
- # definite. For that we can consult its minimum eigenvalue, which
- # should be greater than zero. Since ``A`` is symmetric, its
- # eigenvalues are guaranteed to be real.
- if A.is_zero():
- # A is trivial... so trivially positive-definite.
- return True
- else:
- return min(A.eigenvalues()) > 0
-
-
def random_symmetric_pd(V):
r"""
Generate a random symmetric positive-definite matrix over the
from sage.all import *
-def is_symmetric_psd(A):
- """
- Determine whether or not the matrix ``A`` is symmetric
- positive-semidefinite.
-
- INPUT:
-
- - ``A`` - The matrix in question
-
- OUTPUT:
-
- Either ``True`` if ``A`` is symmetric positive-semidefinite, or
- ``False`` otherwise.
-
- SETUP::
-
- sage: from mjo.cone.symmetric_psd import is_symmetric_psd
-
- EXAMPLES:
-
- Every completely positive matrix is symmetric
- positive-semidefinite::
-
- sage: set_random_seed()
- sage: v = vector(map(abs, random_vector(ZZ, 10)))
- sage: A = v.column() * v.row()
- sage: is_symmetric_psd(A)
- True
-
- The following matrix is symmetric but not positive semidefinite::
-
- sage: A = matrix(ZZ, [[1, 2], [2, 1]])
- sage: is_symmetric_psd(A)
- False
-
- This matrix isn't even symmetric::
-
- sage: A = matrix(ZZ, [[1, 2], [3, 4]])
- sage: is_symmetric_psd(A)
- False
-
- The trivial matrix in a trivial space is trivially symmetric and
- positive-semidefinite::
-
- sage: A = matrix(QQ, 0,0)
- sage: is_symmetric_psd(A)
- True
-
- """
-
- if A.base_ring() == SR:
- msg = 'The matrix ``A`` cannot be symbolic.'
- raise ValueError.new(msg)
-
- # First make sure that ``A`` is symmetric.
- if not A.is_symmetric():
- return False
-
- # If ``A`` is symmetric, we only need to check that it is positive
- # semidefinite. For that we can consult its minimum eigenvalue,
- # which should be zero or greater. Since ``A`` is symmetric, its
- # eigenvalues are guaranteed to be real.
- if A.is_zero():
- # A is trivial... so trivially positive-semudefinite.
- return True
- else:
- return min(A.eigenvalues()) >= 0
-
-
def unit_eigenvectors(A):
"""
Return the unit eigenvectors of a symmetric positive-definite matrix.
SETUP::
- sage: from mjo.cone.symmetric_psd import (is_symmetric_psd,
- ....: random_symmetric_psd)
+ sage: from mjo.cone.symmetric_psd import random_symmetric_psd
EXAMPLES:
sage: A = random_symmetric_psd(V)
sage: A.matrix_space()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
- sage: is_symmetric_psd(A)
+ sage: A.is_positive_semidefinite()
True
A matrix with the desired rank is returned::
projects / sage.d.git / commitdiff