Actually it looks like a lot of the logic is already there. For anyone
interested in fixing this, we just need to add a refine handler for
MatrixElement that handles symmetry. One already exists for Transpose
>>> X = MatrixSymbol("X", n, m)
>>> refine(Transpose(X), Q.symmetric(X))
X
I opened https://github.com/sympy/sympy/issues/16234, which should be
easy to fix.
Aaron Meurer
On Mon, Mar 11, 2019 at 1:16 PM Aaron Meurer <[email protected]> wrote:
>
> I don't think that exists yet, aside from the assumptions in the new
> assumptions Q.symmetric and Q.positive_definite. It would be useful to
> be able to bind those assumptions to the matrix object itself.
>
> Also ideally you would be able to put it under a "with
> assuming(Q.symmetric(q))" block (where q is your matrix Q, not to be
> confused with Q the SymPy object that hold the assumptions
> predicates), but that doesn't work yet either. As far as I know, the
> matrix assumptions so far only do anything for deduction (ask()).
>
> For now, you can use replace to make the resulting matrix symmetric
>
> >>> from sympy.matrices.expressions.matexpr import MatrixElement
> >>> p, Q = sospoly(Matrix([1, 2, 3]), 'Q')
> >>> p
> Q[0, 0] + 2*Q[0, 1] + 3*Q[0, 2] + 2*Q[1, 0] + 4*Q[1, 1] + 6*Q[1, 2] +
> 3*Q[2, 0] + 6*Q[2, 1] + 9*Q[2, 2]
> >>> p.replace(lambda x: isinstance(x, MatrixElement) and x.args[0] == Q,
> >>> lambda x: x if x.args[1] <= x.args[2] else x.args[0][x.args[2],x.args[1]])
> Q[0, 0] + 4*Q[0, 1] + 6*Q[0, 2] + 4*Q[1, 1] + 12*Q[1, 2] + 9*Q[2, 2]
>
> Or you could use a custom MatrixSymbol subclass
>
> >>> class SymmetricMatrixSymbol(MatrixSymbol):
> ... def _entry(self, i, j):
> ... i, j = sympify((i, j))
> ... if (i - j).could_extract_minus_sign():
> ... i, j = j, i
> ... return MatrixElement(self, i, j)
> >>> Q = SymmetricMatrixSymbol("Q", 3, 3)
> >>> Q[0, 1]
> Q[1, 0]
> >>> Q[1, 0]
> Q[1, 0]
> >>> x = Matrix([1, 2, 3])
> >>> (x.T*Matrix(Q)*x)[0]
> Q[0, 0] + 4*Q[1, 0] + 4*Q[1, 1] + 6*Q[2, 0] + 12*Q[2, 1] + 9*Q[2, 2]
>
> I used could_extract_minus_sign() instead of i < j because it handles
> symbolic entries as well.
>
> >>> i, j = symbols('i j')
> >>> Q[i, j]
> Q[i, j]
> >>> Q[j, i]
> Q[i, j]
>
> Aaron Meurer
>
> On Mon, Mar 11, 2019 at 12:04 PM <[email protected]> wrote:
> >
> > Hi,
> >
> > I was wondering how I can create symbolic symmetric positive definite
> > (s.p.d.) Matrices.
> >
> > I would like to get a symbolic (symbolic in the Q-matrix) quadratic form
> >
> > def sospoly(x, varname):
> > N = x.shape[0]
> > Q = MatrixSymbol(varname, N, N)
> > p = (x.T * Matrix(Q) * x)[0]
> > return p, Q
> >
> >
> > where Q is s.p.d.
> > It would be enough to enforce symmetry actually ( making sure that Q[i,j]
> > == Q[j,i]).
> > If there is no "SymmetricMatrixSymbol" is there an easy way to create a
> > symbolic lower triangular matrix?
> >
> > Thank you!
> >
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