Common Matrices

MatrixCommon Class Reference

class sympy.matrices.common.MatrixCommon

All common matrix operations including basic arithmetic, shaping, and special matrices like \(zeros\), and \(eye\).

property C

By-element conjugation

property H

Return Hermite conjugate.

Examples

>>> from sympy import Matrix, I
>>> m = Matrix((0, 1 + I, 2, 3))
>>> m
Matrix([
[    0],
[1 + I],
[    2],
[    3]])
>>> m.H
Matrix([[0, 1 - I, 2, 3]])

See also

conjugate

By-element conjugation

sympy.matrices.matrices.MatrixBase.D

Dirac conjugation

property T

Matrix transposition

__abs__()

Returns a new matrix with entry-wise absolute values.

__add__(other)

Return self + other, raising ShapeError if shapes don’t match.

__getitem__(key)

Implementations of __getitem__ should accept ints, in which case the matrix is indexed as a flat list, tuples (i,j) in which case the (i,j) entry is returned, slices, or mixed tuples (a,b) where a and b are any combintion of slices and integers.

__len__()

The total number of entries in the matrix.

__mul__(other)

Return self*other where other is either a scalar or a matrix of compatible dimensions.

Examples

>>> from sympy.matrices import Matrix
>>> A = Matrix([[1, 2, 3], [4, 5, 6]])
>>> 2*A == A*2 == Matrix([[2, 4, 6], [8, 10, 12]])
True
>>> B = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
>>> A*B
Matrix([
[30, 36, 42],
[66, 81, 96]])
>>> B*A
Traceback (most recent call last):
...
ShapeError: Matrices size mismatch.
>>>

See also

matrix_multiply_elementwise

__pow__(exp)

Return self**exp a scalar or symbol.

__weakref__

list of weak references to the object (if defined)

adjoint()

Conjugate transpose or Hermitian conjugation.

applyfunc(f)

Apply a function to each element of the matrix.

Examples

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

as_real_imag(deep=True, **hints)

Returns a tuple containing the (real, imaginary) part of matrix.

atoms(*types)

Returns the atoms that form the current object.

Examples

>>> from sympy.abc import x, y
>>> from sympy.matrices import Matrix
>>> Matrix([[x]])
Matrix([[x]])
>>> _.atoms()
{x}
>>> Matrix([[x, y], [y, x]])
Matrix([
[x, y],
[y, x]])
>>> _.atoms()
{x, y}

col(j)

Elementary column selector.

Examples

>>> from sympy import eye
>>> eye(2).col(0)
Matrix([
[1],
[0]])

See also

row, col_del, col_join, col_insert

col_del(col)

Delete the specified column.

col_insert(pos, other)

Insert one or more columns at the given column position.

Examples

>>> from sympy import zeros, ones
>>> M = zeros(3)
>>> V = ones(3, 1)
>>> M.col_insert(1, V)
Matrix([
[0, 1, 0, 0],
[0, 1, 0, 0],
[0, 1, 0, 0]])

See also

col, row_insert

col_join(other)

Concatenates two matrices along self’s last and other’s first row.

Examples

>>> from sympy import zeros, ones
>>> M = zeros(3)
>>> V = ones(1, 3)
>>> M.col_join(V)
Matrix([
[0, 0, 0],
[0, 0, 0],
[0, 0, 0],
[1, 1, 1]])

See also

col, row_join

classmethod companion(poly)

Returns a companion matrix of a polynomial.

Examples

>>> from sympy import Matrix, Poly, Symbol, symbols
>>> x = Symbol('x')
>>> c0, c1, c2, c3, c4 = symbols('c0:5')
>>> p = Poly(c0 + c1*x + c2*x**2 + c3*x**3 + c4*x**4 + x**5, x)
>>> Matrix.companion(p)
Matrix([
[0, 0, 0, 0, -c0],
[1, 0, 0, 0, -c1],
[0, 1, 0, 0, -c2],
[0, 0, 1, 0, -c3],
[0, 0, 0, 1, -c4]])

conjugate()

Return the by-element conjugation.

Examples

>>> from sympy.matrices import SparseMatrix
>>> from sympy import I
>>> a = SparseMatrix(((1, 2 + I), (3, 4), (I, -I)))
>>> a
Matrix([
[1, 2 + I],
[3,     4],
[I,    -I]])
>>> a.C
Matrix([
[ 1, 2 - I],
[ 3,     4],
[-I,     I]])

See also

transpose

Matrix transposition

H

Hermite conjugation

sympy.matrices.matrices.MatrixBase.D

Dirac conjugation

classmethod diag(*args, strict=False, unpack=True, rows=None, cols=None, **kwargs)

Returns a matrix with the specified diagonal. If matrices are passed, a block-diagonal matrix is created (i.e. the “direct sum” of the matrices).

Kwargs

rowsrows of the resulting matrix; computed if

not given.

colscolumns of the resulting matrix; computed if

not given.

cls : class for the resulting matrix

unpack : bool which, when True (default), unpacks a single sequence rather than interpreting it as a Matrix.

strict : bool which, when False (default), allows Matrices to have variable-length rows.

Examples

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

The current default is to unpack a single sequence. If this is not desired, set \(unpack=False\) and it will be interpreted as a matrix.

>>> Matrix.diag([1, 2, 3]) == Matrix.diag(1, 2, 3)
True

When more than one element is passed, each is interpreted as something to put on the diagonal. Lists are converted to matrices. Filling of the diagonal always continues from the bottom right hand corner of the previous item: this will create a block-diagonal matrix whether the matrices are square or not.

>>> col = [1, 2, 3]
>>> row = [[4, 5]]
>>> Matrix.diag(col, row)
Matrix([
[1, 0, 0],
[2, 0, 0],
[3, 0, 0],
[0, 4, 5]])

When \(unpack\) is False, elements within a list need not all be of the same length. Setting \(strict\) to True would raise a ValueError for the following:

>>> Matrix.diag([[1, 2, 3], [4, 5], [6]], unpack=False)
Matrix([
[1, 2, 3],
[4, 5, 0],
[6, 0, 0]])

The type of the returned matrix can be set with the cls keyword.

>>> from sympy.matrices import ImmutableMatrix
>>> from sympy.utilities.misc import func_name
>>> func_name(Matrix.diag(1, cls=ImmutableMatrix))
'ImmutableDenseMatrix'

A zero dimension matrix can be used to position the start of the filling at the start of an arbitrary row or column:

>>> from sympy import ones
>>> r2 = ones(0, 2)
>>> Matrix.diag(r2, 1, 2)
Matrix([
[0, 0, 1, 0],
[0, 0, 0, 2]])

See also

eye, diagonal, dense.diag, expressions.blockmatrix.BlockMatrix, sparsetools.banded

diagonal(k=0)

Returns the kth diagonal of self. The main diagonal corresponds to \(k=0\); diagonals above and below correspond to \(k > 0\) and \(k < 0\), respectively. The values of \(self[i, j]\) for which \(j - i = k\), are returned in order of increasing \(i + j\), starting with \(i + j = |k|\).

Examples

>>> from sympy import Matrix
>>> m = Matrix(3, 3, lambda i, j: j - i); m
Matrix([
[ 0,  1, 2],
[-1,  0, 1],
[-2, -1, 0]])
>>> _.diagonal()
Matrix([[0, 0, 0]])
>>> m.diagonal(1)
Matrix([[1, 1]])
>>> m.diagonal(-2)
Matrix([[-2]])

Even though the diagonal is returned as a Matrix, the element retrieval can be done with a single index:

>>> Matrix.diag(1, 2, 3).diagonal()[1]  # instead of [0, 1]
2

See also

diag

evalf(n=15, subs=None, maxn=100, chop=False, strict=False, quad=None, verbose=False)

Apply evalf() to each element of self.

expand(deep=True, modulus=None, power_base=True, power_exp=True, mul=True, log=True, multinomial=True, basic=True, **hints)

Apply core.function.expand to each entry of the matrix.

Examples

>>> from sympy.abc import x
>>> from sympy.matrices import Matrix
>>> Matrix(1, 1, [x*(x+1)])
Matrix([[x*(x + 1)]])
>>> _.expand()
Matrix([[x**2 + x]])

extract(rowsList, colsList)

Return 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.

Examples

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

Rows or columns can be repeated:

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

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

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

RowsList or colsList can also be a list of booleans, in which case the rows or columns corresponding to the True values will be selected:

>>> m.extract([0, 1, 2, 3], [True, False, True])
Matrix([
[0,  2],
[3,  5],
[6,  8],
[9, 11]])

classmethod eye(rows, cols=None, **kwargs)

Returns an identity matrix.

Args

rows : rows of the matrix cols : cols of the matrix (if None, cols=rows)

Kwargs

cls : class of the returned matrix

property free_symbols

Returns the free symbols within the matrix.

Examples

>>> from sympy.abc import x
>>> from sympy.matrices import Matrix
>>> Matrix([[x], [1]]).free_symbols
{x}

get_diag_blocks()

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
>>> 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
Matrix([
[1,    3],
[y, z**2]])
>>> a2
Matrix([[x]])
>>> a3
Matrix([[0]])

has(*patterns)

Test whether any subexpression matches any of the patterns.

Examples

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

classmethod hstack(*args)

Return a matrix formed by joining args horizontally (i.e. by repeated application of row_join).

Examples

>>> from sympy.matrices import Matrix, eye
>>> Matrix.hstack(eye(2), 2*eye(2))
Matrix([
[1, 0, 2, 0],
[0, 1, 0, 2]])

is_anti_symmetric(simplify=True)

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
Matrix([
[ 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
Matrix([
[ 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()

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

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

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

See also

is_lower, is_upper, sympy.matrices.matrices.MatrixEigen.is_diagonalizable, diagonalize

property is_hermitian

Checks if the matrix is Hermitian.

In a Hermitian matrix element i,j is the complex conjugate of element j,i.

Examples

>>> from sympy.matrices import Matrix
>>> from sympy import I
>>> from sympy.abc import x
>>> a = Matrix([[1, I], [-I, 1]])
>>> a
Matrix([
[ 1, I],
[-I, 1]])
>>> a.is_hermitian
True
>>> a[0, 0] = 2*I
>>> a.is_hermitian
False
>>> a[0, 0] = x
>>> a.is_hermitian
>>> a[0, 1] = a[1, 0]*I
>>> a.is_hermitian
False

property is_lower

Check if matrix is a lower triangular matrix. True can be returned even if the matrix is not square.

Examples

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

>>> m = Matrix(4, 3, [0, 0, 0, 2, 0, 0, 1, 4 , 0, 6, 6, 5])
>>> m
Matrix([
[0, 0, 0],
[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
Matrix([
[x**2 + y, x + y**2],
[       0,    x + y]])
>>> m.is_lower
False

See also

is_upper, is_diagonal, is_lower_hessenberg

property is_lower_hessenberg

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
Matrix([
[1, 2, 0, 0],
[5, 2, 3, 0],
[3, 4, 3, 7],
[5, 6, 1, 1]])
>>> a.is_lower_hessenberg
True

See also

is_upper_hessenberg, is_lower

property is_square

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

property is_strongly_diagonally_dominant

Tests if the matrix is row strongly diagonally dominant.

Explanation

A \(n, n\) matrix \(A\) is row strongly diagonally dominant if

\[\left|A_{i, i}\right| > \sum_{j = 0, j \neq i}^{n-1} \left|A_{i, j}\right| \quad {\text{for all }} i \in \{ 0, ..., n-1 \}\]

Examples

>>> from sympy.matrices import Matrix
>>> A = Matrix([[3, -2, 1], [1, -3, 2], [-1, 2, 4]])
>>> A.is_strongly_diagonally_dominant
False

>>> A = Matrix([[-2, 2, 1], [1, 3, 2], [1, -2, 0]])
>>> A.is_strongly_diagonally_dominant
False

>>> A = Matrix([[-4, 2, 1], [1, 6, 2], [1, -2, 5]])
>>> A.is_strongly_diagonally_dominant
True

Notes

If you want to test whether a matrix is column diagonally dominant, you can apply the test after transposing the matrix.

is_symbolic()

Checks if any elements contain Symbols.

Examples

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

is_symmetric(simplify=True)

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

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

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

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

property is_upper

Check if matrix is an upper triangular matrix. True can be returned even if the matrix is not square.

Examples

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

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

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

See also

is_lower, is_diagonal, is_upper_hessenberg

property is_upper_hessenberg

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
Matrix([
[1, 4, 2, 3],
[3, 4, 1, 7],
[0, 2, 3, 4],
[0, 0, 1, 3]])
>>> a.is_upper_hessenberg
True

See also

is_lower_hessenberg, is_upper

property is_weakly_diagonally_dominant

Tests if the matrix is row weakly diagonally dominant.

Explanation

A \(n, n\) matrix \(A\) is row weakly diagonally dominant if

\[\left|A_{i, i}\right| \ge \sum_{j = 0, j \neq i}^{n-1} \left|A_{i, j}\right| \quad {\text{for all }} i \in \{ 0, ..., n-1 \}\]

Examples

>>> from sympy.matrices import Matrix
>>> A = Matrix([[3, -2, 1], [1, -3, 2], [-1, 2, 4]])
>>> A.is_weakly_diagonally_dominant
True

>>> A = Matrix([[-2, 2, 1], [1, 3, 2], [1, -2, 0]])
>>> A.is_weakly_diagonally_dominant
False

>>> A = Matrix([[-4, 2, 1], [1, 6, 2], [1, -2, 5]])
>>> A.is_weakly_diagonally_dominant
True

Notes

If you want to test whether a matrix is column diagonally dominant, you can apply the test after transposing the matrix.

property is_zero_matrix

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. For a matrix that may or may not be zero (e.g. contains a symbol), this will be None

Examples

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

classmethod jordan_block(size=None, eigenvalue=None, *, band='upper', **kwargs)

Returns a Jordan block

Parameters

size : Integer, optional

Specifies the shape of the Jordan block matrix.

eigenvalue : Number or Symbol

Specifies the value for the main diagonal of the matrix.

Note

The keyword eigenval is also specified as an alias of this keyword, but it is not recommended to use.

We may deprecate the alias in later release.

band : ‘upper’ or ‘lower’, optional

Specifies the position of the off-diagonal to put \(1\) s on.

cls : Matrix, optional

Specifies the matrix class of the output form.

If it is not specified, the class type where the method is being executed on will be returned.

rows, cols : Integer, optional

Specifies the shape of the Jordan block matrix. See Notes section for the details of how these key works.

Note

This feature will be deprecated in the future.

Returns

Matrix

A Jordan block matrix.

Raises

ValueError

If insufficient arguments are given for matrix size specification, or no eigenvalue is given.

Examples

Creating a default Jordan block:

>>> from sympy import Matrix
>>> from sympy.abc import x
>>> Matrix.jordan_block(4, x)
Matrix([
[x, 1, 0, 0],
[0, x, 1, 0],
[0, 0, x, 1],
[0, 0, 0, x]])

Creating an alternative Jordan block matrix where \(1\) is on lower off-diagonal:

>>> Matrix.jordan_block(4, x, band='lower')
Matrix([
[x, 0, 0, 0],
[1, x, 0, 0],
[0, 1, x, 0],
[0, 0, 1, x]])

Creating a Jordan block with keyword arguments

>>> Matrix.jordan_block(size=4, eigenvalue=x)
Matrix([
[x, 1, 0, 0],
[0, x, 1, 0],
[0, 0, x, 1],
[0, 0, 0, x]])

Notes

Note

This feature will be deprecated in the future.

The keyword arguments size, rows, cols relates to the Jordan block size specifications.

If you want to create a square Jordan block, specify either one of the three arguments.

If you want to create a rectangular Jordan block, specify rows and cols individually.

Arguments Given			Matrix Shape
size	rows	cols	rows	cols
size	Any		size	size
None	None		ValueError
	rows	None	rows	rows
	None	cols	cols	cols
	rows	cols	rows	cols

References

R521

https://en.wikipedia.org/wiki/Jordan_matrix

lower_triangular(k=0)

returns the elements on and below the kth diagonal of a matrix. If k is not specified then simply returns lower-triangular portion of a matrix

Examples

>>> from sympy import ones
>>> A = ones(4)
>>> A.lower_triangular()
Matrix([
[1, 0, 0, 0],
[1, 1, 0, 0],
[1, 1, 1, 0],
[1, 1, 1, 1]])

>>> A.lower_triangular(-2)
Matrix([
[0, 0, 0, 0],
[0, 0, 0, 0],
[1, 0, 0, 0],
[1, 1, 0, 0]])

>>> A.lower_triangular(1)
Matrix([
[1, 1, 0, 0],
[1, 1, 1, 0],
[1, 1, 1, 1],
[1, 1, 1, 1]])

multiply(other, dotprodsimp=None)

Same as __mul__() but with optional simplification.

Parameters

dotprodsimp : bool, optional

Specifies whether intermediate term algebraic simplification is used during matrix multiplications to control expression blowup and thus speed up calculation. Default is off.

multiply_elementwise(other)

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

Examples

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

See also

sympy.matrices.matrices.MatrixBase.cross, sympy.matrices.matrices.MatrixBase.dot, multiply

n(*args, **kwargs)

Apply evalf() to each element of self.

classmethod ones(rows, cols=None, **kwargs)

Returns a matrix of ones.

Args

rows : rows of the matrix cols : cols of the matrix (if None, cols=rows)

Kwargs

cls : class of the returned matrix

permute(perm, orientation='rows', direction='forward')

Permute the rows or columns of a matrix by the given list of swaps.

Parameters

perm : Permutation, list, or list of lists

A representation for the permutation.

If it is Permutation, it is used directly with some resizing with respect to the matrix size.

If it is specified as list of lists, (e.g., [[0, 1], [0, 2]]), then the permutation is formed from applying the product of cycles. The direction how the cyclic product is applied is described in below.

If it is specified as a list, the list should represent an array form of a permutation. (e.g., [1, 2, 0]) which would would form the swapping function \(0 \mapsto 1, 1 \mapsto 2, 2\mapsto 0\).

orientation : ‘rows’, ‘cols’

A flag to control whether to permute the rows or the columns

direction : ‘forward’, ‘backward’

A flag to control whether to apply the permutations from the start of the list first, or from the back of the list first.

For example, if the permutation specification is [[0, 1], [0, 2]],

If the flag is set to 'forward', the cycle would be formed as \(0 \mapsto 2, 2 \mapsto 1, 1 \mapsto 0\).

If the flag is set to 'backward', the cycle would be formed as \(0 \mapsto 1, 1 \mapsto 2, 2 \mapsto 0\).

If the argument perm is not in a form of list of lists, this flag takes no effect.

Examples

>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.permute([[0, 1], [0, 2]], orientation='rows', direction='forward')
Matrix([
[0, 0, 1],
[1, 0, 0],
[0, 1, 0]])

>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.permute([[0, 1], [0, 2]], orientation='rows', direction='backward')
Matrix([
[0, 1, 0],
[0, 0, 1],
[1, 0, 0]])

Notes

If a bijective function \(\sigma : \mathbb{N}_0 \rightarrow \mathbb{N}_0\) denotes the permutation.

If the matrix \(A\) is the matrix to permute, represented as a horizontal or a vertical stack of vectors:

\[\begin{split}A = \begin{bmatrix} a_0 \\ a_1 \\ \vdots \\ a_{n-1} \end{bmatrix} = \begin{bmatrix} \alpha_0 & \alpha_1 & \cdots & \alpha_{n-1} \end{bmatrix}\end{split}\]

If the matrix \(B\) is the result, the permutation of matrix rows is defined as:

\[\begin{split}B := \begin{bmatrix} a_{\sigma(0)} \\ a_{\sigma(1)} \\ \vdots \\ a_{\sigma(n-1)} \end{bmatrix}\end{split}\]

And the permutation of matrix columns is defined as:

\[B := \begin{bmatrix} \alpha_{\sigma(0)} & \alpha_{\sigma(1)} & \cdots & \alpha_{\sigma(n-1)} \end{bmatrix}\]

permute_cols(swaps, direction='forward')

Alias for self.permute(swaps, orientation='cols', direction=direction)

See also

permute

permute_rows(swaps, direction='forward')

Alias for self.permute(swaps, orientation='rows', direction=direction)

See also

permute

pow(exp, method=None)

Return self**exp a scalar or symbol.

Parameters

method : multiply, mulsimp, jordan, cayley

If multiply then it returns exponentiation using recursion. If jordan then Jordan form exponentiation will be used. If cayley then the exponentiation is done using Cayley-Hamilton theorem. If mulsimp then the exponentiation is done using recursion with dotprodsimp. This specifies whether intermediate term algebraic simplification is used during naive matrix power to control expression blowup and thus speed up calculation. If None, then it heuristically decides which method to use.

refine(assumptions=True)

Apply refine to each element of the matrix.

Examples

>>> from sympy import Symbol, Matrix, Abs, sqrt, Q
>>> x = Symbol('x')
>>> Matrix([[Abs(x)**2, sqrt(x**2)],[sqrt(x**2), Abs(x)**2]])
Matrix([
[ Abs(x)**2, sqrt(x**2)],
[sqrt(x**2),  Abs(x)**2]])
>>> _.refine(Q.real(x))
Matrix([
[  x**2, Abs(x)],
[Abs(x),   x**2]])

replace(F, G, map=False, simultaneous=True, exact=None)

Replaces Function F in Matrix entries with Function G.

Examples

>>> from sympy import symbols, Function, Matrix
>>> F, G = symbols('F, G', cls=Function)
>>> M = Matrix(2, 2, lambda i, j: F(i+j)) ; M
Matrix([
[F(0), F(1)],
[F(1), F(2)]])
>>> N = M.replace(F,G)
>>> N
Matrix([
[G(0), G(1)],
[G(1), G(2)]])

reshape(rows, cols)

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

Examples

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

rmultiply(other, dotprodsimp=None)

Same as __rmul__() but with optional simplification.

Parameters

dotprodsimp : bool, optional

Specifies whether intermediate term algebraic simplification is used during matrix multiplications to control expression blowup and thus speed up calculation. Default is off.

rot90(k=1)

Rotates Matrix by 90 degrees

Parameters

k : int

Specifies how many times the matrix is rotated by 90 degrees (clockwise when positive, counter-clockwise when negative).

Examples

>>> from sympy import Matrix, symbols
>>> A = Matrix(2, 2, symbols('a:d'))
>>> A
Matrix([
[a, b],
[c, d]])

Rotating the matrix clockwise one time:

>>> A.rot90(1)
Matrix([
[c, a],
[d, b]])

Rotating the matrix anticlockwise two times:

>>> A.rot90(-2)
Matrix([
[d, c],
[b, a]])

row(i)

Elementary row selector.

Examples

>>> from sympy import eye
>>> eye(2).row(0)
Matrix([[1, 0]])

See also

col, row_del, row_join, row_insert

row_del(row)

Delete the specified row.

row_insert(pos, other)

Insert one or more rows at the given row position.

Examples

>>> from sympy import zeros, ones
>>> M = zeros(3)
>>> V = ones(1, 3)
>>> M.row_insert(1, V)
Matrix([
[0, 0, 0],
[1, 1, 1],
[0, 0, 0],
[0, 0, 0]])

See also

row, col_insert

row_join(other)

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

Examples

>>> from sympy import zeros, ones
>>> M = zeros(3)
>>> V = ones(3, 1)
>>> M.row_join(V)
Matrix([
[0, 0, 0, 1],
[0, 0, 0, 1],
[0, 0, 0, 1]])

See also

row, col_join

property shape

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

Examples

>>> from sympy.matrices import zeros
>>> M = zeros(2, 3)
>>> M.shape
(2, 3)
>>> M.rows
2
>>> M.cols
3

simplify(**kwargs)

Apply simplify to each element of the matrix.

Examples

>>> from sympy.abc import x, y
>>> from sympy import sin, cos
>>> from sympy.matrices import SparseMatrix
>>> SparseMatrix(1, 1, [x*sin(y)**2 + x*cos(y)**2])
Matrix([[x*sin(y)**2 + x*cos(y)**2]])
>>> _.simplify()
Matrix([[x]])

subs(*args, **kwargs)

Return a new matrix with subs applied to each entry.

Examples

>>> from sympy.abc import x, y
>>> from sympy.matrices import SparseMatrix, Matrix
>>> SparseMatrix(1, 1, [x])
Matrix([[x]])
>>> _.subs(x, y)
Matrix([[y]])
>>> Matrix(_).subs(y, x)
Matrix([[x]])

todod()

Returns matrix as dict of dicts containing non-zero elements of the Matrix

Examples

>>> from sympy import Matrix
>>> A = Matrix([[0, 1],[0, 3]])
>>> A
Matrix([
[0, 1],
[0, 3]])
>>> A.todod()
{0: {1: 1}, 1: {1: 3}}

todok()

Return the matrix as dictionary of keys.

Examples

>>> from sympy import Matrix
>>> M = Matrix.eye(3)
>>> M.todok()
{(0, 0): 1, (1, 1): 1, (2, 2): 1}

tolist()

Return the Matrix as a nested Python list.

Examples

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

When there are no rows then it will not be possible to tell how many columns were in the original matrix:

>>> ones(0, 3).tolist()
[]

trace()

Returns the trace of a square matrix i.e. the sum of the diagonal elements.

Examples

>>> from sympy import Matrix
>>> A = Matrix(2, 2, [1, 2, 3, 4])
>>> A.trace()
5

transpose()

Returns the transpose of the matrix.

Examples

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

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

See also

conjugate

By-element conjugation

upper_triangular(k=0)

returns the elements on and above the kth diagonal of a matrix. If k is not specified then simply returns upper-triangular portion of a matrix

Examples

>>> from sympy import ones
>>> A = ones(4)
>>> A.upper_triangular()
Matrix([
[1, 1, 1, 1],
[0, 1, 1, 1],
[0, 0, 1, 1],
[0, 0, 0, 1]])

>>> A.upper_triangular(2)
Matrix([
[0, 0, 1, 1],
[0, 0, 0, 1],
[0, 0, 0, 0],
[0, 0, 0, 0]])

>>> A.upper_triangular(-1)
Matrix([
[1, 1, 1, 1],
[1, 1, 1, 1],
[0, 1, 1, 1],
[0, 0, 1, 1]])

values()

Return non-zero values of self.

vec()

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

Examples

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

See also

vech

vech(diagonal=True, check_symmetry=True)

Reshapes the matrix into a column vector by stacking the elements in the lower triangle.

Parameters

diagonal : bool, optional

If True, it includes the diagonal elements.

check_symmetry : bool, optional

If True, it checks whether the matrix is symmetric.

Examples

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

Notes

This should work for symmetric matrices and vech can represent symmetric matrices in vector form with less size than vec.

See also

vec

classmethod vstack(*args)

Return a matrix formed by joining args vertically (i.e. by repeated application of col_join).

Examples

>>> from sympy.matrices import Matrix, eye
>>> Matrix.vstack(eye(2), 2*eye(2))
Matrix([
[1, 0],
[0, 1],
[2, 0],
[0, 2]])

classmethod wilkinson(n, **kwargs)

Returns two square Wilkinson Matrix of size 2*n + 1 \(W_{2n + 1}^-, W_{2n + 1}^+ =\) Wilkinson(n)

Examples

>>> from sympy.matrices import Matrix
>>> wminus, wplus = Matrix.wilkinson(3)
>>> wminus
Matrix([
[-3,  1,  0, 0, 0, 0, 0],
[ 1, -2,  1, 0, 0, 0, 0],
[ 0,  1, -1, 1, 0, 0, 0],
[ 0,  0,  1, 0, 1, 0, 0],
[ 0,  0,  0, 1, 1, 1, 0],
[ 0,  0,  0, 0, 1, 2, 1],
[ 0,  0,  0, 0, 0, 1, 3]])
>>> wplus
Matrix([
[3, 1, 0, 0, 0, 0, 0],
[1, 2, 1, 0, 0, 0, 0],
[0, 1, 1, 1, 0, 0, 0],
[0, 0, 1, 0, 1, 0, 0],
[0, 0, 0, 1, 1, 1, 0],
[0, 0, 0, 0, 1, 2, 1],
[0, 0, 0, 0, 0, 1, 3]])

References

R522

https://blogs.mathworks.com/cleve/2013/04/15/wilkinsons-matrices-2/

R523

10. 8. Wilkinson, The Algebraic Eigenvalue Problem, Claredon Press, Oxford, 1965, 662 pp.

xreplace(rule)

Return a new matrix with xreplace applied to each entry.

Examples

>>> from sympy.abc import x, y
>>> from sympy.matrices import SparseMatrix, Matrix
>>> SparseMatrix(1, 1, [x])
Matrix([[x]])
>>> _.xreplace({x: y})
Matrix([[y]])
>>> Matrix(_).xreplace({y: x})
Matrix([[x]])

classmethod zeros(rows, cols=None, **kwargs)

Returns a matrix of zeros.

Args

rows : rows of the matrix cols : cols of the matrix (if None, cols=rows)

Kwargs

cls : class of the returned matrix