.. _matrices:
==========
Matrices
==========
>>> from sympy import *
>>> init_printing(use_unicode=True)
To make a matrix in SymPy, use the ``Matrix`` object. A matrix is constructed
by providing a list of row vectors that make up the matrix. For example,
to construct the matrix
.. math::
\left[\begin{array}{cc}1 & -1\\3 & 4\\0 & 2\end{array}\right]
use
>>> Matrix([[1, -1], [3, 4], [0, 2]])
⎡1 -1⎤
⎢ ⎥
⎢3 4 ⎥
⎢ ⎥
⎣0 2 ⎦
To make it easy to make column vectors, a list of elements is considered to be
a column vector.
>>> Matrix([1, 2, 3])
⎡1⎤
⎢ ⎥
⎢2⎥
⎢ ⎥
⎣3⎦
Matrices are manipulated just like any other object in SymPy or Python.
>>> M = Matrix([[1, 2, 3], [3, 2, 1]])
>>> N = Matrix([0, 1, 1])
>>> M*N
⎡5⎤
⎢ ⎥
⎣3⎦
One important thing to note about SymPy matrices is that, unlike every other
object in SymPy, they are mutable. This means that they can be modified in
place, as we will see below. The downside to this is that ``Matrix`` cannot
be used in places that require immutability, such as inside other SymPy
expressions or as keys to dictionaries. If you need an immutable version of
``Matrix``, use ``ImmutableMatrix``.
Basic Operations
================
Here are some basic operations on ``Matrix``.
Shape
-----
To get the shape of a matrix, use :func:`~.shape` function.
>>> from sympy import shape
>>> M = Matrix([[1, 2, 3], [-2, 0, 4]])
>>> M
⎡1 2 3⎤
⎢ ⎥
⎣-2 0 4⎦
>>> shape(M)
(2, 3)
Accessing Rows and Columns
--------------------------
To get an individual row or column of a matrix, use ``row`` or ``col``. For
example, ``M.row(0)`` will get the first row. ``M.col(-1)`` will get the last
column.
>>> M.row(0)
[1 2 3]
>>> M.col(-1)
⎡3⎤
⎢ ⎥
⎣4⎦
Deleting and Inserting Rows and Columns
---------------------------------------
To delete a row or column, use ``row_del`` or ``col_del``. These operations
will modify the Matrix **in place**.
>>> M.col_del(0)
>>> M
⎡2 3⎤
⎢ ⎥
⎣0 4⎦
>>> M.row_del(1)
>>> M
[2 3]
.. TODO: This is a mess. See issue 6992.
To insert rows or columns, use ``row_insert`` or ``col_insert``. These
operations **do not** operate in place.
>>> 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⎦
Unless explicitly stated, the methods mentioned below do not operate in
place. In general, a method that does not operate in place will return a new
``Matrix`` and a method that does operate in place will return ``None``.
Basic Methods
=============
As noted above, simple operations like addition, multiplication and power are
done just by using ``+``, ``*``, and ``**``. To find the inverse of a matrix,
just raise it to the ``-1`` power.
>>> M = Matrix([[1, 3], [-2, 3]])
>>> N = 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):
...
NonInvertibleMatrixError: Matrix det == 0; not invertible.
To take the transpose of a Matrix, use ``T``.
>>> M = Matrix([[1, 2, 3], [4, 5, 6]])
>>> M
⎡1 2 3⎤
⎢ ⎥
⎣4 5 6⎦
>>> M.T
⎡1 4⎤
⎢ ⎥
⎢2 5⎥
⎢ ⎥
⎣3 6⎦
Matrix Constructors
===================
Several constructors exist for creating common matrices. To create an
identity matrix, use ``eye``. ``eye(n)`` will create an `n\times n` identity matrix.
>>> 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 ``zeros``. ``zeros(n, m)`` creates an
`n\times m` matrix of `0`\ s.
>>> zeros(2, 3)
⎡0 0 0⎤
⎢ ⎥
⎣0 0 0⎦
Similarly, ``ones`` creates a matrix of ones.
>>> ones(3, 2)
⎡1 1⎤
⎢ ⎥
⎢1 1⎥
⎢ ⎥
⎣1 1⎦
To create diagonal matrices, use ``diag``. The arguments to ``diag`` can be
either numbers or matrices. A number is interpreted as a `1\times 1`
matrix. The matrices are stacked diagonally. The remaining elements are
filled with `0`\ s.
>>> 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
================
Determinant
-----------
To compute the determinant of a matrix, use ``det``.
>>> M = Matrix([[1, 0, 1], [2, -1, 3], [4, 3, 2]])
>>> M
⎡1 0 1⎤
⎢ ⎥
⎢2 -1 3⎥
⎢ ⎥
⎣4 3 2⎦
>>> M.det()
-1
RREF
----
To put a matrix into reduced row echelon form, use ``rref``. ``rref`` returns
a tuple of two elements. The first is the reduced row echelon form, and the
second is a tuple of indices of the pivot columns.
>>> M = Matrix([[1, 0, 1, 3], [2, 3, 4, 7], [-1, -3, -3, -4]])
>>> M
⎡1 0 1 3 ⎤
⎢ ⎥
⎢2 3 4 7 ⎥
⎢ ⎥
⎣-1 -3 -3 -4⎦
>>> M.rref()
⎛⎡1 0 1 3 ⎤ ⎞
⎜⎢ ⎥ ⎟
⎜⎢0 1 2/3 1/3⎥, (0, 1)⎟
⎜⎢ ⎥ ⎟
⎝⎣0 0 0 0 ⎦ ⎠
.. Note:: The first element of the tuple returned by ``rref`` is of type
``Matrix``. The second is of type ``tuple``.
Nullspace
---------
To find the nullspace of a matrix, use ``nullspace``. ``nullspace`` returns a
``list`` of column vectors that span the nullspace of the matrix.
>>> M = Matrix([[1, 2, 3, 0, 0], [4, 10, 0, 0, 1]])
>>> M
⎡1 2 3 0 0⎤
⎢ ⎥
⎣4 10 0 0 1⎦
>>> M.nullspace()
⎡⎡-15⎤ ⎡0⎤ ⎡ 1 ⎤⎤
⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎥
⎢⎢ 6 ⎥ ⎢0⎥ ⎢-1/2⎥⎥
⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎥
⎢⎢ 1 ⎥, ⎢0⎥, ⎢ 0 ⎥⎥
⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎥
⎢⎢ 0 ⎥ ⎢1⎥ ⎢ 0 ⎥⎥
⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎥
⎣⎣ 0 ⎦ ⎣0⎦ ⎣ 1 ⎦⎦
Columnspace
-----------
To find the columnspace of a matrix, use ``columnspace``. ``columnspace`` returns a
``list`` of column vectors that span the columnspace of the matrix.
>>> M = Matrix([[1, 1, 2], [2 ,1 , 3], [3 , 1, 4]])
>>> M
⎡1 1 2⎤
⎢ ⎥
⎢2 1 3⎥
⎢ ⎥
⎣3 1 4⎦
>>> M.columnspace()
⎡⎡1⎤ ⎡1⎤⎤
⎢⎢ ⎥ ⎢ ⎥⎥
⎢⎢2⎥, ⎢1⎥⎥
⎢⎢ ⎥ ⎢ ⎥⎥
⎣⎣3⎦ ⎣1⎦⎦
Eigenvalues, Eigenvectors, and Diagonalization
----------------------------------------------
To find the eigenvalues of a matrix, use ``eigenvals``. ``eigenvals``
returns a dictionary of ``eigenvalue: algebraic_multiplicity`` pairs (similar to the
output of :ref:`roots `).
>>> 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.
To find the eigenvectors of a matrix, use ``eigenvects``. ``eigenvects``
returns a list of tuples of the form ``(eigenvalue, algebraic_multiplicity,
[eigenvectors])``.
>>> M.eigenvects()
⎡⎛ ⎡⎡0⎤⎤⎞ ⎛ ⎡⎡1⎤⎤⎞ ⎛ ⎡⎡1⎤ ⎡0 ⎤⎤⎞⎤
⎢⎜ ⎢⎢ ⎥⎥⎟ ⎜ ⎢⎢ ⎥⎥⎟ ⎜ ⎢⎢ ⎥ ⎢ ⎥⎥⎟⎥
⎢⎜ ⎢⎢1⎥⎥⎟ ⎜ ⎢⎢1⎥⎥⎟ ⎜ ⎢⎢1⎥ ⎢-1⎥⎥⎟⎥
⎢⎜-2, 1, ⎢⎢ ⎥⎥⎟, ⎜3, 1, ⎢⎢ ⎥⎥⎟, ⎜5, 2, ⎢⎢ ⎥, ⎢ ⎥⎥⎟⎥
⎢⎜ ⎢⎢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 ``diagonalize``. ``diagonalize`` returns a tuple
`(P, D)`, where `D` is diagonal and `M = PDP^{-1}`.
>>> P, D = M.diagonalize()
>>> P
⎡0 1 1 0 ⎤
⎢ ⎥
⎢1 1 1 -1⎥
⎢ ⎥
⎢1 1 1 0 ⎥
⎢ ⎥
⎣1 1 0 1 ⎦
>>> D
⎡-2 0 0 0⎤
⎢ ⎥
⎢0 3 0 0⎥
⎢ ⎥
⎢0 0 5 0⎥
⎢ ⎥
⎣0 0 0 5⎦
>>> P*D*P**-1
⎡3 -2 4 -2⎤
⎢ ⎥
⎢5 3 -3 -2⎥
⎢ ⎥
⎢5 -2 2 -2⎥
⎢ ⎥
⎣5 -2 -3 3 ⎦
>>> P*D*P**-1 == M
True
.. sidebar:: Quick Tip
``lambda`` is a reserved keyword in Python, so to create a Symbol called
`\lambda`, while using the same names for SymPy Symbols and Python
variables, use ``lamda`` (without the ``b``). It will still pretty print
as `\lambda`.
Note that since ``eigenvects`` also includes the eigenvalues, you should use
it instead of ``eigenvals`` if you also want the eigenvectors. However, as
computing the eigenvectors may often be costly, ``eigenvals`` should be
preferred if you only wish to find the eigenvalues.
If all you want is the characteristic polynomial, use ``charpoly``. This is
more efficient than ``eigenvals``, because sometimes symbolic roots can be
expensive to calculate.
>>> lamda = symbols('lamda')
>>> p = M.charpoly(lamda)
>>> factor(p.as_expr())
2
(λ - 5) ⋅(λ - 3)⋅(λ + 2)
.. TODO: Add an example for ``jordan_form``, once it is fully implemented.
Possible Issues
===============
Zero Testing
------------
If your matrix operations are failing or returning wrong answers,
the common reasons would likely be from zero testing.
If there is an expression not properly zero-tested,
it can possibly bring issues in finding pivots for gaussian elimination,
or deciding whether the matrix is inversible,
or any high level functions which relies on the prior procedures.
Currently, the SymPy's default method of zero testing ``_iszero`` is only
guaranteed to be accurate in some limited domain of numerics and symbols,
and any complicated expressions beyond its decidability are treated as ``None``,
which behaves similarly to logical ``False``.
The list of methods using zero testing procedures are as follows:
``echelon_form`` , ``is_echelon`` , ``rank`` , ``rref`` , ``nullspace`` ,
``eigenvects`` , ``inverse_ADJ`` , ``inverse_GE`` , ``inverse_LU`` ,
``LUdecomposition`` , ``LUdecomposition_Simple`` , ``LUsolve``
They have property ``iszerofunc`` opened up for user to specify zero testing
method, which can accept any function with single input and boolean output,
while being defaulted with ``_iszero``.
Here is an example of solving an issue caused by undertested zero. While the
output for this particular matrix has since been improved, the technique
below is still of interest.
[#zerotestexampleidea-fn]_ [#zerotestexamplediscovery-fn]_
[#zerotestexampleimproved-fn]_
>>> from sympy import *
>>> q = Symbol("q", positive = True)
>>> m = Matrix([
... [-2*cosh(q/3), exp(-q), 1],
... [ exp(q), -2*cosh(q/3), 1],
... [ 1, 1, -2*cosh(q/3)]])
>>> m.nullspace() # doctest: +SKIP
[]
You can trace down which expression is being underevaluated,
by injecting a custom zero test with warnings enabled.
>>> import warnings
>>>
>>> def my_iszero(x):
... result = x.is_zero
...
... # Warnings if evaluated into None
... if result is None:
... warnings.warn("Zero testing of {} evaluated into None".format(x))
... return result
...
>>> m.nullspace(iszerofunc=my_iszero) # doctest: +SKIP
__main__:9: UserWarning: Zero testing of 4*cosh(q/3)**2 - 1 evaluated into None
__main__:9: UserWarning: Zero testing of (-exp(q) - 2*cosh(q/3))*(-2*cosh(q/3) - exp(-q)) - (4*cosh(q/3)**2 - 1)**2 evaluated into None
__main__:9: UserWarning: Zero testing of 2*exp(q)*cosh(q/3) - 16*cosh(q/3)**4 + 12*cosh(q/3)**2 + 2*exp(-q)*cosh(q/3) evaluated into None
__main__:9: UserWarning: Zero testing of -(4*cosh(q/3)**2 - 1)*exp(-q) - 2*cosh(q/3) - exp(-q) evaluated into None
[]
In this case,
``(-exp(q) - 2*cosh(q/3))*(-2*cosh(q/3) - exp(-q)) - (4*cosh(q/3)**2 - 1)**2``
should yield zero, but the zero testing had failed to catch.
possibly meaning that a stronger zero test should be introduced.
For this specific example, rewriting to exponentials and applying simplify would
make zero test stronger for hyperbolics,
while being harmless to other polynomials or transcendental functions.
>>> def my_iszero(x):
... result = x.rewrite(exp).simplify().is_zero
...
... # Warnings if evaluated into None
... if result is None:
... warnings.warn("Zero testing of {} evaluated into None".format(x))
... return result
...
>>> m.nullspace(iszerofunc=my_iszero) # doctest: +SKIP
__main__:9: UserWarning: Zero testing of -2*cosh(q/3) - exp(-q) evaluated into None
⎡⎡ ⎛ q ⎛q⎞⎞ -q 2⎛q⎞ ⎤⎤
⎢⎢- ⎜- ℯ - 2⋅cosh⎜─⎟⎟⋅ℯ + 4⋅cosh ⎜─⎟ - 1⎥⎥
⎢⎢ ⎝ ⎝3⎠⎠ ⎝3⎠ ⎥⎥
⎢⎢─────────────────────────────────────────⎥⎥
⎢⎢ ⎛ 2⎛q⎞ ⎞ ⎛q⎞ ⎥⎥
⎢⎢ 2⋅⎜4⋅cosh ⎜─⎟ - 1⎟⋅cosh⎜─⎟ ⎥⎥
⎢⎢ ⎝ ⎝3⎠ ⎠ ⎝3⎠ ⎥⎥
⎢⎢ ⎥⎥
⎢⎢ ⎛ q ⎛q⎞⎞ ⎥⎥
⎢⎢ -⎜- ℯ - 2⋅cosh⎜─⎟⎟ ⎥⎥
⎢⎢ ⎝ ⎝3⎠⎠ ⎥⎥
⎢⎢ ──────────────────── ⎥⎥
⎢⎢ 2⎛q⎞ ⎥⎥
⎢⎢ 4⋅cosh ⎜─⎟ - 1 ⎥⎥
⎢⎢ ⎝3⎠ ⎥⎥
⎢⎢ ⎥⎥
⎣⎣ 1 ⎦⎦
You can clearly see ``nullspace`` returning proper result, after injecting an
alternative zero test.
Note that this approach is only valid for some limited cases of matrices
containing only numerics, hyperbolics, and exponentials.
For other matrices, you should use different method opted for their domains.
Possible suggestions would be either taking advantage of rewriting and
simplifying, with tradeoff of speed [#zerotestsimplifysolution-fn]_ ,
or using random numeric testing, with tradeoff of accuracy
[#zerotestnumerictestsolution-fn]_ .
If you wonder why there is no generic algorithm for zero testing that can work
with any symbolic entities,
it's because of the constant problem stating that zero testing is undecidable
[#constantproblemwikilink-fn]_ ,
and not only the SymPy, but also other computer algebra systems
[#mathematicazero-fn]_ [#matlabzero-fn]_
would face the same fundamental issue.
However, discovery of any zero test failings can provide some good examples to
improve SymPy,
so if you have encountered one, you can report the issue to
SymPy issue tracker [#sympyissues-fn]_ to get detailed help from the community.
.. rubric:: Footnotes
.. [#zerotestexampleidea-fn] Inspired by https://gitter.im/sympy/sympy?at=5b7c3e8ee5b40332abdb206c
.. [#zerotestexamplediscovery-fn] Discovered from https://github.com/sympy/sympy/issues/15141
.. [#zerotestexampleimproved-fn] Improved by https://github.com/sympy/sympy/pull/19548
.. [#zerotestsimplifysolution-fn] Suggested from https://github.com/sympy/sympy/issues/10120
.. [#zerotestnumerictestsolution-fn] Suggested from https://github.com/sympy/sympy/issues/10279
.. [#constantproblemwikilink-fn] https://en.wikipedia.org/wiki/Constant_problem
.. [#mathematicazero-fn] How mathematica tests zero https://reference.wolfram.com/language/ref/PossibleZeroQ.html
.. [#matlabzero-fn] How matlab tests zero https://web.archive.org/web/20200307091449/https://www.mathworks.com/help/symbolic/mupad_ref/iszero.html
.. [#sympyissues-fn] https://github.com/sympy/sympy/issues