# Matrices (linear algebra)¶

## Creating Matrices¶

The linear algebra module is designed to be as simple as possible. First, we import and declare our first Matrix object:

>>> from sympy.interactive.printing import init_printing
>>> init_printing(use_unicode=False, wrap_line=False, no_global=True)
>>> from sympy.matrices import *
>>> Matrix([[1,0], [0,1]])
[1  0]
[    ]
[0  1]
>>> Matrix((
...   Matrix((
...     (1, 0, 0),
...     (0, 0, 0)
...   )),
...   (0, 0, -1)
... ))
[1  0  0 ]
[        ]
[0  0  0 ]
[        ]
[0  0  -1]
>>> Matrix([[1, 2, 3]])
[1 2 3]
>>> Matrix([1, 2, 3])
[1]
[ ]
[2]
[ ]
[3]


This is the standard manner one creates a matrix, i.e. with a list of appropriately-sizes lists and/or matrices. SymPy also supports more advanced methods of matrix creation including a single list of values and dimension inputs:

>>> Matrix(2, 3, [1, 2, 3, 4, 5, 6])
[1  2  3]
[       ]
[4  5  6]


More interestingly (and usefully), we can use a 2-variable function (or lambda) to make one. Here we create an indicator function which is 1 on the diagonal and then use it to make the identity matrix:

>>> def f(i,j):
...     if i == j:
...             return 1
...     else:
...             return 0
...
>>> Matrix(4, 4, f)
[1  0  0  0]
[          ]
[0  1  0  0]
[          ]
[0  0  1  0]
[          ]
[0  0  0  1]


Finally let’s use lambda to create a 1-line matrix with 1’s in the even permutation entries:

>>> Matrix(3, 4, lambda i,j: 1 - (i+j) % 2)
[1  0  1  0]
[          ]
[0  1  0  1]
[          ]
[1  0  1  0]


There are also a couple of special constructors for quick matrix construction - eye is the identity matrix, zeros and ones for matrices of all zeros and ones, respectively:

>>> eye(4)
[1  0  0  0]
[          ]
[0  1  0  0]
[          ]
[0  0  1  0]
[          ]
[0  0  0  1]
>>> zeros(2)
[0  0]
[    ]
[0  0]
>>> zeros(2, 5)
[0  0  0  0  0]
[             ]
[0  0  0  0  0]
>>> ones(3)
[1  1  1]
[       ]
[1  1  1]
[       ]
[1  1  1]
>>> ones(1, 3)
[1  1  1]


## Basic Manipulation¶

While learning to work with matrices, let’s choose one where the entries are readily identifiable. One useful thing to know is that while matrices are 2-dimensional, the storage is not and so it is allowable - though one should be careful - to access the entries as if they were a 1-d list.

>>> M = Matrix(2, 3, [1, 2, 3, 4, 5, 6])
>>> M[4]
5


Now, the more standard entry access is a pair of indices:

>>> M[1,2]
6
>>> M[0,0]
1
>>> M[1,1]
5


Since this is Python we’re also able to slice submatrices:

>>> M[0:2,0:2]
[1  2]
[    ]
[4  5]
>>> M[1:2,2]
[6]
>>> M[:,2]
[3]
[ ]
[6]


Remember in the 2nd example above that slicing 2:2 gives an empty range and that, as in python, a 4 column list is indexed from 0 to 3. In particular, this mean a quick way to create a copy of the matrix is:

>>> M2 = M[:,:]
>>> M2[0,0] = 100
>>> M
[1  2  3]
[       ]
[4  5  6]


See? Changing M2 didn’t change M. Since we can slice, we can also assign entries:

>>> M = Matrix(([1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]))
>>> M
[1   2   3   4 ]
[              ]
[5   6   7   8 ]
[              ]
[9   10  11  12]
[              ]
[13  14  15  16]
>>> M[2,2] = M[0,3] = 0
>>> M
[1   2   3   0 ]
[              ]
[5   6   7   8 ]
[              ]
[9   10  0   12]
[              ]
[13  14  15  16]


as well as assign slices:

>>> M = Matrix(([1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]))
>>> M[2:,2:] = Matrix(2,2,lambda i,j: 0)
>>> M
[1   2   3  4]
[            ]
[5   6   7  8]
[            ]
[9   10  0  0]
[            ]
[13  14  0  0]


All the standard arithmetic operations are supported:

>>> M = Matrix(([1,2,3],[4,5,6],[7,8,9]))
>>> M - M
[0  0  0]
[       ]
[0  0  0]
[       ]
[0  0  0]
>>> M + M
[2   4   6 ]
[          ]
[8   10  12]
[          ]
[14  16  18]
>>> M * M
[30   36   42 ]
[             ]
[66   81   96 ]
[             ]
[102  126  150]
>>> M2 = Matrix(3,1,[1,5,0])
>>> M*M2
[11]
[  ]
[29]
[  ]
[47]
>>> M**2
[30   36   42 ]
[             ]
[66   81   96 ]
[             ]
[102  126  150]


As well as some useful vector operations:

>>> M.row_del(0)
>>> M
[4  5  6]
[       ]
[7  8  9]
>>> M.col_del(1)
>>> M
[4  6]
[    ]
[7  9]
>>> v1 = Matrix([1,2,3])
>>> v2 = Matrix([4,5,6])
>>> v3 = v1.cross(v2)
>>> v1.dot(v2)
32
>>> v2.dot(v3)
0
>>> v1.dot(v3)
0


Recall that the row_del() and col_del() operations don’t return a value - they simply change the matrix object. We can also ‘’glue’’ together matrices of the appropriate size:

>>> M1 = eye(3)
>>> M2 = zeros(3, 4)
>>> M1.row_join(M2)
[1  0  0  0  0  0  0]
[                   ]
[0  1  0  0  0  0  0]
[                   ]
[0  0  1  0  0  0  0]
>>> M3 = zeros(4, 3)
>>> M1.col_join(M3)
[1  0  0]
[       ]
[0  1  0]
[       ]
[0  0  1]
[       ]
[0  0  0]
[       ]
[0  0  0]
[       ]
[0  0  0]
[       ]
[0  0  0]


## Operations on entries¶

We are not restricted to having multiplication between two matrices:

>>> M = eye(3)
>>> 2*M
[2  0  0]
[       ]
[0  2  0]
[       ]
[0  0  2]
>>> 3*M
[3  0  0]
[       ]
[0  3  0]
[       ]
[0  0  3]


but we can also apply functions to our matrix entries using applyfunc(). Here we’ll declare a function that double any input number. Then we apply it to the 3x3 identity matrix:

>>> f = lambda x: 2*x
>>> eye(3).applyfunc(f)
[2  0  0]
[       ]
[0  2  0]
[       ]
[0  0  2]


One more useful matrix-wide entry application function is the substitution function. Let’s declare a matrix with symbolic entries then substitute a value. Remember we can substitute anything - even another symbol!:

>>> from sympy import Symbol
>>> x = Symbol('x')
>>> M = eye(3) * x
>>> M
[x  0  0]
[       ]
[0  x  0]
[       ]
[0  0  x]
>>> M.subs(x, 4)
[4  0  0]
[       ]
[0  4  0]
[       ]
[0  0  4]
>>> y = Symbol('y')
>>> M.subs(x, y)
[y  0  0]
[       ]
[0  y  0]
[       ]
[0  0  y]


## Linear algebra¶

Now that we have the basics out of the way, let’s see what we can do with the actual matrices. Of course the first things that come to mind are the basics like the determinant:

>>> M = Matrix(( [1, 2, 3], [3, 6, 2], [2, 0, 1] ))
>>> M.det()
-28
>>> M2 = eye(3)
>>> M2.det()
1
>>> M3 = Matrix(( [1, 0, 0], [1, 0, 0], [1, 0, 0] ))
>>> M3.det()
0


and the inverse. In SymPy the inverse is computed by Gaussian elimination by default but we can specify it be done by LU decomposition as well:

>>> M2.inv()
[1  0  0]
[       ]
[0  1  0]
[       ]
[0  0  1]
>>> M2.inv("LU")
[1  0  0]
[       ]
[0  1  0]
[       ]
[0  0  1]
>>> M.inv("LU")
[-3/14  1/14  1/2 ]
[                 ]
[-1/28  5/28  -1/4]
[                 ]
[ 3/7   -1/7   0  ]
>>> M * M.inv("LU")
[1  0  0]
[       ]
[0  1  0]
[       ]
[0  0  1]


We can perform a QR factorization which is handy for solving systems:

>>> A = Matrix([[1,1,1],[1,1,3],[2,3,4]])
>>> Q, R = A.QRdecomposition()
>>> Q
[  ___     ___     ___]
[\/ 6   -\/ 3   -\/ 2 ]
[-----  ------  ------]
[  6      3       2   ]
[                     ]
[  ___     ___    ___ ]
[\/ 6   -\/ 3   \/ 2  ]
[-----  ------  ----- ]
[  6      3       2   ]
[                     ]
[  ___    ___         ]
[\/ 6   \/ 3          ]
[-----  -----     0   ]
[  3      3           ]
>>> R
[           ___         ]
[  ___  4*\/ 6       ___]
[\/ 6   -------  2*\/ 6 ]
[          3            ]
[                       ]
[          ___          ]
[        \/ 3           ]
[  0     -----      0   ]
[          3            ]
[                       ]
[                   ___ ]
[  0       0      \/ 2  ]
>>> Q*R
[1  1  1]
[       ]
[1  1  3]
[       ]
[2  3  4]


In addition to the solvers in the solver.py file, we can solve the system Ax=b by passing the b vector to the matrix A’s LUsolve function. Here we’ll cheat a little choose A and x then multiply to get b. Then we can solve for x and check that it’s correct:

>>> A = Matrix([ [2, 3, 5], [3, 6, 2], [8, 3, 6] ])
>>> x = Matrix(3,1,[3,7,5])
>>> b = A*x
>>> soln = A.LUsolve(b)
>>> soln
[3]
[ ]
[7]
[ ]
[5]


There’s also a nice Gram-Schmidt orthogonalizer which will take a set of vectors and orthogonalize then with respect to another another. There is an optional argument which specifies whether or not the output should also be normalized, it defaults to False. Let’s take some vectors and orthogonalize them - one normalized and one not:

>>> L = [Matrix([2,3,5]), Matrix([3,6,2]), Matrix([8,3,6])]
>>> out1 = GramSchmidt(L)
>>> out2 = GramSchmidt(L, True)


Let’s take a look at the vectors:

>>> for i in out1:
...     print i
...
[2]
[3]
[5]
[ 23/19]
[ 63/19]
[-47/19]
[ 1692/353]
[-1551/706]
[ -423/706]
>>> for i in out2:
...      print i
...
[  sqrt(38)/19]
[3*sqrt(38)/38]
[5*sqrt(38)/38]
[ 23*sqrt(6707)/6707]
[ 63*sqrt(6707)/6707]
[-47*sqrt(6707)/6707]
[ 12*sqrt(706)/353]
[-11*sqrt(706)/706]
[ -3*sqrt(706)/706]


We can spot-check their orthogonality with dot() and their normality with norm():

>>> out1[0].dot(out1[1])
0
>>> out1[0].dot(out1[2])
0
>>> out1[1].dot(out1[2])
0
>>> out2[0].norm()
1
>>> out2[1].norm()
1
>>> out2[2].norm()
1


So there is quite a bit that can be done with the module including eigenvalues, eigenvectors, nullspace calculation, cofactor expansion tools, and so on. From here one might want to look over the matrices.py file for all functionality.

## Matrix Class Reference¶

class sympy.matrices.matrices.MatrixBase[source]
C

By-element conjugation.

D[source]

Dirac conjugation.

conjugate
By-element conjugation
H
Hermite conjugation
H[source]

Hermite conjugation.

>>> from sympy import Matrix, I
>>> m=Matrix(((1,2+I),(3,4)))
>>> m
[1, 2 + I]
[3,     4]
>>> m.H
[    1, 3]
[2 - I, 4]


conjugate
By-element conjugation
D
Dirac conjugation
LDLdecomposition()[source]

Returns the LDL Decomposition (L,D) of matrix A, such that L * D * L.T == A This method eliminates the use of square root. Further this ensures that all the diagonal entries of L are 1. A must be a square, symmetric, positive-definite and non-singular matrix.

>>> from sympy.matrices import Matrix, eye
>>> A = Matrix(((25,15,-5),(15,18,0),(-5,0,11)))
>>> L, D = A.LDLdecomposition()
>>> L
[   1,   0, 0]
[ 3/5,   1, 0]
[-1/5, 1/3, 1]
>>> D
[25, 0, 0]
[ 0, 9, 0]
[ 0, 0, 9]
>>> L * D * L.T * A.inv() == eye(A.rows)
True


LDLsolve(rhs)[source]

Solves Ax = B using LDL decomposition, for a general square and non-singular matrix.

For a non-square matrix with rows > cols, the least squares solution is returned.

LUdecomposition(iszerofunc=<function _iszero at 0x1076d5848>)[source]

Returns the decomposition LU and the row swaps p.

Examples

>>> from sympy import Matrix
>>> a = Matrix([[4, 3], [6, 3]])
>>> L, U, _ = a.LUdecomposition()
>>> L
[  1, 0]
[3/2, 1]
>>> U
[4,    3]
[0, -3/2]

LUdecompositionFF()[source]

Compute a fraction-free LU decomposition.

Returns 4 matrices P, L, D, U such that PA = L D**-1 U. If the elements of the matrix belong to some integral domain I, then all elements of L, D and U are guaranteed to belong to I.

Reference
• W. Zhou & D.J. Jeffrey, “Fraction-free matrix factors: new forms for LU and QR factors”. Frontiers in Computer Science in China, Vol 2, no. 1, pp. 67-80, 2008.

LUdecomposition_Simple(iszerofunc=<function _iszero at 0x1076d5848>)[source]

Returns A comprised of L,U (L’s diag entries are 1) and p which is the list of the row swaps (in order).

LUsolve(rhs, iszerofunc=<function _iszero at 0x1076d5848>)[source]

Solve the linear system Ax = b for x. self is the coefficient matrix A and rhs is the right side b.

This is for symbolic matrices, for real or complex ones use sympy.mpmath.lu_solve or sympy.mpmath.qr_solve.

QRdecomposition()[source]

Return Q,R where A = Q*R, Q is orthogonal and R is upper triangular.

Examples

This is the example from wikipedia:

>>> from sympy import Matrix, eye
>>> A = Matrix([[12,-51,4],[6,167,-68],[-4,24,-41]])
>>> Q, R = A.QRdecomposition()
>>> Q
[ 6/7, -69/175, -58/175]
[ 3/7, 158/175,   6/175]
[-2/7,    6/35,  -33/35]
>>> R
[14,  21, -14]
[ 0, 175, -70]
[ 0,   0,  35]
>>> A == Q*R
True


QR factorization of an identity matrix:

>>> A = Matrix([[1,0,0],[0,1,0],[0,0,1]])
>>> Q, R = A.QRdecomposition()
>>> Q
[1, 0, 0]
[0, 1, 0]
[0, 0, 1]
>>> R
[1, 0, 0]
[0, 1, 0]
[0, 0, 1]

QRsolve(b)[source]

Solve the linear system ‘Ax = b’.

‘self’ is the matrix ‘A’, the method argument is the vector ‘b’. The method returns the solution vector ‘x’. If ‘b’ is a matrix, the system is solved for each column of ‘b’ and the return value is a matrix of the same shape as ‘b’.

This method is slower (approximately by a factor of 2) but more stable for floating-point arithmetic than the LUsolve method. However, LUsolve usually uses an exact arithmetic, so you don’t need to use QRsolve.

This is mainly for educational purposes and symbolic matrices, for real (or complex) matrices use sympy.mpmath.qr_solve.

T

Matrix transposition.

Return self+b

Conjugate transpose or Hermitian conjugation.

Adjugate matrix is the transpose of the cofactor matrix.

applyfunc(f)[source]

Apply a function to each element of the matrix.

>>> from sympy import Matrix
>>> m = Matrix(2,2,lambda i,j: i*2+j)
>>> m
[0, 1]
[2, 3]
>>> m.applyfunc(lambda i: 2*i)
[0, 2]
[4, 6]

as_immutable()[source]

Returns an Immutable version of this Matrix

as_mutable()[source]

Returns a Mutable version of this Matrix

>>> from sympy import ImmutableMatrix
>>> X = ImmutableMatrix([[1,2],[3,4]])
>>> Y = X.as_mutable()
>>> Y[1,1] = 5 # Can set values in Y
>>> Y
[1, 2]
[3, 5]

berkowitz()[source]

The Berkowitz algorithm.

Given N x N matrix with symbolic content, compute efficiently coefficients of characteristic polynomials of ‘self’ and all its square sub-matrices composed by removing both i-th row and column, without division in the ground domain.

This method is particularly useful for computing determinant, principal minors and characteristic polynomial, when ‘self’ has complicated coefficients e.g. polynomials. Semi-direct usage of this algorithm is also important in computing efficiently sub-resultant PRS.

Assuming that M is a square matrix of dimension N x N and I is N x N identity matrix, then the following following definition of characteristic polynomial is begin used:

charpoly(M) = det(t*I - M)

As a consequence, all polynomials generated by Berkowitz algorithm are monic.

>>> from sympy import Matrix
>>> from sympy.abc import x, y, z

>>> M = Matrix([[x,y,z], [1,0,0], [y,z,x]])

>>> p, q, r = M.berkowitz()

>>> p # 1 x 1 M's sub-matrix
(1, -x)

>>> q # 2 x 2 M's sub-matrix
(1, -x, -y)

>>> r # 3 x 3 M's sub-matrix
(1, -2*x, x**2 - y*z - y, x*y - z**2)


[1] S.J. Berkowitz, On computing the determinant in small
parallel time using a small number of processors, ACM, Information Processing Letters 18, 1984, pp. 147-150
[2] M. Keber, Division-Free computation of sub-resultants
using Bezout matrices, Tech. Report MPI-I-2006-1-006, Saarbrucken, 2006
berkowitz_charpoly(x=_lambda, simplify=<function simplify at 0x10764b758>)[source]

Computes characteristic polynomial minors using Berkowitz method.

A PurePoly is returned so using different variables for x does not affect the comparison or the polynomials:

>>> from sympy import Matrix
>>> from sympy.abc import x, y
>>> A = Matrix([[1, 3], [2, 0]])
>>> A.berkowitz_charpoly(x) == A.berkowitz_charpoly(y)
True


Specifying x is optional; a Dummy with name lambda is used by default (which looks good when pretty-printed in unicode):

>>> A.berkowitz_charpoly().as_expr()
_lambda**2 - _lambda - 6


No test is done to see that x doesn’t clash with an existing symbol, so using the default (lambda) or your own Dummy symbol is the safest option:

>>> A = Matrix([[1, 2], [x, 0]])
>>> A.charpoly().as_expr()
_lambda**2 - _lambda - 2*x
>>> A.charpoly(x).as_expr()
x**2 - 3*x


berkowitz

berkowitz_det()[source]

Computes determinant using Berkowitz method.

berkowitz_eigenvals(**flags)[source]

Computes eigenvalues of a Matrix using Berkowitz method.

berkowitz

berkowitz_minors()[source]

Computes principal minors using Berkowitz method.

berkowitz

charpoly(x=_lambda, simplify=<function simplify at 0x10764b758>)

Computes characteristic polynomial minors using Berkowitz method.

A PurePoly is returned so using different variables for x does not affect the comparison or the polynomials:

>>> from sympy import Matrix
>>> from sympy.abc import x, y
>>> A = Matrix([[1, 3], [2, 0]])
>>> A.berkowitz_charpoly(x) == A.berkowitz_charpoly(y)
True


Specifying x is optional; a Dummy with name lambda is used by default (which looks good when pretty-printed in unicode):

>>> A.berkowitz_charpoly().as_expr()
_lambda**2 - _lambda - 6


No test is done to see that x doesn’t clash with an existing symbol, so using the default (lambda) or your own Dummy symbol is the safest option:

>>> A = Matrix([[1, 2], [x, 0]])
>>> A.charpoly().as_expr()
_lambda**2 - _lambda - 2*x
>>> A.charpoly(x).as_expr()
x**2 - 3*x


berkowitz

cholesky()[source]

Returns the Cholesky Decomposition L of a Matrix A such that L * L.T = A

A must be a square, symmetric, positive-definite and non-singular matrix

>>> from sympy.matrices import Matrix
>>> A = Matrix(((25,15,-5),(15,18,0),(-5,0,11)))
>>> A.cholesky()
[ 5, 0, 0]
[ 3, 3, 0]
[-1, 1, 3]
>>> A.cholesky() * A.cholesky().T
[25, 15, -5]
[15, 18,  0]
[-5,  0, 11]


cholesky_solve(rhs)[source]

Solves Ax = B using Cholesky decomposition, for a general square non-singular matrix. For a non-square matrix with rows > cols, the least squares solution is returned.

clone()[source]

Create a shallow copy of this matrix.

cofactor(i, j, method='berkowitz')[source]

Calculate the cofactor of an element.

cofactorMatrix(method='berkowitz')[source]

Return a matrix containing the cofactor of each element.

col_insert(pos, mti)[source]

Insert a column at the given position.

>>> from sympy import Matrix, zeros
>>> M = Matrix(3,3,lambda i,j: i+j)
>>> M
[0, 1, 2]
[1, 2, 3]
[2, 3, 4]
>>> V = zeros(3, 1)
>>> V
[0]
[0]
[0]
>>> M.col_insert(1,V)
[0, 0, 1, 2]
[1, 0, 2, 3]
[2, 0, 3, 4]


col, row_insert

col_join(bott)[source]

Concatenates two matrices along self’s last and bott’s first row

>>> from sympy import Matrix, ones
>>> M = ones(3, 3)
>>> V = Matrix([[7,7,7]])
>>> M.col_join(V)
[1, 1, 1]
[1, 1, 1]
[1, 1, 1]
[7, 7, 7]


col, row_join

condition_number()[source]

Returns the condition number of a matrix.

This is the maximum singular value divided by the minimum singular value

>>> from sympy import Matrix, S
>>> A = Matrix([[1, 0, 0], [0, 10, 0], [0,0,S.One/10]])
>>> A.condition_number()
100


singular_values

conjugate()[source]

By-element conjugation.

transpose
Matrix transposition
H
Hermite conjugation
D
Dirac conjugation
cross(b)[source]

Calculate the cross product of self and b.

delRowCol(i, j)[source]

Creates a copy of the matrix with the given row and column deleted.

row_del, col_del

Examples

>>> import sympy
>>> I = sympy.matrices.eye(4)
>>> I.delRowCol(1,2)
[1, 0, 0]
[0, 0, 0]
[0, 0, 1]

det(method='bareis')[source]

Computes the matrix determinant using the method “method”.

Possible values for “method”:
bareis ... det_bareis berkowitz ... berkowitz_det lu_decomposition ... det_LU

det_bareis, berkowitz_det, det_LU

det_LU_decomposition()[source]

Compute matrix determinant using LU decomposition

Note that this method fails if the LU decomposition itself fails. In particular, if the matrix has no inverse this method will fail.

TODO: Implement algorithm for sparse matrices (SFF).

det_bareis()[source]

Compute matrix determinant using Bareis’ fraction-free algorithm which is an extension of the well known Gaussian elimination method. This approach is best suited for dense symbolic matrices and will result in a determinant with minimal number of fractions. It means that less term rewriting is needed on resulting formulae.

TODO: Implement algorithm for sparse matrices (SFF).

diagonal_solve(rhs)[source]

Solves Ax = B efficiently, where A is a diagonal Matrix, with non-zero diagonal entries.

diagonalize(reals_only=False)[source]

Return diagonalized matrix D and transformation P such as

D = P^-1 * M * P

where M is current matrix.

Examples

>>> from sympy import Matrix
>>> m = Matrix(3,3,[1, 2, 0, 0, 3, 0, 2, -4, 2])
>>> m
[1,  2, 0]
[0,  3, 0]
[2, -4, 2]
>>> (P, D) = m.diagonalize()
>>> D
[1, 0, 0]
[0, 2, 0]
[0, 0, 3]
>>> P
[-1, 0, -1]
[ 0, 0, -1]
[ 2, 1,  2]
>>> P.inv() * m * P
[1, 0, 0]
[0, 2, 0]
[0, 0, 3]

diff(*args)[source]

Calculate the derivative of each element in the matrix.

Examples

>>> import sympy
>>> from sympy.abc import x, y
>>> M = sympy.matrices.Matrix([[x, y], [1, 0]])
>>> M.diff(x)
[1, 0]
[0, 0]

dot(b)[source]

Return the dot product of Matrix self and b relaxing the condition of compatible dimensions: if either the number of rows or columns are the same as the length of b then the dot product is returned. If self is a row or column vector, a scalar is returned. Otherwise, a list of results is returned (and in that case the number of columns in self must match the length of b).

>>> from sympy import Matrix
>>> M = Matrix([[1,2,3], [4,5,6], [7,8,9]])
>>> v = [1, 1, 1]
>>> M.row(0).dot(v)
6
>>> M.col(0).dot(v)
12
>>> M.dot(v)
[6, 15, 24]


dual()[source]

Returns the dual of a matrix, which is:

$$(1/2)*levicivita(i,j,k,l)*M(k,l)$$ summed over indices $$k$$ and $$l$$

Since the levicivita method is anti_symmetric for any pairwise exchange of indices, the dual of a symmetric matrix is the zero matrix. Strictly speaking the dual defined here assumes that the ‘matrix’ $$M$$ is a contravariant anti_symmetric second rank tensor, so that the dual is a covariant second rank tensor.

eigenvals(**flags)[source]

Return eigen values using the berkowitz_eigenvals routine.

Since the roots routine doesn’t always work well with Floats, they will be replaced with Rationals before calling that routine. If this is not desired, set flag rational to False.

eigenvects(**flags)[source]

Return list of triples (eigenval, multiplicity, basis).

The flag simplify has two effects:
1) if bool(simplify) is True, as_content_primitive() will be used to tidy up normalization artifacts; 2) if nullspace needs simplification to compute the basis, the simplify flag will be passed on to the nullspace routine which will interpret it there.

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. If it is desired to removed small imaginary portions during the evalf step, pass a value for the chop flag.

evalf(prec=None, **options)[source]

Evaluate each element of the matrix as a float.

exp()[source]

Returns the exponentiation of a matrix

matrix_multiply

expand(**hints)[source]

Expand each element of the matrix by calling expand().

extract(rowsList, colsList)[source]

Extract a submatrix by specifying a list of rows and columns. Negative indices can be given. All indices must be in the range -n <= i < n where n is the number of rows or columns.

submatrix

Examples

>>> from sympy import Matrix
>>> m = Matrix(4, 3, range(12))
>>> m
[0,  1,  2]
[3,  4,  5]
[6,  7,  8]
[9, 10, 11]
>>> m.extract([0,1,3],[0,1])
[0,  1]
[3,  4]
[9, 10]


Rows or columns can be repeated:

>>> m.extract([0,0,1], [-1])
[2]
[2]
[5]


Every other row can be taken by using range to provide the indices:

>>> m.extract(range(0, m.rows, 2),[-1])
[2]
[8]

classmethod eye(n)[source]

Returns the identity matrix of size n.

get_diag_blocks()[source]

Obtains the square sub-matrices on the main diagonal of a square matrix.

Useful for inverting symbolic matrices or solving systems of linear equations which may be decoupled by having a block diagonal structure.

Examples

>>> from sympy import Matrix, symbols
>>> from sympy.abc import x, y, z
>>> A = Matrix([[1, 3, 0, 0], [y, z*z, 0, 0], [0, 0, x, 0], [0, 0, 0, 0]])
>>> a1, a2, a3 = A.get_diag_blocks()
>>> a1
[1,    3]
[y, z**2]
>>> a2
[x]
>>> a3
[0]
>>>

has(*patterns)[source]

Test whether any subexpression matches any of the patterns.

Examples

>>> from sympy import Matrix, Float
>>> from sympy.abc import x, y
>>> A = Matrix(((1, x), (0.2, 3)))
>>> A.has(x)
True
>>> A.has(y)
False
>>> A.has(Float)
True

hash()[source]

Compute a hash every time, because the matrix elements could change.

integrate(*args)[source]

Integrate each element of the matrix.

Examples

>>> import sympy
>>> from sympy.abc import x, y
>>> M = sympy.matrices.Matrix([[x, y], [1, 0]])
>>> M.integrate((x,))
[x**2/2, x*y]
[     x,   0]
>>> M.integrate((x, 0, 2))
[2, 2*y]
[2,   0]

inv(method='GE', iszerofunc=<function _iszero at 0x1076d5848>, try_block_diag=False)[source]

Calculates the matrix inverse.

According to the “method” parameter, it calls the appropriate method:

According to the “try_block_diag” parameter, it will try to form block diagonal matrices using the method get_diag_blocks(), invert these individually, and then reconstruct the full inverse matrix.

Note, the GE and LU methods may require the matrix to be simplified before it is inverted in order to properly detect zeros during pivoting. In difficult cases a custom zero detection function can be provided by setting the iszerosfunc argument to a function that should return True if its argument is zero. The ADJ routine computes the determinant and uses that to detect singular matrices in addition to testing for zeros on the diagonal.

Calculates the inverse using the adjugate matrix and a determinant.

inverse_GE(iszerofunc=<function _iszero at 0x1076d5848>)[source]

Calculates the inverse using Gaussian elimination.

inverse_LU(iszerofunc=<function _iszero at 0x1076d5848>)[source]

Calculates the inverse using LU decomposition.

is_anti_symmetric(simplify=True)[source]

Check if matrix M is an antisymmetric matrix, that is, M is a square matrix with all M[i, j] == -M[j, i].

When simplify=True (default), the sum M[i, j] + M[j, i] is simplified before testing to see if it is zero. By default, the SymPy simplify function is used. To use a custom function set simplify to a function that accepts a single argument which returns a simplified expression. To skip simplification, set simplify to False but note that although this will be faster, it may induce false negatives.

Examples

>>> from sympy import Matrix, symbols
>>> m = Matrix(2,2,[0, 1, -1, 0])
>>> m
[ 0, 1]
[-1, 0]
>>> m.is_anti_symmetric()
True
>>> x, y = symbols('x y')
>>> m = Matrix(2,3,[0, 0, x, -y, 0, 0])
>>> m
[ 0, 0, x]
[-y, 0, 0]
>>> m.is_anti_symmetric()
False

>>> from sympy.abc import x, y
>>> m = Matrix(3, 3, [0, x**2 + 2*x + 1, y,
...                   -(x + 1)**2 , 0, x*y,
...                   -y, -x*y, 0])


Simplification of matrix elements is done by default so even though two elements which should be equal and opposite wouldn’t pass an equality test, the matrix is still reported as anti-symmetric:

>>> m[0, 1] == -m[1, 0]
False
>>> m.is_anti_symmetric()
True


If ‘simplify=False’ is used for the case when a Matrix is already simplified, this will speed things up. Here, we see that without simplification the matrix does not appear anti-symmetric:

>>> m.is_anti_symmetric(simplify=False)
False


But if the matrix were already expanded, then it would appear anti-symmetric and simplification in the is_anti_symmetric routine is not needed:

>>> m = m.expand()
>>> m.is_anti_symmetric(simplify=False)
True

is_diagonal()[source]

Check if matrix is diagonal, that is matrix in which the entries outside the main diagonal are all zero.

Examples

>>> from sympy import Matrix, diag
>>> m = Matrix(2,2,[1, 0, 0, 2])
>>> m
[1, 0]
[0, 2]
>>> m.is_diagonal()
True

>>> m = Matrix(2,2,[1, 1, 0, 2])
>>> m
[1, 1]
[0, 2]
>>> m.is_diagonal()
False

>>> m = diag(1, 2, 3)
>>> m
[1, 0, 0]
[0, 2, 0]
[0, 0, 3]
>>> m.is_diagonal()
True

is_diagonalizable(reals_only=False, clear_subproducts=True)[source]

Check if matrix is diagonalizable.

If reals_only==True then check that diagonalized matrix consists of the only not complex values.

Some subproducts could be used further in other methods to avoid double calculations, By default (if clear_subproducts==True) they will be deleted.

Examples

>>> from sympy import Matrix
>>> m = Matrix(3,3,[1, 2, 0, 0, 3, 0, 2, -4, 2])
>>> m
[1,  2, 0]
[0,  3, 0]
[2, -4, 2]
>>> m.is_diagonalizable()
True
>>> m = Matrix(2,2,[0, 1, 0, 0])
>>> m
[0, 1]
[0, 0]
>>> m.is_diagonalizable()
False
>>> m = Matrix(2,2,[0, 1, -1, 0])
>>> m
[ 0, 1]
[-1, 0]
>>> m.is_diagonalizable()
True
>>> m.is_diagonalizable(True)
False

is_lower()[source]

Check if matrix is a lower triangular matrix.

Examples

>>> from sympy import Matrix
>>> m = Matrix(2,2,[1, 0, 0, 1])
>>> m
[1, 0]
[0, 1]
>>> m.is_lower()
True

>>> m = Matrix(3,3,[2, 0, 0, 1, 4 , 0, 6, 6, 5])
>>> m
[2, 0, 0]
[1, 4, 0]
[6, 6, 5]
>>> m.is_lower()
True

>>> from sympy.abc import x, y
>>> m = Matrix(2,2,[x**2 + y, y**2 + x, 0, x + y])
>>> m
[x**2 + y, x + y**2]
[       0,    x + y]
>>> m.is_lower()
False

is_lower_hessenberg()[source]

Checks if the matrix is in the lower hessenberg form.

The lower hessenberg matrix has zero entries above the first superdiagonal.

Examples

>>> from sympy.matrices import Matrix
>>> a = Matrix([[1,2,0,0],[5,2,3,0],[3,4,3,7],[5,6,1,1]])
>>> a
[1, 2, 0, 0]
[5, 2, 3, 0]
[3, 4, 3, 7]
[5, 6, 1, 1]
>>> a.is_lower_hessenberg()
True

is_nilpotent()[source]

Checks if a matrix is nilpotent.

A matrix B is nilpotent if for some integer k, B**k is a zero matrix.

Examples

>>> from sympy import Matrix
>>> a = Matrix([[0,0,0],[1,0,0],[1,1,0]])
>>> a.is_nilpotent()
True

>>> a = Matrix([[1,0,1],[1,0,0],[1,1,0]])
>>> a.is_nilpotent()
False

is_square[source]

Checks if a matrix is square.

A matrix is square if the number of rows equals the number of columns. The empty matrix is square by definition, since the number of rows and the number of columns are both zero.

Examples

>>> from sympy import Matrix
>>> a = Matrix([[1, 2, 3], [4, 5, 6]])
>>> b = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
>>> c = Matrix([])
>>> a.is_square
False
>>> b.is_square
True
>>> c.is_square
True

is_symbolic()[source]

Checks if any elements are symbols.

Examples

>>> import sympy
>>> from sympy.abc import x, y
>>> M = sympy.matrices.Matrix([[x, y], [1, 0]])
>>> M.is_symbolic()
True

is_symmetric(simplify=True)[source]

Check if matrix is symmetric matrix, that is square matrix and is equal to its transpose.

By default, simplifications occur before testing symmetry. They can be skipped using ‘simplify=False’; while speeding things a bit, this may however induce false negatives.

Examples

>>> from sympy import Matrix
>>> m = Matrix(2,2,[0, 1, 1, 2])
>>> m
[0, 1]
[1, 2]
>>> m.is_symmetric()
True

>>> m = Matrix(2,2,[0, 1, 2, 0])
>>> m
[0, 1]
[2, 0]
>>> m.is_symmetric()
False

>>> m = Matrix(2,3,[0, 0, 0, 0, 0, 0])
>>> m
[0, 0, 0]
[0, 0, 0]
>>> m.is_symmetric()
False

>>> from sympy.abc import x, y
>>> m = Matrix(3,3,[1, x**2 + 2*x + 1, y, (x + 1)**2 , 2, 0, y, 0, 3])
>>> m
[         1, x**2 + 2*x + 1, y]
[(x + 1)**2,              2, 0]
[         y,              0, 3]
>>> m.is_symmetric()
True


If the matrix is already simplified, you may speed-up is_symmetric() test by using ‘simplify=False’.

>>> m.is_symmetric(simplify=False)
False
>>> m1 = m.expand()
>>> m1.is_symmetric(simplify=False)
True

is_upper()[source]

Check if matrix is an upper triangular matrix.

Examples

>>> from sympy import Matrix
>>> m = Matrix(2,2,[1, 0, 0, 1])
>>> m
[1, 0]
[0, 1]
>>> m.is_upper()
True

>>> m = Matrix(3,3,[5, 1, 9, 0, 4 , 6, 0, 0, 5])
>>> m
[5, 1, 9]
[0, 4, 6]
[0, 0, 5]
>>> m.is_upper()
True

>>> m = Matrix(2,3,[4, 2, 5, 6, 1, 1])
>>> m
[4, 2, 5]
[6, 1, 1]
>>> m.is_upper()
False

is_upper_hessenberg()[source]

Checks if the matrix is the upper hessenberg form.

The upper hessenberg matrix has zero entries below the first subdiagonal.

Examples

>>> from sympy.matrices import Matrix
>>> a = Matrix([[1,4,2,3],[3,4,1,7],[0,2,3,4],[0,0,1,3]])
>>> a
[1, 4, 2, 3]
[3, 4, 1, 7]
[0, 2, 3, 4]
[0, 0, 1, 3]
>>> a.is_upper_hessenberg()
True

is_zero[source]

Checks if a matrix is a zero matrix.

A matrix is zero if every element is zero. A matrix need not be square to be considered zero. The empty matrix is zero by the principle of vacuous truth.

Examples

>>> from sympy import Matrix, zeros
>>> a = Matrix([[0, 0], [0, 0]])
>>> b = zeros(3, 4)
>>> c = Matrix([[0, 1], [0, 0]])
>>> d = Matrix([])
>>> a.is_zero
True
>>> b.is_zero
True
>>> c.is_zero
False
>>> d.is_zero
True

jacobian(X)[source]

Calculates the Jacobian matrix (derivative of a vectorial function).

self
A vector of expressions representing functions f_i(x_1, ..., x_n).
X
The set of x_i’s in order, it can be a list or a Matrix

Both self and X can be a row or a column matrix in any order (jacobian() should always work).

Examples

>>> from sympy import sin, cos, Matrix
>>> from sympy.abc import rho, phi
>>> X = Matrix([rho*cos(phi), rho*sin(phi), rho**2])
>>> Y = Matrix([rho, phi])
>>> X.jacobian(Y)
[cos(phi), -rho*sin(phi)]
[sin(phi),  rho*cos(phi)]
[   2*rho,             0]
>>> X = Matrix([rho*cos(phi), rho*sin(phi)])
>>> X.jacobian(Y)
[cos(phi), -rho*sin(phi)]
[sin(phi),  rho*cos(phi)]

jordan_cells(calc_transformation=True)[source]

Return a list of Jordan cells of current matrix. This list shape Jordan matrix J.

If calc_transformation is specified as False, then transformation P such that

J = P^-1 * M * P

will not be calculated.

jordan_form

Notes

Calculation of transformation P is not implemented yet.

Examples

>>> from sympy import Matrix
>>> m = Matrix(4, 4, [6, 5, -2, -3, -3, -1, 3, 3, 2, 1, -2, -3, -1, 1, 5, 5])
>>> m
[ 6,  5, -2, -3]
[-3, -1,  3,  3]
[ 2,  1, -2, -3]
[-1,  1,  5,  5]

>>> (P, Jcells) = m.jordan_cells()
>>> Jcells[0]
[2, 1]
[0, 2]
>>> Jcells[1]
[2, 1]
[0, 2]

jordan_form(calc_transformation=True)[source]

Return Jordan form J of current matrix.

If calc_transformation is specified as False, then transformation P such that

J = P^-1 * M * P

will not be calculated.

jordan_cells

Notes

Calculation of transformation P is not implemented yet.

Examples

>>> from sympy import Matrix
>>> m = Matrix(4, 4, [6, 5, -2, -3, -3, -1, 3, 3, 2, 1, -2, -3, -1, 1, 5, 5])
>>> m
[ 6,  5, -2, -3]
[-3, -1,  3,  3]
[ 2,  1, -2, -3]
[-1,  1,  5,  5]

>>> (P, J) = m.jordan_form()
>>> J
[2, 1, 0, 0]
[0, 2, 0, 0]
[0, 0, 2, 1]
[0, 0, 0, 2]

key2bounds(keys)[source]

Converts a key with potentially mixed types of keys (integer and slice) into a tuple of ranges and raises an error if any index is out of self’s range.

key2ij(key)[source]

Converts key=(4,6) to 4,6 and ensures the key is correct. Negative indices are also supported and are remapped to positives provided they are valid indices.

limit(*args)[source]

Calculate the limit of each element in the matrix.

Examples

>>> import sympy
>>> from sympy.abc import x, y
>>> M = sympy.matrices.Matrix([[x, y], [1, 0]])
>>> M.limit(x, 2)
[2, y]
[1, 0]

lower_triangular_solve(rhs)[source]

Solves Ax = B, where A is a lower triangular matrix.

minorEntry(i, j, method='berkowitz')[source]

Calculate the minor of an element.

minorMatrix(i, j)[source]

Creates the minor matrix of a given element.

multiply(b)[source]

Returns self*b

multiply_elementwise(b)[source]

Return the Hadamard product (elementwise product) of A and B

>>> from sympy.matrices.matrices import Matrix
>>> A = Matrix([[0, 1, 2], [3, 4, 5]])
>>> B = Matrix([[1, 10, 100], [100, 10, 1]])
>>> A.multiply_elementwise(B)
[  0, 10, 200]
[300, 40,   5]


n(prec=None, **options)

Evaluate each element of the matrix as a float.

norm(ord=None)[source]

Return the Norm of a Matrix or Vector. In the simplest case this is the geometric size of the vector Other norms can be specified by the ord parameter

ord norm for matrices norm for vectors
None Frobenius norm 2-norm
‘fro’ Frobenius norm
• does not exist
inf max(abs(x))
-inf min(abs(x))
1 as below
-1 as below
2 2-norm (largest sing. value) as below
-2 smallest singular value as below
other
• does not exist
sum(abs(x)**ord)**(1./ord)
>>> from sympy import Matrix, Symbol, trigsimp, cos, sin
>>> x = Symbol('x', real=True)
>>> v = Matrix([cos(x), sin(x)])
>>> trigsimp( v.norm() )
1
>>> v.norm(10)
(sin(x)**10 + cos(x)**10)**(1/10)
>>> A = Matrix([[1,1], [1,1]])
>>> A.norm(2)# Spectral norm (max of |Ax|/|x| under 2-vector-norm)
2
>>> A.norm(-2) # Inverse spectral norm (smallest singular value)
0
>>> A.norm() # Frobenius Norm
2


normalized

normalized()[source]

Return the normalized version of self.

norm

nullspace(simplified=False, simplify=False)[source]

Returns list of vectors (Matrix objects) that span nullspace of self

permuteBkwd(perm)[source]

Permute the rows of the matrix with the given permutation in reverse.

>>> import sympy
>>> M = sympy.matrices.eye(3)
>>> M.permuteBkwd([[0,1],[0,2]])
[0, 1, 0]
[0, 0, 1]
[1, 0, 0]


permuteFwd

permuteFwd(perm)[source]

Permute the rows of the matrix with the given permutation.

>>> import sympy
>>> M = sympy.matrices.eye(3)
>>> M.permuteFwd([[0,1],[0,2]])
[0, 0, 1]
[1, 0, 0]
[0, 1, 0]


permuteBkwd

print_nonzero(symb='X')[source]

Shows location of non-zero entries for fast shape lookup.

Examples

>>> from sympy import Matrix, matrices
>>> m = Matrix(2,3,lambda i,j: i*3+j)
>>> m
[0, 1, 2]
[3, 4, 5]
>>> m.print_nonzero()
[ XX]
[XXX]
>>> m = matrices.eye(4)
>>> m.print_nonzero("x")
[x   ]
[ x  ]
[  x ]
[   x]

project(v)[source]

Return the projection of self onto the line containing v.

>>> from sympy import Matrix, S, sqrt
>>> V = Matrix([sqrt(3)/2,S.Half])
>>> x = Matrix([[1, 0]])
>>> V.project(x)
[sqrt(3)/2, 0]
>>> V.project(-x)
[sqrt(3)/2, 0]

reshape(_rows, _cols)[source]

Reshape the matrix. Total number of elements must remain the same.

>>> from sympy import Matrix
>>> m = Matrix(2,3,lambda i,j: 1)
>>> m
[1, 1, 1]
[1, 1, 1]
>>> m.reshape(1,6)
[1, 1, 1, 1, 1, 1]
>>> m.reshape(3,2)
[1, 1]
[1, 1]
[1, 1]

row_insert(pos, mti)[source]

Insert a row at the given position.

>>> from sympy import Matrix, zeros
>>> M = Matrix(3,3,lambda i,j: i+j)
>>> M
[0, 1, 2]
[1, 2, 3]
[2, 3, 4]
>>> V = zeros(1, 3)
>>> V
[0, 0, 0]
>>> M.row_insert(1,V)
[0, 1, 2]
[0, 0, 0]
[1, 2, 3]
[2, 3, 4]


row, col_insert

row_join(rhs)[source]

Concatenates two matrices along self’s last and rhs’s first column

>>> from sympy import Matrix
>>> M = Matrix(3,3,lambda i,j: i+j)
>>> V = Matrix(3,1,lambda i,j: 3+i+j)
>>> M.row_join(V)
[0, 1, 2, 3]
[1, 2, 3, 4]
[2, 3, 4, 5]


row, col_join

rref(simplified=False, iszerofunc=<function _iszero at 0x1076d5848>, simplify=False)[source]

Return reduced row-echelon form of matrix and indices of pivot vars.

To simplify elements before finding nonzero pivots set simplify=True (to use the default SymPy simplify function) or pass a custom simplify function.

>>> from sympy import Matrix
>>> from sympy.abc import x
>>> m = Matrix([[1, 2], [x, 1 - 1/x]])
>>> m.rref()
([1, 0]
[0, 1], [0, 1])

shape[source]

The shape (dimensions) of the matrix as the 2-tuple (rows, cols).

singular_values()[source]

Compute the singular values of a Matrix

>>> from sympy import Matrix, Symbol, eye
>>> x = Symbol('x', real=True)
>>> A = Matrix([[0, 1, 0], [0, x, 0], [-1, 0, 0]])
>>> A.singular_values()
[sqrt(x**2 + 1), 1, 0]


condition_number

slice2bounds(key, defmax)[source]

Takes slice or number and returns (min,max) for iteration Takes a default maxval to deal with the slice ‘:’ which is (none, none)

submatrix(keys)[source]

Get a slice/submatrix of the matrix using the given slice.

>>> from sympy import Matrix
>>> m = Matrix(4,4,lambda i,j: i+j)
>>> m
[0, 1, 2, 3]
[1, 2, 3, 4]
[2, 3, 4, 5]
[3, 4, 5, 6]
>>> m[:1, 1]
[1]
>>> m[:2, :1]
[0]
[1]
>>> m[2:4, 2:4]
[4, 5]
[5, 6]


extract

subs(*args, **kwargs)[source]

Create substituted expressions for each element with Expr.subs.

tolist()[source]

Return the Matrix converted in a python list.

>>> from sympy import Matrix
>>> m=Matrix(3, 3, range(9))
>>> m
[0, 1, 2]
[3, 4, 5]
[6, 7, 8]
>>> m.tolist()
[[0, 1, 2], [3, 4, 5], [6, 7, 8]]

trace()[source]

Calculate the trace of a (square) matrix.

>>> import sympy
>>> M = sympy.matrices.eye(3)
>>> M.trace()
3

transpose()[source]

Matrix transposition.

>>> from sympy import Matrix, I
>>> m=Matrix(((1,2+I),(3,4)))
>>> m
[1, 2 + I]
[3,     4]
>>> m.transpose()
[    1, 3]
[2 + I, 4]
>>> m.T == m.transpose()
True


conjugate
By-element conjugation
upper_triangular_solve(rhs)[source]

Solves Ax = B, where A is an upper triangular matrix.

vec()[source]

Return the Matrix converted into a one column matrix by stacking columns

>>> from sympy import Matrix
>>> m=Matrix([[1,3], [2,4]])
>>> m
[1, 3]
[2, 4]
>>> m.vec()
[1]
[2]
[3]
[4]


vech

vech(diagonal=True, check_symmetry=True)[source]

Return the unique elements of a symmetric Matrix as a one column matrix by stacking the elements in the lower triangle.

Arguments: diagonal – include the diagonal cells of self or not check_symmetry – checks symmetry of self but not completely reliably

>>> from sympy import Matrix
>>> m=Matrix([[1,2], [2,3]])
>>> m
[1, 2]
[2, 3]
>>> m.vech()
[1]
[2]
[3]
>>> m.vech(diagonal=False)
[2]


vec

classmethod zeros(r, c=None)[source]

Returns a matrix of zeros with r rows and c columns; if c is omitted a square matrix will be returned.

ones, fill

sympy.matrices.matrices.Matrix

alias of MutableMatrix

class sympy.matrices.matrices.MutableMatrix[source]
col(j, f=None)[source]

Elementary column selector (default) or operation using functor which is a function two args interpreted as (self[i, j], i).

>>> from sympy import ones
>>> I = ones(3)
>>> I.col(0, lambda v, i: v*3)
>>> I
[3, 1, 1]
[3, 1, 1]
[3, 1, 1]
>>> I.col(0)
[3]
[3]
[3]


row, col_swap, col_del, col_join, col_insert, delRowCol

col_del(i)[source]

Delete the given column.

>>> import sympy
>>> M = sympy.matrices.eye(3)
>>> M.col_del(1)
>>> M
[1, 0]
[0, 0]
[0, 1]


col_swap(i, j)[source]

Swap the two given columns of the matrix in-place.

>>> from sympy.matrices import Matrix
>>> M = Matrix([[1,0],[1,0]])
>>> M
[1, 0]
[1, 0]
>>> M.col_swap(0, 1)
>>> M
[0, 1]
[0, 1]


copyin_list(key, value)[source]

Copy in elements from a list.

Parameters : key : slice The section of this matrix to replace. value : iterable The iterable to copy values from.

copyin_matrix

Examples

>>> import sympy
>>> I = sympy.matrices.eye(5)
>>> I.copyin_list((slice(0,2), slice(0,1)), [1,2])
>>> I
[1, 0, 0, 0, 0]
[2, 1, 0, 0, 0]
[0, 0, 1, 0, 0]
[0, 0, 0, 1, 0]
[0, 0, 0, 0, 1]

copyin_matrix(key, value)[source]

Copy in values from a matrix into the given bounds.

Parameters : key : slice The section of this matrix to replace. value : Matrix The matrix to copy values from.

copyin_list

Examples

>>> import sympy
>>> M = sympy.matrices.Matrix([[0,1],[2,3]])
>>> I = sympy.matrices.eye(5)
>>> I.copyin_matrix((slice(0,2), slice(0,2)),M)
>>> I
[0, 1, 0, 0, 0]
[2, 3, 0, 0, 0]
[0, 0, 1, 0, 0]
[0, 0, 0, 1, 0]
[0, 0, 0, 0, 1]

fill(value)[source]

Fill the matrix with the scalar value.

row(i, f=None)[source]

Elementary row selector (default) or operation using functor which is a function two args interpreted as (self[i, j], j).

>>> from sympy import ones
>>> I = ones(3)
>>> I.row(1, lambda v,i: v*3)
>>> I
[1, 1, 1]
[3, 3, 3]
[1, 1, 1]
>>> I.row(1)
[3, 3, 3]


col, row_swap, row_del, row_join, row_insert, delRowCol

row_del(i)[source]

Delete the given row.

>>> import sympy
>>> M = sympy.matrices.eye(3)
>>> M.row_del(1)
>>> M
[1, 0, 0]
[0, 0, 1]


row_swap(i, j)[source]

Swap the two given rows of the matrix in-place.

>>> from sympy.matrices import Matrix
>>> M = Matrix([[0,1],[1,0]])
>>> M
[0, 1]
[1, 0]
>>> M.row_swap(0, 1)
>>> M
[1, 0]
[0, 1]


simplify(ratio=1.7, measure=<function count_ops at 0x1072f1b18>)[source]

Applies simplify to the elements of a matrix in place.

This is a shortcut for M.applyfunc(lambda x: simplify(x, ratio, measure))

class sympy.matrices.matrices.SparseMatrix(*args)[source]

A sparse matrix (a matrix with a large number of zero elements).

Matrix

CL

Alternate faster representation

RL

Alternate faster representation

T

Matrix transposition.

Add two sparse matrices with dictionary representation.

>>> from sympy.matrices.matrices import SparseMatrix
>>> A = SparseMatrix(5, 5, lambda i, j: i * j + i)
>>> A
[0, 0,  0,  0,  0]
[1, 2,  3,  4,  5]
[2, 4,  6,  8, 10]
[3, 6,  9, 12, 15]
[4, 8, 12, 16, 20]
>>> B = SparseMatrix(5, 5, lambda i, j: i + 2 * j)
>>> B
[0, 2, 4,  6,  8]
[1, 3, 5,  7,  9]
[2, 4, 6,  8, 10]
[3, 5, 7,  9, 11]
[4, 6, 8, 10, 12]
>>> A + B
[0,  2,  4,  6,  8]
[2,  5,  8, 11, 14]
[4,  8, 12, 16, 20]
[6, 11, 16, 21, 26]
[8, 14, 20, 26, 32]


multiply

col_del(k)[source]

Delete the given column of the matrix.

row_del

Examples

>>> import sympy
>>> M = sympy.matrices.SparseMatrix([[0,0],[0,1]])
>>> M
[0, 0]
[0, 1]
>>> M.col_del(0)
>>> M
[0]
[1]

col_list()[source]

Returns a Column-sorted list of non-zero elements of the matrix. >>> from sympy.matrices import SparseMatrix >>> a=SparseMatrix(((1,2),(3,4))) >>> a [1, 2] [3, 4] >>> a.CL [(0, 0, 1), (1, 0, 3), (0, 1, 2), (1, 1, 4)]

row_list

multiply(b)[source]

Returns self*b

row_del(k)[source]

Delete the given row of the matrix.

col_del

Examples

>>> import sympy
>>> M = sympy.matrices.SparseMatrix([[0,0],[0,1]])
>>> M
[0, 0]
[0, 1]
>>> M.row_del(0)
>>> M
[0, 1]

row_list()[source]

Returns a Row-sorted list of non-zero elements of the matrix.

>>> from sympy.matrices import SparseMatrix
>>> a=SparseMatrix(((1,2),(3,4)))
>>> a
[1, 2]
[3, 4]
>>> a.RL
[(0, 0, 1), (0, 1, 2), (1, 0, 3), (1, 1, 4)]


col_list

rowdecomp(num)[source]

Perform a row decomposition on the matrix.

toMatrix()[source]

Convert this sparse matrix into a matrix.

transpose()[source]

Returns the transposed SparseMatrix of this SparseMatrix >>> from sympy.matrices import SparseMatrix >>> a = SparseMatrix(((1,2),(3,4))) >>> a [1, 2] [3, 4] >>> a.T [1, 3] [2, 4]

classmethod zeros(r, c=None)[source]

Returns a matrix of zeros with r rows and c columns; if c is omitted a square matrix will be returned.

## Matrix Exceptions Reference¶

class sympy.matrices.matrices.MatrixError[source]
class sympy.matrices.matrices.ShapeError[source]

Wrong matrix shape

class sympy.matrices.matrices.NonSquareMatrixError[source]

## Matrix Functions Reference¶

sympy.matrices.matrices.matrix_multiply(A, B)[source]

Matrix product A*B.

A and B must be of appropriate dimensions. If A is an m x k matrix, and B is a k x n matrix, the product will be an m x n matrix.

Examples

>>> from sympy import Matrix
>>> A = Matrix([[1, 2, 3], [4, 5, 6]])
>>> B = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
>>> A*B
[30, 36, 42]
[66, 81, 96]
>>> B*A
Traceback (most recent call last):
...
ShapeError: Matrices size mismatch.
>>>

sympy.matrices.matrices.matrix_multiply_elementwise(A, B)[source]

Return the Hadamard product (elementwise product) of A and B

>>> from sympy.matrices.matrices import Matrix, matrix_multiply_elementwise
>>> A = Matrix([[0, 1, 2], [3, 4, 5]])
>>> B = Matrix([[1, 10, 100], [100, 10, 1]])
>>> matrix_multiply_elementwise(A, B)
[  0, 10, 200]
[300, 40,   5]


Return A + B

sympy.matrices.matrices.zeros(r, c=None, cls=<class 'sympy.matrices.matrices.MutableMatrix'>)[source]

Returns a matrix of zeros with r rows and c columns; if c is omitted a square matrix will be returned.

sympy.matrices.matrices.ones(r, c=None)[source]

Returns a matrix of ones with r rows and c columns; if c is omitted a square matrix will be returned.

sympy.matrices.matrices.eye(n, cls=<class 'sympy.matrices.matrices.MutableMatrix'>)[source]

Create square identity matrix n x n

sympy.matrices.matrices.diag(*values)[source]

Create diagonal matrix from a list as a diagonal values.

Arguments might be matrices too, in case of it they are fitted in result matrix

eye

Examples

>>> from sympy.matrices import diag, Matrix
>>> diag(1, 2, 3)
[1, 0, 0]
[0, 2, 0]
[0, 0, 3]

>>> from sympy.abc import x, y, z
>>> a = Matrix([x, y, z])
>>> b = Matrix([[1, 2], [3, 4]])
>>> c = Matrix([[5, 6]])
>>> diag(a, 7, b, c)
[x, 0, 0, 0, 0, 0]
[y, 0, 0, 0, 0, 0]
[z, 0, 0, 0, 0, 0]
[0, 7, 0, 0, 0, 0]
[0, 0, 1, 2, 0, 0]
[0, 0, 3, 4, 0, 0]
[0, 0, 0, 0, 5, 6]

sympy.matrices.matrices.randMatrix(r, c=None, min=0, max=99, seed=None, symmetric=False)[source]

Create random matrix with dimensions r x c. If c is omitted the matrix will be square. If symmetric is True the matrix must be square.

Examples

>>> from sympy.matrices import randMatrix
>>> randMatrix(3)
[25, 45, 27]
[44, 54,  9]
[23, 96, 46]
>>> randMatrix(3, 2)
[87, 29]
[23, 37]
[90, 26]
>>> randMatrix(3, 3, 0, 2)
[0, 2, 0]
[2, 0, 1]
[0, 0, 1]
>>> randMatrix(3, symmetric=True)
[85, 26, 29]
[26, 71, 43]
[29, 43, 57]
>>> A = randMatrix(3, seed=1)
>>> B = randMatrix(3, seed=2)
>>> A == B
False
>>> A == randMatrix(3, seed=1)
True

sympy.matrices.matrices.hessian(f, varlist)[source]

Compute Hessian matrix for a function f

sympy.matrices.matrices.Matrix.jacobian, wronskian

sympy.matrices.matrices.GramSchmidt(vlist, orthog=False)[source]

Apply the Gram-Schmidt process to a set of vectors.

sympy.matrices.matrices.wronskian(functions, var, method='bareis')[source]

Compute Wronskian for [] of functions

               | f1       f2        ...   fn      |
| f1'      f2'       ...   fn'     |
|  .        .        .      .      |
W(f1,...,fn) = |  .        .         .     .      |
|  .        .          .    .      |
|  (n)      (n)            (n)     |
| D   (f1) D   (f2)  ...  D   (fn) |

sympy.matrices.matrices.Matrix.jacobian, hessian

sympy.matrices.matrices.casoratian(seqs, n, zero=True)[source]

Given linear difference operator L of order ‘k’ and homogeneous equation Ly = 0 we want to compute kernel of L, which is a set of ‘k’ sequences: a(n), b(n), ... z(n).

Solutions of L are linearly independent iff their Casoratian, denoted as C(a, b, ..., z), do not vanish for n = 0.

Casoratian is defined by k x k determinant:

+  a(n)     b(n)     . . . z(n)     +
|  a(n+1)   b(n+1)   . . . z(n+1)   |
|    .         .     .        .     |
|    .         .       .      .     |
|    .         .         .    .     |
+  a(n+k-1) b(n+k-1) . . . z(n+k-1) +

It proves very useful in rsolve_hyper() where it is applied to a generating set of a recurrence to factor out linearly dependent solutions and return a basis:

>>> from sympy import Symbol, casoratian, factorial
>>> n = Symbol('n', integer=True)


Exponential and factorial are linearly independent:

>>> casoratian([2**n, factorial(n)], n) != 0
True


## Numpy Utility Functions Reference¶

sympy.matrices.matrices.list2numpy(l)[source]

Converts python list of SymPy expressions to a NumPy array.

matrix2numpy

sympy.matrices.matrices.matrix2numpy(m)[source]

Converts SymPy’s matrix to a NumPy array.

list2numpy

sympy.matrices.matrices.a2idx(a)[source]

Tries to convert “a” to an index, returns None on failure.

The result of a2idx() (if not None) can be safely used as an index to arrays/matrices.

sympy.matrices.matrices.symarray(prefix, shape)[source]

Create a numpy ndarray of symbols (as an object array).

The created symbols are named prefix_i1_i2_... You should thus provide a non-empty prefix if you want your symbols to be unique for different output arrays, as SymPy symbols with identical names are the same object.

Parameters : prefix : string A prefix prepended to the name of every symbol. shape : int or tuple Shape of the created array. If an int, the array is one-dimensional; for more than one dimension the shape must be a tuple.

Examples

These doctests require numpy.

>>> from sympy import symarray
>>> symarray('', 3)
[_0, _1, _2]


If you want multiple symarrays to contain distinct symbols, you must provide unique prefixes:

>>> a = symarray('', 3)
>>> b = symarray('', 3)
>>> a[0] is b[0]
True
>>> a = symarray('a', 3)
>>> b = symarray('b', 3)
>>> a[0] is b[0]
False


Creating symarrays with a prefix:

>>> symarray('a', 3)
[a_0, a_1, a_2]


For more than one dimension, the shape must be given as a tuple:

>>> symarray('a', (2, 3))
[[a_0_0, a_0_1, a_0_2],
[a_1_0, a_1_1, a_1_2]]
>>> symarray('a', (2, 3, 2))
[[[a_0_0_0, a_0_0_1],
[a_0_1_0, a_0_1_1],
[a_0_2_0, a_0_2_1]],

[[a_1_0_0, a_1_0_1],
[a_1_1_0, a_1_1_1],
[a_1_2_0, a_1_2_1]]]

sympy.matrices.matrices.rot_axis1(theta)[source]

Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis.

rot_axis2
Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis
rot_axis3
Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis

Examples

>>> from sympy import pi
>>> from sympy.matrices import rot_axis1


A rotation of pi/3 (60 degrees):

>>> theta = pi/3
>>> rot_axis1(theta)
[1,          0,         0]
[0,        1/2, sqrt(3)/2]
[0, -sqrt(3)/2,       1/2]


If we rotate by pi/2 (90 degrees):

>>> rot_axis1(pi/2)
[1,  0, 0]
[0,  0, 1]
[0, -1, 0]

sympy.matrices.matrices.rot_axis2(theta)[source]

Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis.

rot_axis1
Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis
rot_axis3
Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis

Examples

>>> from sympy import pi
>>> from sympy.matrices import rot_axis2


A rotation of pi/3 (60 degrees):

>>> theta = pi/3
>>> rot_axis2(theta)
[      1/2, 0, -sqrt(3)/2]
[        0, 1,          0]
[sqrt(3)/2, 0,        1/2]


If we rotate by pi/2 (90 degrees):

>>> rot_axis2(pi/2)
[0, 0, -1]
[0, 1,  0]
[1, 0,  0]

sympy.matrices.matrices.rot_axis3(theta)[source]

Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis.

rot_axis1
Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis
rot_axis2
Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis

Examples

>>> from sympy import pi
>>> from sympy.matrices import rot_axis3


A rotation of pi/3 (60 degrees):

>>> theta = pi/3
>>> rot_axis3(theta)
[       1/2, sqrt(3)/2, 0]
[-sqrt(3)/2,       1/2, 0]
[         0,         0, 1]


If we rotate by pi/2 (90 degrees):

>>> rot_axis3(pi/2)
[ 0, 1, 0]
[-1, 0, 0]
[ 0, 0, 1]