from types import FunctionType from collections import Counter from mpmath import mp, workprec from mpmath.libmp.libmpf import prec_to_dps from sympy.core.sorting import default_sort_key from sympy.core.evalf import DEFAULT_MAXPREC, PrecisionExhausted from sympy.core.logic import fuzzy_and, fuzzy_or from sympy.core.numbers import Float from sympy.core.sympify import _sympify from sympy.functions.elementary.miscellaneous import sqrt from sympy.polys import roots, CRootOf, ZZ, QQ, EX from sympy.polys.matrices import DomainMatrix from sympy.polys.matrices.eigen import dom_eigenvects, dom_eigenvects_to_sympy from sympy.simplify import nsimplify, simplify as _simplify from sympy.utilities.exceptions import sympy_deprecation_warning from .common import MatrixError, NonSquareMatrixError from .determinant import _find_reasonable_pivot from .utilities import _iszero def _eigenvals_eigenvects_mpmath(M): norm2 = lambda v: mp.sqrt(sum(i**2 for i in v)) v1 = None prec = max([x._prec for x in M.atoms(Float)]) eps = 2**-prec while prec < DEFAULT_MAXPREC: with workprec(prec): A = mp.matrix(M.evalf(n=prec_to_dps(prec))) E, ER = mp.eig(A) v2 = norm2([i for e in E for i in (mp.re(e), mp.im(e))]) if v1 is not None and mp.fabs(v1 - v2) < eps: return E, ER v1 = v2 prec *= 2 # we get here because the next step would have taken us # past MAXPREC or because we never took a step; in case # of the latter, we refuse to send back a solution since # it would not have been verified; we also resist taking # a small step to arrive exactly at MAXPREC since then # the two calculations might be artificially close. raise PrecisionExhausted def _eigenvals_mpmath(M, multiple=False): """Compute eigenvalues using mpmath""" E, _ = _eigenvals_eigenvects_mpmath(M) result = [_sympify(x) for x in E] if multiple: return result return dict(Counter(result)) def _eigenvects_mpmath(M): E, ER = _eigenvals_eigenvects_mpmath(M) result = [] for i in range(M.rows): eigenval = _sympify(E[i]) eigenvect = _sympify(ER[:, i]) result.append((eigenval, 1, [eigenvect])) return result # This function is a candidate for caching if it gets implemented for matrices. def _eigenvals( M, error_when_incomplete=True, *, simplify=False, multiple=False, rational=False, **flags): r"""Compute eigenvalues of the matrix. Parameters ========== error_when_incomplete : bool, optional If it is set to ``True``, it will raise an error if not all eigenvalues are computed. This is caused by ``roots`` not returning a full list of eigenvalues. simplify : bool or function, optional If it is set to ``True``, it attempts to return the most simplified form of expressions returned by applying default simplification method in every routine. If it is set to ``False``, it will skip simplification in this particular routine to save computation resources. If a function is passed to, it will attempt to apply the particular function as simplification method. rational : bool, optional If it is set to ``True``, every floating point numbers would be replaced with rationals before computation. It can solve some issues of ``roots`` routine not working well with floats. multiple : bool, optional If it is set to ``True``, the result will be in the form of a list. If it is set to ``False``, the result will be in the form of a dictionary. Returns ======= eigs : list or dict Eigenvalues of a matrix. The return format would be specified by the key ``multiple``. Raises ====== MatrixError If not enough roots had got computed. NonSquareMatrixError If attempted to compute eigenvalues from a non-square matrix. Examples ======== >>> from sympy import Matrix >>> M = Matrix(3, 3, [0, 1, 1, 1, 0, 0, 1, 1, 1]) >>> M.eigenvals() {-1: 1, 0: 1, 2: 1} See Also ======== MatrixDeterminant.charpoly eigenvects Notes ===== Eigenvalues of a matrix $A$ can be computed by solving a matrix equation $\det(A - \lambda I) = 0$ It's not always possible to return radical solutions for eigenvalues for matrices larger than $4, 4$ shape due to Abel-Ruffini theorem. If there is no radical solution is found for the eigenvalue, it may return eigenvalues in the form of :class:`sympy.polys.rootoftools.ComplexRootOf`. """ if not M: if multiple: return [] return {} if not M.is_square: raise NonSquareMatrixError("{} must be a square matrix.".format(M)) if M._rep.domain not in (ZZ, QQ): # Skip this check for ZZ/QQ because it can be slow if all(x.is_number for x in M) and M.has(Float): return _eigenvals_mpmath(M, multiple=multiple) if rational: M = M.applyfunc( lambda x: nsimplify(x, rational=True) if x.has(Float) else x) if multiple: return _eigenvals_list( M, error_when_incomplete=error_when_incomplete, simplify=simplify, **flags) return _eigenvals_dict( M, error_when_incomplete=error_when_incomplete, simplify=simplify, **flags) eigenvals_error_message = \ "It is not always possible to express the eigenvalues of a matrix " + \ "of size 5x5 or higher in radicals. " + \ "We have CRootOf, but domains other than the rationals are not " + \ "currently supported. " + \ "If there are no symbols in the matrix, " + \ "it should still be possible to compute numeric approximations " + \ "of the eigenvalues using " + \ "M.evalf().eigenvals() or M.charpoly().nroots()." def _eigenvals_list( M, error_when_incomplete=True, simplify=False, **flags): iblocks = M.strongly_connected_components() all_eigs = [] is_dom = M._rep.domain in (ZZ, QQ) for b in iblocks: # Fast path for a 1x1 block: if is_dom and len(b) == 1: index = b[0] val = M[index, index] all_eigs.append(val) continue block = M[b, b] if isinstance(simplify, FunctionType): charpoly = block.charpoly(simplify=simplify) else: charpoly = block.charpoly() eigs = roots(charpoly, multiple=True, **flags) if len(eigs) != block.rows: degree = int(charpoly.degree()) f = charpoly.as_expr() x = charpoly.gen try: eigs = [CRootOf(f, x, idx) for idx in range(degree)] except NotImplementedError: if error_when_incomplete: raise MatrixError(eigenvals_error_message) else: eigs = [] all_eigs += eigs if not simplify: return all_eigs if not isinstance(simplify, FunctionType): simplify = _simplify return [simplify(value) for value in all_eigs] def _eigenvals_dict( M, error_when_incomplete=True, simplify=False, **flags): iblocks = M.strongly_connected_components() all_eigs = {} is_dom = M._rep.domain in (ZZ, QQ) for b in iblocks: # Fast path for a 1x1 block: if is_dom and len(b) == 1: index = b[0] val = M[index, index] all_eigs[val] = all_eigs.get(val, 0) + 1 continue block = M[b, b] if isinstance(simplify, FunctionType): charpoly = block.charpoly(simplify=simplify) else: charpoly = block.charpoly() eigs = roots(charpoly, multiple=False, **flags) if sum(eigs.values()) != block.rows: degree = int(charpoly.degree()) f = charpoly.as_expr() x = charpoly.gen try: eigs = {CRootOf(f, x, idx): 1 for idx in range(degree)} except NotImplementedError: if error_when_incomplete: raise MatrixError(eigenvals_error_message) else: eigs = {} for k, v in eigs.items(): if k in all_eigs: all_eigs[k] += v else: all_eigs[k] = v if not simplify: return all_eigs if not isinstance(simplify, FunctionType): simplify = _simplify return {simplify(key): value for key, value in all_eigs.items()} def _eigenspace(M, eigenval, iszerofunc=_iszero, simplify=False): """Get a basis for the eigenspace for a particular eigenvalue""" m = M - M.eye(M.rows) * eigenval ret = m.nullspace(iszerofunc=iszerofunc) # The nullspace for a real eigenvalue should be non-trivial. # If we didn't find an eigenvector, try once more a little harder if len(ret) == 0 and simplify: ret = m.nullspace(iszerofunc=iszerofunc, simplify=True) if len(ret) == 0: raise NotImplementedError( "Can't evaluate eigenvector for eigenvalue {}".format(eigenval)) return ret def _eigenvects_DOM(M, **kwargs): DOM = DomainMatrix.from_Matrix(M, field=True, extension=True) DOM = DOM.to_dense() if DOM.domain != EX: rational, algebraic = dom_eigenvects(DOM) eigenvects = dom_eigenvects_to_sympy( rational, algebraic, M.__class__, **kwargs) eigenvects = sorted(eigenvects, key=lambda x: default_sort_key(x[0])) return eigenvects return None def _eigenvects_sympy(M, iszerofunc, simplify=True, **flags): eigenvals = M.eigenvals(rational=False, **flags) # Make sure that we have all roots in radical form for x in eigenvals: if x.has(CRootOf): raise MatrixError( "Eigenvector computation is not implemented if the matrix have " "eigenvalues in CRootOf form") eigenvals = sorted(eigenvals.items(), key=default_sort_key) ret = [] for val, mult in eigenvals: vects = _eigenspace(M, val, iszerofunc=iszerofunc, simplify=simplify) ret.append((val, mult, vects)) return ret # This functions is a candidate for caching if it gets implemented for matrices. def _eigenvects(M, error_when_incomplete=True, iszerofunc=_iszero, *, chop=False, **flags): """Compute eigenvectors of the matrix. Parameters ========== error_when_incomplete : bool, optional Raise an error when not all eigenvalues are computed. This is caused by ``roots`` not returning a full list of eigenvalues. iszerofunc : function, optional Specifies a zero testing function to be used in ``rref``. Default value is ``_iszero``, which uses SymPy's naive and fast default assumption handler. It can also accept any user-specified zero testing function, if it is formatted as a function which accepts a single symbolic argument and returns ``True`` if it is tested as zero and ``False`` if it is tested as non-zero, and ``None`` if it is undecidable. simplify : bool or function, optional If ``True``, ``as_content_primitive()`` will be used to tidy up normalization artifacts. It will also be used by the ``nullspace`` routine. chop : bool or positive number, optional If the matrix contains any Floats, they will be changed to Rationals for computation purposes, but the answers will be returned after being evaluated with evalf. The ``chop`` flag is passed to ``evalf``. When ``chop=True`` a default precision will be used; a number will be interpreted as the desired level of precision. Returns ======= ret : [(eigenval, multiplicity, eigenspace), ...] A ragged list containing tuples of data obtained by ``eigenvals`` and ``nullspace``. ``eigenspace`` is a list containing the ``eigenvector`` for each eigenvalue. ``eigenvector`` is a vector in the form of a ``Matrix``. e.g. a vector of length 3 is returned as ``Matrix([a_1, a_2, a_3])``. Raises ====== NotImplementedError If failed to compute nullspace. Examples ======== >>> from sympy import Matrix >>> M = Matrix(3, 3, [0, 1, 1, 1, 0, 0, 1, 1, 1]) >>> M.eigenvects() [(-1, 1, [Matrix([ [-1], [ 1], [ 0]])]), (0, 1, [Matrix([ [ 0], [-1], [ 1]])]), (2, 1, [Matrix([ [2/3], [1/3], [ 1]])])] See Also ======== eigenvals MatrixSubspaces.nullspace """ simplify = flags.get('simplify', True) primitive = flags.get('simplify', False) flags.pop('simplify', None) # remove this if it's there flags.pop('multiple', None) # remove this if it's there if not isinstance(simplify, FunctionType): simpfunc = _simplify if simplify else lambda x: x has_floats = M.has(Float) if has_floats: if all(x.is_number for x in M): return _eigenvects_mpmath(M) M = M.applyfunc(lambda x: nsimplify(x, rational=True)) ret = _eigenvects_DOM(M) if ret is None: ret = _eigenvects_sympy(M, iszerofunc, simplify=simplify, **flags) if primitive: # if the primitive flag is set, get rid of any common # integer denominators def denom_clean(l): from sympy.polys.polytools import gcd return [(v / gcd(list(v))).applyfunc(simpfunc) for v in l] ret = [(val, mult, denom_clean(es)) for val, mult, es in ret] if has_floats: # if we had floats to start with, turn the eigenvectors to floats ret = [(val.evalf(chop=chop), mult, [v.evalf(chop=chop) for v in es]) for val, mult, es in ret] return ret def _is_diagonalizable_with_eigen(M, reals_only=False): """See _is_diagonalizable. This function returns the bool along with the eigenvectors to avoid calculating them again in functions like ``diagonalize``.""" if not M.is_square: return False, [] eigenvecs = M.eigenvects(simplify=True) for val, mult, basis in eigenvecs: if reals_only and not val.is_real: # if we have a complex eigenvalue return False, eigenvecs if mult != len(basis): # if the geometric multiplicity doesn't equal the algebraic return False, eigenvecs return True, eigenvecs def _is_diagonalizable(M, reals_only=False, **kwargs): """Returns ``True`` if a matrix is diagonalizable. Parameters ========== reals_only : bool, optional If ``True``, it tests whether the matrix can be diagonalized to contain only real numbers on the diagonal. If ``False``, it tests whether the matrix can be diagonalized at all, even with numbers that may not be real. Examples ======== Example of a diagonalizable matrix: >>> from sympy import Matrix >>> M = Matrix([[1, 2, 0], [0, 3, 0], [2, -4, 2]]) >>> M.is_diagonalizable() True Example of a non-diagonalizable matrix: >>> M = Matrix([[0, 1], [0, 0]]) >>> M.is_diagonalizable() False Example of a matrix that is diagonalized in terms of non-real entries: >>> M = Matrix([[0, 1], [-1, 0]]) >>> M.is_diagonalizable(reals_only=False) True >>> M.is_diagonalizable(reals_only=True) False See Also ======== is_diagonal diagonalize """ if 'clear_cache' in kwargs: sympy_deprecation_warning( """ The 'clear_cache' keyword to Matrix.is_diagonalizable is deprecated. It does nothing and should be omitted. """, deprecated_since_version="1.4", active_deprecations_target="deprecated-matrix-is_diagonalizable-cache", stacklevel=4, ) if 'clear_subproducts' in kwargs: sympy_deprecation_warning( """ The 'clear_subproducts' keyword to Matrix.is_diagonalizable is deprecated. It does nothing and should be omitted. """, deprecated_since_version="1.4", active_deprecations_target="deprecated-matrix-is_diagonalizable-cache", stacklevel=4, ) if not M.is_square: return False if all(e.is_real for e in M) and M.is_symmetric(): return True if all(e.is_complex for e in M) and M.is_hermitian: return True return _is_diagonalizable_with_eigen(M, reals_only=reals_only)[0] #G&VL, Matrix Computations, Algo 5.4.2 def _householder_vector(x): if not x.cols == 1: raise ValueError("Input must be a column matrix") v = x.copy() v_plus = x.copy() v_minus = x.copy() q = x[0, 0] / abs(x[0, 0]) norm_x = x.norm() v_plus[0, 0] = x[0, 0] + q * norm_x v_minus[0, 0] = x[0, 0] - q * norm_x if x[1:, 0].norm() == 0: bet = 0 v[0, 0] = 1 else: if v_plus.norm() <= v_minus.norm(): v = v_plus else: v = v_minus v = v / v[0] bet = 2 / (v.norm() ** 2) return v, bet def _bidiagonal_decmp_hholder(M): m = M.rows n = M.cols A = M.as_mutable() U, V = A.eye(m), A.eye(n) for i in range(min(m, n)): v, bet = _householder_vector(A[i:, i]) hh_mat = A.eye(m - i) - bet * v * v.H A[i:, i:] = hh_mat * A[i:, i:] temp = A.eye(m) temp[i:, i:] = hh_mat U = U * temp if i + 1 <= n - 2: v, bet = _householder_vector(A[i, i+1:].T) hh_mat = A.eye(n - i - 1) - bet * v * v.H A[i:, i+1:] = A[i:, i+1:] * hh_mat temp = A.eye(n) temp[i+1:, i+1:] = hh_mat V = temp * V return U, A, V def _eval_bidiag_hholder(M): m = M.rows n = M.cols A = M.as_mutable() for i in range(min(m, n)): v, bet = _householder_vector(A[i:, i]) hh_mat = A.eye(m-i) - bet * v * v.H A[i:, i:] = hh_mat * A[i:, i:] if i + 1 <= n - 2: v, bet = _householder_vector(A[i, i+1:].T) hh_mat = A.eye(n - i - 1) - bet * v * v.H A[i:, i+1:] = A[i:, i+1:] * hh_mat return A def _bidiagonal_decomposition(M, upper=True): """ Returns $(U,B,V.H)$ for $$A = UBV^{H}$$ where $A$ is the input matrix, and $B$ is its Bidiagonalized form Note: Bidiagonal Computation can hang for symbolic matrices. Parameters ========== upper : bool. Whether to do upper bidiagnalization or lower. True for upper and False for lower. References ========== .. [1] Algorithm 5.4.2, Matrix computations by Golub and Van Loan, 4th edition .. [2] Complex Matrix Bidiagonalization, https://github.com/vslobody/Householder-Bidiagonalization """ if not isinstance(upper, bool): raise ValueError("upper must be a boolean") if upper: return _bidiagonal_decmp_hholder(M) X = _bidiagonal_decmp_hholder(M.H) return X[2].H, X[1].H, X[0].H def _bidiagonalize(M, upper=True): """ Returns $B$, the Bidiagonalized form of the input matrix. Note: Bidiagonal Computation can hang for symbolic matrices. Parameters ========== upper : bool. Whether to do upper bidiagnalization or lower. True for upper and False for lower. References ========== .. [1] Algorithm 5.4.2, Matrix computations by Golub and Van Loan, 4th edition .. [2] Complex Matrix Bidiagonalization : https://github.com/vslobody/Householder-Bidiagonalization """ if not isinstance(upper, bool): raise ValueError("upper must be a boolean") if upper: return _eval_bidiag_hholder(M) return _eval_bidiag_hholder(M.H).H def _diagonalize(M, reals_only=False, sort=False, normalize=False): """ Return (P, D), where D is diagonal and D = P^-1 * M * P where M is current matrix. Parameters ========== reals_only : bool. Whether to throw an error if complex numbers are need to diagonalize. (Default: False) sort : bool. Sort the eigenvalues along the diagonal. (Default: False) normalize : bool. If True, normalize the columns of P. (Default: False) Examples ======== >>> from sympy import Matrix >>> M = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, -4, 2]) >>> M Matrix([ [1, 2, 0], [0, 3, 0], [2, -4, 2]]) >>> (P, D) = M.diagonalize() >>> D Matrix([ [1, 0, 0], [0, 2, 0], [0, 0, 3]]) >>> P Matrix([ [-1, 0, -1], [ 0, 0, -1], [ 2, 1, 2]]) >>> P.inv() * M * P Matrix([ [1, 0, 0], [0, 2, 0], [0, 0, 3]]) See Also ======== is_diagonal is_diagonalizable """ if not M.is_square: raise NonSquareMatrixError() is_diagonalizable, eigenvecs = _is_diagonalizable_with_eigen(M, reals_only=reals_only) if not is_diagonalizable: raise MatrixError("Matrix is not diagonalizable") if sort: eigenvecs = sorted(eigenvecs, key=default_sort_key) p_cols, diag = [], [] for val, mult, basis in eigenvecs: diag += [val] * mult p_cols += basis if normalize: p_cols = [v / v.norm() for v in p_cols] return M.hstack(*p_cols), M.diag(*diag) def _fuzzy_positive_definite(M): positive_diagonals = M._has_positive_diagonals() if positive_diagonals is False: return False if positive_diagonals and M.is_strongly_diagonally_dominant: return True return None def _fuzzy_positive_semidefinite(M): nonnegative_diagonals = M._has_nonnegative_diagonals() if nonnegative_diagonals is False: return False if nonnegative_diagonals and M.is_weakly_diagonally_dominant: return True return None def _is_positive_definite(M): if not M.is_hermitian: if not M.is_square: return False M = M + M.H fuzzy = _fuzzy_positive_definite(M) if fuzzy is not None: return fuzzy return _is_positive_definite_GE(M) def _is_positive_semidefinite(M): if not M.is_hermitian: if not M.is_square: return False M = M + M.H fuzzy = _fuzzy_positive_semidefinite(M) if fuzzy is not None: return fuzzy return _is_positive_semidefinite_cholesky(M) def _is_negative_definite(M): return _is_positive_definite(-M) def _is_negative_semidefinite(M): return _is_positive_semidefinite(-M) def _is_indefinite(M): if M.is_hermitian: eigen = M.eigenvals() args1 = [x.is_positive for x in eigen.keys()] any_positive = fuzzy_or(args1) args2 = [x.is_negative for x in eigen.keys()] any_negative = fuzzy_or(args2) return fuzzy_and([any_positive, any_negative]) elif M.is_square: return (M + M.H).is_indefinite return False def _is_positive_definite_GE(M): """A division-free gaussian elimination method for testing positive-definiteness.""" M = M.as_mutable() size = M.rows for i in range(size): is_positive = M[i, i].is_positive if is_positive is not True: return is_positive for j in range(i+1, size): M[j, i+1:] = M[i, i] * M[j, i+1:] - M[j, i] * M[i, i+1:] return True def _is_positive_semidefinite_cholesky(M): """Uses Cholesky factorization with complete pivoting References ========== .. [1] http://eprints.ma.man.ac.uk/1199/1/covered/MIMS_ep2008_116.pdf .. [2] https://www.value-at-risk.net/cholesky-factorization/ """ M = M.as_mutable() for k in range(M.rows): diags = [M[i, i] for i in range(k, M.rows)] pivot, pivot_val, nonzero, _ = _find_reasonable_pivot(diags) if nonzero: return None if pivot is None: for i in range(k+1, M.rows): for j in range(k, M.cols): iszero = M[i, j].is_zero if iszero is None: return None elif iszero is False: return False return True if M[k, k].is_negative or pivot_val.is_negative: return False elif not (M[k, k].is_nonnegative and pivot_val.is_nonnegative): return None if pivot > 0: M.col_swap(k, k+pivot) M.row_swap(k, k+pivot) M[k, k] = sqrt(M[k, k]) M[k, k+1:] /= M[k, k] M[k+1:, k+1:] -= M[k, k+1:].H * M[k, k+1:] return M[-1, -1].is_nonnegative _doc_positive_definite = \ r"""Finds out the definiteness of a matrix. Explanation =========== A square real matrix $A$ is: - A positive definite matrix if $x^T A x > 0$ for all non-zero real vectors $x$. - A positive semidefinite matrix if $x^T A x \geq 0$ for all non-zero real vectors $x$. - A negative definite matrix if $x^T A x < 0$ for all non-zero real vectors $x$. - A negative semidefinite matrix if $x^T A x \leq 0$ for all non-zero real vectors $x$. - An indefinite matrix if there exists non-zero real vectors $x, y$ with $x^T A x > 0 > y^T A y$. A square complex matrix $A$ is: - A positive definite matrix if $\text{re}(x^H A x) > 0$ for all non-zero complex vectors $x$. - A positive semidefinite matrix if $\text{re}(x^H A x) \geq 0$ for all non-zero complex vectors $x$. - A negative definite matrix if $\text{re}(x^H A x) < 0$ for all non-zero complex vectors $x$. - A negative semidefinite matrix if $\text{re}(x^H A x) \leq 0$ for all non-zero complex vectors $x$. - An indefinite matrix if there exists non-zero complex vectors $x, y$ with $\text{re}(x^H A x) > 0 > \text{re}(y^H A y)$. A matrix need not be symmetric or hermitian to be positive definite. - A real non-symmetric matrix is positive definite if and only if $\frac{A + A^T}{2}$ is positive definite. - A complex non-hermitian matrix is positive definite if and only if $\frac{A + A^H}{2}$ is positive definite. And this extension can apply for all the definitions above. However, for complex cases, you can restrict the definition of $\text{re}(x^H A x) > 0$ to $x^H A x > 0$ and require the matrix to be hermitian. But we do not present this restriction for computation because you can check ``M.is_hermitian`` independently with this and use the same procedure. Examples ======== An example of symmetric positive definite matrix: .. plot:: :context: reset :format: doctest :include-source: True >>> from sympy import Matrix, symbols >>> from sympy.plotting import plot3d >>> a, b = symbols('a b') >>> x = Matrix([a, b]) >>> A = Matrix([[1, 0], [0, 1]]) >>> A.is_positive_definite True >>> A.is_positive_semidefinite True >>> p = plot3d((x.T*A*x)[0, 0], (a, -1, 1), (b, -1, 1)) An example of symmetric positive semidefinite matrix: .. plot:: :context: close-figs :format: doctest :include-source: True >>> A = Matrix([[1, -1], [-1, 1]]) >>> A.is_positive_definite False >>> A.is_positive_semidefinite True >>> p = plot3d((x.T*A*x)[0, 0], (a, -1, 1), (b, -1, 1)) An example of symmetric negative definite matrix: .. plot:: :context: close-figs :format: doctest :include-source: True >>> A = Matrix([[-1, 0], [0, -1]]) >>> A.is_negative_definite True >>> A.is_negative_semidefinite True >>> A.is_indefinite False >>> p = plot3d((x.T*A*x)[0, 0], (a, -1, 1), (b, -1, 1)) An example of symmetric indefinite matrix: .. plot:: :context: close-figs :format: doctest :include-source: True >>> A = Matrix([[1, 2], [2, -1]]) >>> A.is_indefinite True >>> p = plot3d((x.T*A*x)[0, 0], (a, -1, 1), (b, -1, 1)) An example of non-symmetric positive definite matrix. .. plot:: :context: close-figs :format: doctest :include-source: True >>> A = Matrix([[1, 2], [-2, 1]]) >>> A.is_positive_definite True >>> A.is_positive_semidefinite True >>> p = plot3d((x.T*A*x)[0, 0], (a, -1, 1), (b, -1, 1)) Notes ===== Although some people trivialize the definition of positive definite matrices only for symmetric or hermitian matrices, this restriction is not correct because it does not classify all instances of positive definite matrices from the definition $x^T A x > 0$ or $\text{re}(x^H A x) > 0$. For instance, ``Matrix([[1, 2], [-2, 1]])`` presented in the example above is an example of real positive definite matrix that is not symmetric. However, since the following formula holds true; .. math:: \text{re}(x^H A x) > 0 \iff \text{re}(x^H \frac{A + A^H}{2} x) > 0 We can classify all positive definite matrices that may or may not be symmetric or hermitian by transforming the matrix to $\frac{A + A^T}{2}$ or $\frac{A + A^H}{2}$ (which is guaranteed to be always real symmetric or complex hermitian) and we can defer most of the studies to symmetric or hermitian positive definite matrices. But it is a different problem for the existance of Cholesky decomposition. Because even though a non symmetric or a non hermitian matrix can be positive definite, Cholesky or LDL decomposition does not exist because the decompositions require the matrix to be symmetric or hermitian. References ========== .. [1] https://en.wikipedia.org/wiki/Definiteness_of_a_matrix#Eigenvalues .. [2] http://mathworld.wolfram.com/PositiveDefiniteMatrix.html .. [3] Johnson, C. R. "Positive Definite Matrices." Amer. Math. Monthly 77, 259-264 1970. """ _is_positive_definite.__doc__ = _doc_positive_definite _is_positive_semidefinite.__doc__ = _doc_positive_definite _is_negative_definite.__doc__ = _doc_positive_definite _is_negative_semidefinite.__doc__ = _doc_positive_definite _is_indefinite.__doc__ = _doc_positive_definite def _jordan_form(M, calc_transform=True, *, chop=False): """Return $(P, J)$ where $J$ is a Jordan block matrix and $P$ is a matrix such that $M = P J P^{-1}$ Parameters ========== calc_transform : bool If ``False``, then only $J$ is returned. chop : bool All matrices are converted to exact types when computing eigenvalues and eigenvectors. As a result, there may be approximation errors. If ``chop==True``, these errors will be truncated. Examples ======== >>> from sympy import Matrix >>> M = Matrix([[ 6, 5, -2, -3], [-3, -1, 3, 3], [ 2, 1, -2, -3], [-1, 1, 5, 5]]) >>> P, J = M.jordan_form() >>> J Matrix([ [2, 1, 0, 0], [0, 2, 0, 0], [0, 0, 2, 1], [0, 0, 0, 2]]) See Also ======== jordan_block """ if not M.is_square: raise NonSquareMatrixError("Only square matrices have Jordan forms") mat = M has_floats = M.has(Float) if has_floats: try: max_prec = max(term._prec for term in M.values() if isinstance(term, Float)) except ValueError: # if no term in the matrix is explicitly a Float calling max() # will throw a error so setting max_prec to default value of 53 max_prec = 53 # setting minimum max_dps to 15 to prevent loss of precision in # matrix containing non evaluated expressions max_dps = max(prec_to_dps(max_prec), 15) def restore_floats(*args): """If ``has_floats`` is `True`, cast all ``args`` as matrices of floats.""" if has_floats: args = [m.evalf(n=max_dps, chop=chop) for m in args] if len(args) == 1: return args[0] return args # cache calculations for some speedup mat_cache = {} def eig_mat(val, pow): """Cache computations of ``(M - val*I)**pow`` for quick retrieval""" if (val, pow) in mat_cache: return mat_cache[(val, pow)] if (val, pow - 1) in mat_cache: mat_cache[(val, pow)] = mat_cache[(val, pow - 1)].multiply( mat_cache[(val, 1)], dotprodsimp=None) else: mat_cache[(val, pow)] = (mat - val*M.eye(M.rows)).pow(pow) return mat_cache[(val, pow)] # helper functions def nullity_chain(val, algebraic_multiplicity): """Calculate the sequence [0, nullity(E), nullity(E**2), ...] until it is constant where ``E = M - val*I``""" # mat.rank() is faster than computing the null space, # so use the rank-nullity theorem cols = M.cols ret = [0] nullity = cols - eig_mat(val, 1).rank() i = 2 while nullity != ret[-1]: ret.append(nullity) if nullity == algebraic_multiplicity: break nullity = cols - eig_mat(val, i).rank() i += 1 # Due to issues like #7146 and #15872, SymPy sometimes # gives the wrong rank. In this case, raise an error # instead of returning an incorrect matrix if nullity < ret[-1] or nullity > algebraic_multiplicity: raise MatrixError( "SymPy had encountered an inconsistent " "result while computing Jordan block: " "{}".format(M)) return ret def blocks_from_nullity_chain(d): """Return a list of the size of each Jordan block. If d_n is the nullity of E**n, then the number of Jordan blocks of size n is 2*d_n - d_(n-1) - d_(n+1)""" # d[0] is always the number of columns, so skip past it mid = [2*d[n] - d[n - 1] - d[n + 1] for n in range(1, len(d) - 1)] # d is assumed to plateau with "d[ len(d) ] == d[-1]", so # 2*d_n - d_(n-1) - d_(n+1) == d_n - d_(n-1) end = [d[-1] - d[-2]] if len(d) > 1 else [d[0]] return mid + end def pick_vec(small_basis, big_basis): """Picks a vector from big_basis that isn't in the subspace spanned by small_basis""" if len(small_basis) == 0: return big_basis[0] for v in big_basis: _, pivots = M.hstack(*(small_basis + [v])).echelon_form( with_pivots=True) if pivots[-1] == len(small_basis): return v # roots doesn't like Floats, so replace them with Rationals if has_floats: mat = mat.applyfunc(lambda x: nsimplify(x, rational=True)) # first calculate the jordan block structure eigs = mat.eigenvals() # Make sure that we have all roots in radical form for x in eigs: if x.has(CRootOf): raise MatrixError( "Jordan normal form is not implemented if the matrix have " "eigenvalues in CRootOf form") # most matrices have distinct eigenvalues # and so are diagonalizable. In this case, don't # do extra work! if len(eigs.keys()) == mat.cols: blocks = list(sorted(eigs.keys(), key=default_sort_key)) jordan_mat = mat.diag(*blocks) if not calc_transform: return restore_floats(jordan_mat) jordan_basis = [eig_mat(eig, 1).nullspace()[0] for eig in blocks] basis_mat = mat.hstack(*jordan_basis) return restore_floats(basis_mat, jordan_mat) block_structure = [] for eig in sorted(eigs.keys(), key=default_sort_key): algebraic_multiplicity = eigs[eig] chain = nullity_chain(eig, algebraic_multiplicity) block_sizes = blocks_from_nullity_chain(chain) # if block_sizes = = [a, b, c, ...], then the number of # Jordan blocks of size 1 is a, of size 2 is b, etc. # create an array that has (eig, block_size) with one # entry for each block size_nums = [(i+1, num) for i, num in enumerate(block_sizes)] # we expect larger Jordan blocks to come earlier size_nums.reverse() block_structure.extend( [(eig, size) for size, num in size_nums for _ in range(num)]) jordan_form_size = sum(size for eig, size in block_structure) if jordan_form_size != M.rows: raise MatrixError( "SymPy had encountered an inconsistent result while " "computing Jordan block. : {}".format(M)) blocks = (mat.jordan_block(size=size, eigenvalue=eig) for eig, size in block_structure) jordan_mat = mat.diag(*blocks) if not calc_transform: return restore_floats(jordan_mat) # For each generalized eigenspace, calculate a basis. # We start by looking for a vector in null( (A - eig*I)**n ) # which isn't in null( (A - eig*I)**(n-1) ) where n is # the size of the Jordan block # # Ideally we'd just loop through block_structure and # compute each generalized eigenspace. However, this # causes a lot of unneeded computation. Instead, we # go through the eigenvalues separately, since we know # their generalized eigenspaces must have bases that # are linearly independent. jordan_basis = [] for eig in sorted(eigs.keys(), key=default_sort_key): eig_basis = [] for block_eig, size in block_structure: if block_eig != eig: continue null_big = (eig_mat(eig, size)).nullspace() null_small = (eig_mat(eig, size - 1)).nullspace() # we want to pick something that is in the big basis # and not the small, but also something that is independent # of any other generalized eigenvectors from a different # generalized eigenspace sharing the same eigenvalue. vec = pick_vec(null_small + eig_basis, null_big) new_vecs = [eig_mat(eig, i).multiply(vec, dotprodsimp=None) for i in range(size)] eig_basis.extend(new_vecs) jordan_basis.extend(reversed(new_vecs)) basis_mat = mat.hstack(*jordan_basis) return restore_floats(basis_mat, jordan_mat) def _left_eigenvects(M, **flags): """Returns left eigenvectors and eigenvalues. This function returns the list of triples (eigenval, multiplicity, basis) for the left eigenvectors. Options are the same as for eigenvects(), i.e. the ``**flags`` arguments gets passed directly to eigenvects(). Examples ======== >>> from sympy import Matrix >>> M = Matrix([[0, 1, 1], [1, 0, 0], [1, 1, 1]]) >>> M.eigenvects() [(-1, 1, [Matrix([ [-1], [ 1], [ 0]])]), (0, 1, [Matrix([ [ 0], [-1], [ 1]])]), (2, 1, [Matrix([ [2/3], [1/3], [ 1]])])] >>> M.left_eigenvects() [(-1, 1, [Matrix([[-2, 1, 1]])]), (0, 1, [Matrix([[-1, -1, 1]])]), (2, 1, [Matrix([[1, 1, 1]])])] """ eigs = M.transpose().eigenvects(**flags) return [(val, mult, [l.transpose() for l in basis]) for val, mult, basis in eigs] def _singular_values(M): """Compute the singular values of a Matrix Examples ======== >>> from sympy import Matrix, Symbol >>> x = Symbol('x', real=True) >>> M = Matrix([[0, 1, 0], [0, x, 0], [-1, 0, 0]]) >>> M.singular_values() [sqrt(x**2 + 1), 1, 0] See Also ======== condition_number """ if M.rows >= M.cols: valmultpairs = M.H.multiply(M).eigenvals() else: valmultpairs = M.multiply(M.H).eigenvals() # Expands result from eigenvals into a simple list vals = [] for k, v in valmultpairs.items(): vals += [sqrt(k)] * v # dangerous! same k in several spots! # Pad with zeros if singular values are computed in reverse way, # to give consistent format. if len(vals) < M.cols: vals += [M.zero] * (M.cols - len(vals)) # sort them in descending order vals.sort(reverse=True, key=default_sort_key) return vals