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
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 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]
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 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
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)
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. [1] [2] [3]
>>> 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()
[]
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)
__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)
__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 [4] , or using random numeric testing, with tradeoff of accuracy [5] .
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 [6] , and not only the SymPy, but also other computer algebra systems [7] [8] 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 [9] to get detailed help from the community.
Footnotes