512 lines
12 KiB
Python
512 lines
12 KiB
Python
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from sympy.assumptions import Predicate
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from sympy.multipledispatch import Dispatcher
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class SquarePredicate(Predicate):
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"""
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Square matrix predicate.
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Explanation
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===========
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``Q.square(x)`` is true iff ``x`` is a square matrix. A square matrix
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is a matrix with the same number of rows and columns.
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Examples
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========
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>>> from sympy import Q, ask, MatrixSymbol, ZeroMatrix, Identity
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>>> X = MatrixSymbol('X', 2, 2)
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>>> Y = MatrixSymbol('X', 2, 3)
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>>> ask(Q.square(X))
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True
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>>> ask(Q.square(Y))
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False
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>>> ask(Q.square(ZeroMatrix(3, 3)))
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True
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>>> ask(Q.square(Identity(3)))
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True
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Square_matrix
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"""
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name = 'square'
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handler = Dispatcher("SquareHandler", doc="Handler for Q.square.")
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class SymmetricPredicate(Predicate):
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"""
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Symmetric matrix predicate.
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Explanation
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===========
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``Q.symmetric(x)`` is true iff ``x`` is a square matrix and is equal to
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its transpose. Every square diagonal matrix is a symmetric matrix.
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Examples
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========
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>>> from sympy import Q, ask, MatrixSymbol
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>>> X = MatrixSymbol('X', 2, 2)
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>>> Y = MatrixSymbol('Y', 2, 3)
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>>> Z = MatrixSymbol('Z', 2, 2)
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>>> ask(Q.symmetric(X*Z), Q.symmetric(X) & Q.symmetric(Z))
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True
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>>> ask(Q.symmetric(X + Z), Q.symmetric(X) & Q.symmetric(Z))
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True
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>>> ask(Q.symmetric(Y))
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False
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Symmetric_matrix
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"""
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# TODO: Add handlers to make these keys work with
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# actual matrices and add more examples in the docstring.
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name = 'symmetric'
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handler = Dispatcher("SymmetricHandler", doc="Handler for Q.symmetric.")
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class InvertiblePredicate(Predicate):
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"""
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Invertible matrix predicate.
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Explanation
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===========
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``Q.invertible(x)`` is true iff ``x`` is an invertible matrix.
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A square matrix is called invertible only if its determinant is 0.
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Examples
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========
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>>> from sympy import Q, ask, MatrixSymbol
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>>> X = MatrixSymbol('X', 2, 2)
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>>> Y = MatrixSymbol('Y', 2, 3)
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>>> Z = MatrixSymbol('Z', 2, 2)
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>>> ask(Q.invertible(X*Y), Q.invertible(X))
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False
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>>> ask(Q.invertible(X*Z), Q.invertible(X) & Q.invertible(Z))
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True
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>>> ask(Q.invertible(X), Q.fullrank(X) & Q.square(X))
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True
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Invertible_matrix
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"""
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name = 'invertible'
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handler = Dispatcher("InvertibleHandler", doc="Handler for Q.invertible.")
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class OrthogonalPredicate(Predicate):
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"""
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Orthogonal matrix predicate.
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Explanation
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===========
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``Q.orthogonal(x)`` is true iff ``x`` is an orthogonal matrix.
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A square matrix ``M`` is an orthogonal matrix if it satisfies
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``M^TM = MM^T = I`` where ``M^T`` is the transpose matrix of
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``M`` and ``I`` is an identity matrix. Note that an orthogonal
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matrix is necessarily invertible.
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Examples
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========
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>>> from sympy import Q, ask, MatrixSymbol, Identity
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>>> X = MatrixSymbol('X', 2, 2)
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>>> Y = MatrixSymbol('Y', 2, 3)
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>>> Z = MatrixSymbol('Z', 2, 2)
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>>> ask(Q.orthogonal(Y))
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False
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>>> ask(Q.orthogonal(X*Z*X), Q.orthogonal(X) & Q.orthogonal(Z))
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True
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>>> ask(Q.orthogonal(Identity(3)))
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True
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>>> ask(Q.invertible(X), Q.orthogonal(X))
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True
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Orthogonal_matrix
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"""
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name = 'orthogonal'
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handler = Dispatcher("OrthogonalHandler", doc="Handler for key 'orthogonal'.")
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class UnitaryPredicate(Predicate):
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"""
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Unitary matrix predicate.
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Explanation
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===========
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``Q.unitary(x)`` is true iff ``x`` is a unitary matrix.
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Unitary matrix is an analogue to orthogonal matrix. A square
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matrix ``M`` with complex elements is unitary if :math:``M^TM = MM^T= I``
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where :math:``M^T`` is the conjugate transpose matrix of ``M``.
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Examples
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========
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>>> from sympy import Q, ask, MatrixSymbol, Identity
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>>> X = MatrixSymbol('X', 2, 2)
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>>> Y = MatrixSymbol('Y', 2, 3)
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>>> Z = MatrixSymbol('Z', 2, 2)
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>>> ask(Q.unitary(Y))
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False
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>>> ask(Q.unitary(X*Z*X), Q.unitary(X) & Q.unitary(Z))
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True
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>>> ask(Q.unitary(Identity(3)))
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True
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Unitary_matrix
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"""
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name = 'unitary'
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handler = Dispatcher("UnitaryHandler", doc="Handler for key 'unitary'.")
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class FullRankPredicate(Predicate):
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"""
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Fullrank matrix predicate.
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Explanation
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===========
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``Q.fullrank(x)`` is true iff ``x`` is a full rank matrix.
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A matrix is full rank if all rows and columns of the matrix
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are linearly independent. A square matrix is full rank iff
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its determinant is nonzero.
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Examples
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========
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>>> from sympy import Q, ask, MatrixSymbol, ZeroMatrix, Identity
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>>> X = MatrixSymbol('X', 2, 2)
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>>> ask(Q.fullrank(X.T), Q.fullrank(X))
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True
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>>> ask(Q.fullrank(ZeroMatrix(3, 3)))
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False
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>>> ask(Q.fullrank(Identity(3)))
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True
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"""
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name = 'fullrank'
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handler = Dispatcher("FullRankHandler", doc="Handler for key 'fullrank'.")
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class PositiveDefinitePredicate(Predicate):
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r"""
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Positive definite matrix predicate.
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Explanation
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===========
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If $M$ is a :math:`n \times n` symmetric real matrix, it is said
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to be positive definite if :math:`Z^TMZ` is positive for
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every non-zero column vector $Z$ of $n$ real numbers.
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Examples
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========
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>>> from sympy import Q, ask, MatrixSymbol, Identity
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>>> X = MatrixSymbol('X', 2, 2)
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>>> Y = MatrixSymbol('Y', 2, 3)
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>>> Z = MatrixSymbol('Z', 2, 2)
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>>> ask(Q.positive_definite(Y))
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False
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>>> ask(Q.positive_definite(Identity(3)))
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True
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>>> ask(Q.positive_definite(X + Z), Q.positive_definite(X) &
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... Q.positive_definite(Z))
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True
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Positive-definite_matrix
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"""
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name = "positive_definite"
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handler = Dispatcher("PositiveDefiniteHandler", doc="Handler for key 'positive_definite'.")
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class UpperTriangularPredicate(Predicate):
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"""
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Upper triangular matrix predicate.
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Explanation
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===========
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A matrix $M$ is called upper triangular matrix if :math:`M_{ij}=0`
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for :math:`i<j`.
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Examples
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========
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>>> from sympy import Q, ask, ZeroMatrix, Identity
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>>> ask(Q.upper_triangular(Identity(3)))
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True
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>>> ask(Q.upper_triangular(ZeroMatrix(3, 3)))
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True
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References
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==========
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.. [1] https://mathworld.wolfram.com/UpperTriangularMatrix.html
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"""
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name = "upper_triangular"
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handler = Dispatcher("UpperTriangularHandler", doc="Handler for key 'upper_triangular'.")
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class LowerTriangularPredicate(Predicate):
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"""
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Lower triangular matrix predicate.
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Explanation
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===========
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A matrix $M$ is called lower triangular matrix if :math:`M_{ij}=0`
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for :math:`i>j`.
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Examples
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========
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>>> from sympy import Q, ask, ZeroMatrix, Identity
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>>> ask(Q.lower_triangular(Identity(3)))
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True
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>>> ask(Q.lower_triangular(ZeroMatrix(3, 3)))
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True
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References
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==========
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.. [1] https://mathworld.wolfram.com/LowerTriangularMatrix.html
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"""
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name = "lower_triangular"
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handler = Dispatcher("LowerTriangularHandler", doc="Handler for key 'lower_triangular'.")
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class DiagonalPredicate(Predicate):
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"""
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Diagonal matrix predicate.
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Explanation
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===========
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``Q.diagonal(x)`` is true iff ``x`` is a diagonal matrix. A diagonal
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matrix is a matrix in which the entries outside the main diagonal
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are all zero.
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Examples
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========
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>>> from sympy import Q, ask, MatrixSymbol, ZeroMatrix
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>>> X = MatrixSymbol('X', 2, 2)
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>>> ask(Q.diagonal(ZeroMatrix(3, 3)))
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True
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>>> ask(Q.diagonal(X), Q.lower_triangular(X) &
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... Q.upper_triangular(X))
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True
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Diagonal_matrix
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"""
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name = "diagonal"
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handler = Dispatcher("DiagonalHandler", doc="Handler for key 'diagonal'.")
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class IntegerElementsPredicate(Predicate):
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"""
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Integer elements matrix predicate.
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Explanation
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===========
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``Q.integer_elements(x)`` is true iff all the elements of ``x``
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are integers.
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Examples
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========
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>>> from sympy import Q, ask, MatrixSymbol
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>>> X = MatrixSymbol('X', 4, 4)
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>>> ask(Q.integer(X[1, 2]), Q.integer_elements(X))
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True
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"""
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name = "integer_elements"
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handler = Dispatcher("IntegerElementsHandler", doc="Handler for key 'integer_elements'.")
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class RealElementsPredicate(Predicate):
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"""
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Real elements matrix predicate.
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Explanation
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===========
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``Q.real_elements(x)`` is true iff all the elements of ``x``
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are real numbers.
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Examples
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========
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>>> from sympy import Q, ask, MatrixSymbol
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>>> X = MatrixSymbol('X', 4, 4)
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>>> ask(Q.real(X[1, 2]), Q.real_elements(X))
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True
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"""
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name = "real_elements"
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handler = Dispatcher("RealElementsHandler", doc="Handler for key 'real_elements'.")
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class ComplexElementsPredicate(Predicate):
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"""
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Complex elements matrix predicate.
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Explanation
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===========
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``Q.complex_elements(x)`` is true iff all the elements of ``x``
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are complex numbers.
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Examples
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========
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>>> from sympy import Q, ask, MatrixSymbol
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>>> X = MatrixSymbol('X', 4, 4)
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>>> ask(Q.complex(X[1, 2]), Q.complex_elements(X))
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True
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>>> ask(Q.complex_elements(X), Q.integer_elements(X))
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True
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"""
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name = "complex_elements"
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handler = Dispatcher("ComplexElementsHandler", doc="Handler for key 'complex_elements'.")
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class SingularPredicate(Predicate):
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"""
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Singular matrix predicate.
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A matrix is singular iff the value of its determinant is 0.
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Examples
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========
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>>> from sympy import Q, ask, MatrixSymbol
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>>> X = MatrixSymbol('X', 4, 4)
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>>> ask(Q.singular(X), Q.invertible(X))
|
||
|
|
False
|
||
|
|
>>> ask(Q.singular(X), ~Q.invertible(X))
|
||
|
|
True
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://mathworld.wolfram.com/SingularMatrix.html
|
||
|
|
|
||
|
|
"""
|
||
|
|
name = "singular"
|
||
|
|
handler = Dispatcher("SingularHandler", doc="Predicate fore key 'singular'.")
|
||
|
|
|
||
|
|
|
||
|
|
class NormalPredicate(Predicate):
|
||
|
|
"""
|
||
|
|
Normal matrix predicate.
|
||
|
|
|
||
|
|
A matrix is normal if it commutes with its conjugate transpose.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Q, ask, MatrixSymbol
|
||
|
|
>>> X = MatrixSymbol('X', 4, 4)
|
||
|
|
>>> ask(Q.normal(X), Q.unitary(X))
|
||
|
|
True
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://en.wikipedia.org/wiki/Normal_matrix
|
||
|
|
|
||
|
|
"""
|
||
|
|
name = "normal"
|
||
|
|
handler = Dispatcher("NormalHandler", doc="Predicate fore key 'normal'.")
|
||
|
|
|
||
|
|
|
||
|
|
class TriangularPredicate(Predicate):
|
||
|
|
"""
|
||
|
|
Triangular matrix predicate.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
``Q.triangular(X)`` is true if ``X`` is one that is either lower
|
||
|
|
triangular or upper triangular.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Q, ask, MatrixSymbol
|
||
|
|
>>> X = MatrixSymbol('X', 4, 4)
|
||
|
|
>>> ask(Q.triangular(X), Q.upper_triangular(X))
|
||
|
|
True
|
||
|
|
>>> ask(Q.triangular(X), Q.lower_triangular(X))
|
||
|
|
True
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://en.wikipedia.org/wiki/Triangular_matrix
|
||
|
|
|
||
|
|
"""
|
||
|
|
name = "triangular"
|
||
|
|
handler = Dispatcher("TriangularHandler", doc="Predicate fore key 'triangular'.")
|
||
|
|
|
||
|
|
|
||
|
|
class UnitTriangularPredicate(Predicate):
|
||
|
|
"""
|
||
|
|
Unit triangular matrix predicate.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
A unit triangular matrix is a triangular matrix with 1s
|
||
|
|
on the diagonal.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Q, ask, MatrixSymbol
|
||
|
|
>>> X = MatrixSymbol('X', 4, 4)
|
||
|
|
>>> ask(Q.triangular(X), Q.unit_triangular(X))
|
||
|
|
True
|
||
|
|
|
||
|
|
"""
|
||
|
|
name = "unit_triangular"
|
||
|
|
handler = Dispatcher("UnitTriangularHandler", doc="Predicate fore key 'unit_triangular'.")
|