Python Matrix.eigenvects方法代码示例

# 需要导入模块: from sympy.matrices import Matrix [as 别名]
# 或者: from sympy.matrices.Matrix import eigenvects [as 别名]
def demo_corr_bound(n):
    def corr(i, j):
        if i == j:
            return 1
        else:
            return rho
    Rho = Matrix(n, n, corr)
    print "what's the bound of rho to make below a correlation matrix?\n"
    pprint(Rho)
    print "\ncan be viewed as the sum of 2 matrices Rho = A + B:\n"
    A = (1-rho)*eye(n)
    B = rho*ones(n,n)
    pprint(A)
    pprint(B)
    print "\nthe eigen value and its dimention of first matrix A is:"
    pprint(A.eigenvects())
    print "\nas for the seconde matrix B\n"\
            "it's product of any vector v:"
    v = IndexedBase('v')
    vec = Matrix(n, 1, lambda i, j: v[i+1])
    pprint(vec)
    pprint(B*vec)
    print "\nin order for it's to equal to a linear transform of v\n"\
            "we can see its eigen values, vectors and dimentions are:"
    pprint(B.eigenvects())
    print "\nfor any eigen vector of B v, we have Rho*v :\n"
    pprint(Rho*vec)
    print "\nwhich means v is also an eigen vector of Rho,\n"\
            "the eigen values, vectors and dimentions of Rho are:\n"
    pprint(Rho.eigenvects())
    print "\nsince have no negative eigen values <=> positive semidefinite:\n"\
            "the boundaries for rho are: [1/(%d-1),1]" %n

# 需要导入模块: from sympy.matrices import Matrix [as 别名]
# 或者: from sympy.matrices.Matrix import eigenvects [as 别名]
def demo_eigen_number():
    m = Matrix(3,3,[2,1,0,0,2,1,0,0,2])
    pprint(m)
    print "\nany n*n matrix has n complex eigen values, counted with multiplications\n"\
            "since the characteritic polynomial is n-dim\n"\
            "and Fundamental Theorem of Algebra gives us exactly n roots:"
    pprint(m.charpoly().as_expr())
    print"\neach distinct eigen value as at least 1 at most multiplication eigen vectors\n"\
            "but it's possible to have less than multiplication number of eigen vectors\n"\
            "the given matrix is an example with only 1 eigen vector:\n"
    pprint(m.eigenvects())
    print "to see this for any vector v we can compute M * v:\n"
    v = IndexedBase('v')
    vec = Matrix(m.rows, 1, lambda i, j: v[i+1])
    pprint(vec)
    pprint(m*vec)

# 需要导入模块: from sympy.matrices import Matrix [as 别名]
# 或者: from sympy.matrices.Matrix import eigenvects [as 别名]

#.........这里部分代码省略.........
        [1,     0],
        [0,     1]
    ])

    # symmetric
    assert not a.is_symmetric(simplify=False)

    # test_cofactor
    assert sparse_eye(3) == sparse_eye(3).cofactor_matrix()
    test = SparseMatrix([[1, 3, 2], [2, 6, 3], [2, 3, 6]])
    assert test.cofactor_matrix() == \
        SparseMatrix([[27, -6, -6], [-12, 2, 3], [-3, 1, 0]])
    test = SparseMatrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
    assert test.cofactor_matrix() == \
        SparseMatrix([[-3, 6, -3], [6, -12, 6], [-3, 6, -3]])

    # test_jacobian
    x = Symbol('x')
    y = Symbol('y')
    L = SparseMatrix(1, 2, [x**2*y, 2*y**2 + x*y])
    syms = [x, y]
    assert L.jacobian(syms) == Matrix([[2*x*y, x**2], [y, 4*y + x]])

    L = SparseMatrix(1, 2, [x, x**2*y**3])
    assert L.jacobian(syms) == SparseMatrix([[1, 0], [2*x*y**3, x**2*3*y**2]])

    # test_QR
    A = Matrix([[1, 2], [2, 3]])
    Q, S = A.QRdecomposition()
    R = Rational
    assert Q == Matrix([
        [  5**R(-1, 2),  (R(2)/5)*(R(1)/5)**R(-1, 2)],
        [2*5**R(-1, 2), (-R(1)/5)*(R(1)/5)**R(-1, 2)]])
    assert S == Matrix([
        [5**R(1, 2),     8*5**R(-1, 2)],
        [         0, (R(1)/5)**R(1, 2)]])
    assert Q*S == A
    assert Q.T * Q == sparse_eye(2)

    R = Rational
    # test nullspace
    # first test reduced row-ech form

    M = SparseMatrix([[5, 7, 2, 1],
               [1, 6, 2, -1]])
    out, tmp = M.rref()
    assert out == Matrix([[1, 0, -R(2)/23, R(13)/23],
                          [0, 1,  R(8)/23, R(-6)/23]])

    M = SparseMatrix([[ 1,  3, 0,  2,  6, 3, 1],
                      [-2, -6, 0, -2, -8, 3, 1],
                      [ 3,  9, 0,  0,  6, 6, 2],
                      [-1, -3, 0,  1,  0, 9, 3]])

    out, tmp = M.rref()
    assert out == Matrix([[1, 3, 0, 0, 2, 0, 0],
                          [0, 0, 0, 1, 2, 0, 0],
                          [0, 0, 0, 0, 0, 1, R(1)/3],
                          [0, 0, 0, 0, 0, 0, 0]])
    # now check the vectors
    basis = M.nullspace()
    assert basis[0] == Matrix([-3, 1, 0, 0, 0, 0, 0])
    assert basis[1] == Matrix([0, 0, 1, 0, 0, 0, 0])
    assert basis[2] == Matrix([-2, 0, 0, -2, 1, 0, 0])
    assert basis[3] == Matrix([0, 0, 0, 0, 0, R(-1)/3, 1])

    # test eigen
    x = Symbol('x')
    y = Symbol('y')
    sparse_eye3 = sparse_eye(3)
    assert sparse_eye3.charpoly(x) == PurePoly(((x - 1)**3))
    assert sparse_eye3.charpoly(y) == PurePoly(((y - 1)**3))

    # test values
    M = Matrix([( 0, 1, -1),
                ( 1, 1,  0),
                (-1, 0,  1)])
    vals = M.eigenvals()
    assert sorted(vals.keys()) == [-1, 1, 2]

    R = Rational
    M = Matrix([[1, 0, 0],
                [0, 1, 0],
                [0, 0, 1]])
    assert M.eigenvects() == [(1, 3, [
        Matrix([1, 0, 0]),
        Matrix([0, 1, 0]),
        Matrix([0, 0, 1])])]
    M = Matrix([[5, 0, 2],
                [3, 2, 0],
                [0, 0, 1]])
    assert M.eigenvects() == [(1, 1, [Matrix([R(-1)/2, R(3)/2, 1])]),
                              (2, 1, [Matrix([0, 1, 0])]),
                              (5, 1, [Matrix([1, 1, 0])])]

    assert M.zeros(3, 5) == SparseMatrix(3, 5, {})
    A =  SparseMatrix(10, 10, {(0, 0): 18, (0, 9): 12, (1, 4): 18, (2, 7): 16, (3, 9): 12, (4, 2): 19, (5, 7): 16, (6, 2): 12, (9, 7): 18})
    assert A.row_list() == [(0, 0, 18), (0, 9, 12), (1, 4, 18), (2, 7, 16), (3, 9, 12), (4, 2, 19), (5, 7, 16), (6, 2, 12), (9, 7, 18)]
    assert A.col_list() == [(0, 0, 18), (4, 2, 19), (6, 2, 12), (1, 4, 18), (2, 7, 16), (5, 7, 16), (9, 7, 18), (0, 9, 12), (3, 9, 12)]
    assert SparseMatrix.eye(2).nnz() == 2