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Matrices

To make a matrix in Diofant, use the :class:`~diofant.matrices.Matrix` object. A matrix is constructed by providing a list of row vectors that make up the matrix.

>>> Matrix([[1, -1], [3, 4], [0, 2]])
⎡1  -1⎤
⎢     ⎥
⎢3  4 ⎥
⎢     ⎥
⎣0  2 ⎦

A list of elements is considered to be a column vector.

>>> Matrix([1, 2, 3])
⎡1⎤
⎢ ⎥
⎢2⎥
⎢ ⎥
⎣3⎦

One important thing to note about Diofant matrices is that, unlike every other object in Diofant, they are mutable. This means that they can be modified in place, as we will see below. Use :class:`~diofant.matrices.immutable.ImmutableMatrix` in places that require immutability, such as inside other Diofant expressions or as keys to dictionaries.

Indexing

Diofant matrices support subscription of matrix elements with pair of integers or :class:`slice` instances. In last case, new :class:`~diofant.matrices.Matrix` instances will be returned.

>>> M = Matrix([[1, 2, 3], [4, 5, 6]])
>>> M[0, 1]
2
>>> M[1, 1]
5
>>> M[:, 1]
⎡2⎤
⎢ ⎥
⎣5⎦
>>> M[1, :-1]
[4  5]
>>> M[0, :]
[1  2  3]
>>> M[:, -1]
⎡3⎤
⎢ ⎥
⎣6⎦

It's possible to modify matrix elements.

>>> M[0, 0] = 0
>>> M
⎡0  2  3⎤
⎢       ⎥
⎣4  5  6⎦
>>> M[1, 1:] = Matrix([[0, 0]])
>>> M
⎡0  2  3⎤
⎢       ⎥
⎣4  0  0⎦

Reshape and Rearrange

To get the shape of a matrix use :attr:`~diofant.matrices.matrices.MatrixBase.shape` property

>>> M = Matrix([[1, 2, 3], [-2, 0, 4]])
>>> M
⎡1   2  3⎤
⎢        ⎥
⎣-2  0  4⎦
>>> M.shape
(2, 3)

To delete a row or column, use :keyword:`del`

>>> del M[:, 0]
>>> M
⎡2  3⎤
⎢    ⎥
⎣0  4⎦
>>> del M[1, :]
>>> M
[2  3]

To insert rows or columns, use methods :meth:`~diofant.matrices.matrices.MatrixBase.row_insert` or :meth:`~diofant.matrices.matrices.MatrixBase.col_insert`.

>>> M
[2  3]
>>> M = M.row_insert(1, Matrix([[0, 4]]))
>>> M
⎡2  3⎤
⎢    ⎥
⎣0  4⎦
>>> M = M.col_insert(0, Matrix([1, -2]))
>>> M
⎡1   2  3⎤
⎢        ⎥
⎣-2  0  4⎦

Note

You can see, that these methods will modify the Matrix in place. In general, as a rule, such methods will return None.

To swap two given rows or columns, use methods :meth:`~diofant.matrices.dense.MutableDenseMatrix.row_swap` or :meth:`~diofant.matrices.dense.MutableDenseMatrix.col_swap`.

>>> M.row_swap(0, 1)
>>> M
⎡-2  0  4⎤
⎢        ⎥
⎣1   2  3⎦
>>> M.col_swap(1, 2)
>>> M
⎡-2  4  0⎤
⎢        ⎥
⎣1   3  2⎦

To take the transpose of a Matrix, use :attr:`~diofant.matrices.matrices.MatrixBase.T` property.

>>> M.T
⎡-2  1⎤
⎢     ⎥
⎢4   3⎥
⎢     ⎥
⎣0   2⎦

Algebraic Operations

Simple operations like addition and multiplication are done just by using +, *, and **. To find the inverse of a matrix, just raise it to the -1 power.

>>> M, N = Matrix([[1, 3], [-2, 3]]), Matrix([[0, 3], [0, 7]])
>>> M + N
⎡1   6 ⎤
⎢      ⎥
⎣-2  10⎦
>>> M*N
⎡0  24⎤
⎢     ⎥
⎣0  15⎦
>>> 3*M
⎡3   9⎤
⎢     ⎥
⎣-6  9⎦
>>> M**2
⎡-5  12⎤
⎢      ⎥
⎣-8  3 ⎦
>>> M**-1
⎡1/3  -1/3⎤
⎢         ⎥
⎣2/9  1/9 ⎦
>>> N**-1
Traceback (most recent call last):
...
ValueError: Matrix det == 0; not invertible.

Special Matrices

Several constructors exist for creating common matrices. To create an identity matrix, use :func:`~diofant.matrices.dense.eye` function.

>>> eye(3)
⎡1  0  0⎤
⎢       ⎥
⎢0  1  0⎥
⎢       ⎥
⎣0  0  1⎦
>>> eye(4)
⎡1  0  0  0⎤
⎢          ⎥
⎢0  1  0  0⎥
⎢          ⎥
⎢0  0  1  0⎥
⎢          ⎥
⎣0  0  0  1⎦

To create a matrix of all zeros, use :func:`~diofant.matrices.dense.zeros` function.

>>> zeros(2, 3)
⎡0  0  0⎤
⎢       ⎥
⎣0  0  0⎦

Similarly, function :func:`~diofant.matrices.dense.ones` creates a matrix of ones.

>>> ones(3, 2)
⎡1  1⎤
⎢    ⎥
⎢1  1⎥
⎢    ⎥
⎣1  1⎦

To create diagonal matrices, use function :func:`~diofant.matrices.dense.diag`. Its arguments can be either numbers or matrices. A number is interpreted as a 1times 1 matrix. The matrices are stacked diagonally.

>>> diag(1, 2, 3)
⎡1  0  0⎤
⎢       ⎥
⎢0  2  0⎥
⎢       ⎥
⎣0  0  3⎦
>>> diag(-1, ones(2, 2), Matrix([5, 7, 5]))
⎡-1  0  0  0⎤
⎢           ⎥
⎢0   1  1  0⎥
⎢           ⎥
⎢0   1  1  0⎥
⎢           ⎥
⎢0   0  0  5⎥
⎢           ⎥
⎢0   0  0  7⎥
⎢           ⎥
⎣0   0  0  5⎦

Advanced Methods

To compute the determinant of a matrix, use :meth:`~diofant.matrices.matrices.MatrixBase.det` method.

>>> Matrix([[1, 0, 1], [2, -1, 3], [4, 3, 2]])
⎡1  0   1⎤
⎢        ⎥
⎢2  -1  3⎥
⎢        ⎥
⎣4  3   2⎦
>>> det(_)
-1

To put a matrix into reduced row echelon form, use method :meth:`~diofant.matrices.matrices.MatrixBase.rref`. It returns a tuple of two elements. The first is the reduced row echelon form, and the second is a list of indices of the pivot columns.

>>> Matrix([[1, 0, 1, 3], [2, 3, 4, 7], [-1, -3, -3, -4]])
⎡1   0   1   3 ⎤
⎢              ⎥
⎢2   3   4   7 ⎥
⎢              ⎥
⎣-1  -3  -3  -4⎦
>>> _.rref()
⎛⎡1  0   1    3 ⎤, [0, 1]⎞
⎜⎢              ⎥        ⎟
⎜⎢0  1  2/3  1/3⎥        ⎟
⎜⎢              ⎥        ⎟
⎝⎣0  0   0    0 ⎦        ⎠

To find the nullspace of a matrix, use method :meth:`~diofant.matrices.matrices.MatrixBase.nullspace`. It returns a list of column vectors that span the nullspace of the matrix.

>>> Matrix([[1, 2, 3, 0, 0], [4, 10, 0, 0, 1]])
⎡1  2   3  0  0⎤
⎢              ⎥
⎣4  10  0  0  1⎦
>>> _.nullspace()
⎡⎡-15⎤, ⎡0⎤, ⎡ 1  ⎤⎤
⎢⎢   ⎥  ⎢ ⎥  ⎢    ⎥⎥
⎢⎢ 6 ⎥  ⎢0⎥  ⎢-1/2⎥⎥
⎢⎢   ⎥  ⎢ ⎥  ⎢    ⎥⎥
⎢⎢ 1 ⎥  ⎢0⎥  ⎢ 0  ⎥⎥
⎢⎢   ⎥  ⎢ ⎥  ⎢    ⎥⎥
⎢⎢ 0 ⎥  ⎢1⎥  ⎢ 0  ⎥⎥
⎢⎢   ⎥  ⎢ ⎥  ⎢    ⎥⎥
⎣⎣ 0 ⎦  ⎣0⎦  ⎣ 1  ⎦⎦

To find the eigenvalues of a matrix, use method :meth:`~diofant.matrices.matrices.MatrixBase.eigenvals`. It returns a dictionary of roots including its multiplicity (similar to the output of :func:`~diofant.polys.polyroots.roots` function).

>>> M = Matrix([[3, -2,  4, -2], [5,  3, -3, -2],
...             [5, -2,  2, -2], [5, -2, -3,  3]])
>>> M
⎡3  -2  4   -2⎤
⎢             ⎥
⎢5  3   -3  -2⎥
⎢             ⎥
⎢5  -2  2   -2⎥
⎢             ⎥
⎣5  -2  -3  3 ⎦
>>> M.eigenvals()
{-2: 1, 3: 1, 5: 2}

This means that M has eigenvalues -2, 3, and 5, and that the eigenvalues -2 and 3 have algebraic multiplicity 1 and that the eigenvalue 5 has algebraic multiplicity 2.

Matrices can have symbolic elements.

>>> Matrix([[x, x + y], [y, x]])
⎡x  x + y⎤
⎢        ⎥
⎣y    x  ⎦
>>> _.eigenvals()
⎧      ___________           ___________   ⎫
⎨x - ╲╱ y⋅(x + y) : 1, x + ╲╱ y⋅(x + y) : 1⎬
⎩                                          ⎭

To find the eigenvectors of a matrix, use method :meth:`~diofant.matrices.matrices.MatrixBase.eigenvects`.

>>> M.eigenvects()
⎡⎛-2, 1, ⎡⎡0⎤⎤⎞, ⎛3, 1, ⎡⎡1⎤⎤⎞, ⎛5, 2, ⎡⎡1⎤, ⎡0 ⎤⎤⎞⎤
⎢⎜       ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥  ⎢  ⎥⎥⎟⎥
⎢⎜       ⎢⎢1⎥⎥⎟  ⎜      ⎢⎢1⎥⎥⎟  ⎜      ⎢⎢1⎥  ⎢-1⎥⎥⎟⎥
⎢⎜       ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥  ⎢  ⎥⎥⎟⎥
⎢⎜       ⎢⎢1⎥⎥⎟  ⎜      ⎢⎢1⎥⎥⎟  ⎜      ⎢⎢1⎥  ⎢0 ⎥⎥⎟⎥
⎢⎜       ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥  ⎢  ⎥⎥⎟⎥
⎣⎝       ⎣⎣1⎦⎦⎠  ⎝      ⎣⎣1⎦⎦⎠  ⎝      ⎣⎣0⎦  ⎣1 ⎦⎦⎠⎦

This shows us that, for example, the eigenvalue 5 also has geometric multiplicity 2, because it has two eigenvectors. Because the algebraic and geometric multiplicities are the same for all the eigenvalues, M is diagonalizable.

To diagonalize a matrix, use method :meth:`~diofant.matrices.matrices.MatrixBase.diagonalize`. It returns a tuple (P, D), where D is diagonal and M = PDP^{-1}.

>>> M.diagonalize()
⎛⎡0  1  1  0 ⎤, ⎡-2  0  0  0⎤⎞
⎜⎢           ⎥  ⎢           ⎥⎟
⎜⎢1  1  1  -1⎥  ⎢0   3  0  0⎥⎟
⎜⎢           ⎥  ⎢           ⎥⎟
⎜⎢1  1  1  0 ⎥  ⎢0   0  5  0⎥⎟
⎜⎢           ⎥  ⎢           ⎥⎟
⎝⎣1  1  0  1 ⎦  ⎣0   0  0  5⎦⎠
>>> _[0]*_[1]*_[0]**-1 == M
True

If all you want is the characteristic polynomial, use method :meth:`~diofant.matrices.matrices.MatrixBase.charpoly`. This is more efficient than :meth:`~diofant.matrices.matrices.MatrixBase.eigenvals` method, because sometimes symbolic roots can be expensive to calculate.

>>> M.charpoly(x)
PurePoly(x**4 - 11*x**3 + 29*x**2 + 35*x - 150, x, domain='ZZ')
>>> factor(_)
       2
(x - 5) ⋅(x - 3)⋅(x + 2)

To compute Jordan canonical form J for matrix M and its similarity transformation P (i.e. such that J = P M P^{-1}), use method :meth:`~diofant.matrices.matrices.MatrixBase.jordan_form`.

>>> Matrix([[-2, 4], [1, 3]]).jordan_form()
⎛                              ⎡      ____              ⎤⎞
⎜⎡    -4            -4      ⎤  ⎢1   ╲╱ 41               ⎥⎟
⎜⎢────────────  ────────────⎥, ⎢─ + ──────       0      ⎥⎟
⎜⎢    ____              ____⎥  ⎢2     2                 ⎥⎟
⎜⎢  ╲╱ 41    5    5   ╲╱ 41 ⎥  ⎢                        ⎥⎟
⎜⎢- ────── - ─  - ─ + ──────⎥  ⎢                ____    ⎥⎟
⎜⎢    2      2    2     2   ⎥  ⎢              ╲╱ 41    1⎥⎟
⎜⎢                          ⎥  ⎢    0       - ────── + ─⎥⎟
⎝⎣     1             1      ⎦  ⎣                2      2⎦⎠