# Dense Matrices¶

## Matrix Class Reference¶

class diofant.matrices.dense.DenseMatrix[source]
__getitem__(key)[source]

Return portion of self defined by key. If the key involves a slice then a list will be returned (if key is a single slice) or a matrix (if key was a tuple involving a slice).

Examples

>>> m = Matrix([
... [1, 2 + I],
... [3, 4    ]])


If the key is a tuple that doesn’t involve a slice then that element is returned:

>>> m[1, 0]
3


When a tuple key involves a slice, a matrix is returned. Here, the first column is selected (all rows, column 0):

>>> m[:, 0]
Matrix([
[1],
[3]])


If the slice is not a tuple then it selects from the underlying list of elements that are arranged in row order and a list is returned if a slice is involved:

>>> m[0]
1
>>> m[::2]
[1, 3]

__mul__(other)[source]

Return self*other.

applyfunc(f)[source]

Apply a function to each element of the matrix.

Examples

>>> 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_immutable()[source]

Returns an Immutable version of this Matrix.

as_mutable()[source]

Returns a mutable version of this matrix

Examples

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

equals(other, failing_expression=False)[source]

Applies equals to corresponding elements of the matrices, trying to prove that the elements are equivalent, returning True if they are, False if any pair is not, and None (or the first failing expression if failing_expression is True) if it cannot be decided if the expressions are equivalent or not. This is, in general, an expensive operation.

Examples

>>> A = Matrix([x*(x - 1), 0])
>>> B = Matrix([x**2 - x, 0])
>>> A == B
False
>>> A.simplify() == B.simplify()
True
>>> A.equals(B)
True
>>> A.equals(2)
False

classmethod eye(n)[source]

Return an n x n identity matrix.

reshape(rows, cols)[source]

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

Examples

>>> 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]])

tolist()[source]

Return the Matrix as a nested Python list.

Examples

>>> 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()
[]

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

Return an r x c matrix of zeros, square if c is omitted.

class diofant.matrices.dense.MutableDenseMatrix[source]
col_op(j, f)[source]

In-place operation on col j using two-arg functor whose args are interpreted as (self[i, j], i).

Examples

>>> M = eye(3)
>>> M.col_op(1, lambda v, i: v + 2*M[i, 0]); M
Matrix([
[1, 2, 0],
[0, 1, 0],
[0, 0, 1]])

col_swap(i, j)[source]

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

Examples

>>> M = Matrix([[1, 0], [1, 0]])
>>> M
Matrix([
[1, 0],
[1, 0]])
>>> M.col_swap(0, 1)
>>> M
Matrix([
[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.

Examples

>>> I = eye(3)
>>> I[:2, 0] = [1, 2] # col
>>> I
Matrix([
[1, 0, 0],
[2, 1, 0],
[0, 0, 1]])
>>> I[1, :2] = [[3, 4]]
>>> I
Matrix([
[1, 0, 0],
[3, 4, 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.

Examples

>>> M = Matrix([[0, 1], [2, 3], [4, 5]])
>>> I = eye(3)
>>> I[:3, :2] = M
>>> I
Matrix([
[0, 1, 0],
[2, 3, 0],
[4, 5, 1]])
>>> I[0, 1] = M
>>> I
Matrix([
[0, 0, 1],
[2, 2, 3],
[4, 4, 5]])

fill(value)[source]

Fill the matrix with the scalar value.

row_op(i, f)[source]

In-place operation on row i using two-arg functor whose args are interpreted as (self[i, j], j).

Examples

>>> M = eye(3)
>>> M.row_op(1, lambda v, j: v + 2*M[0, j]); M
Matrix([
[1, 0, 0],
[2, 1, 0],
[0, 0, 1]])

row_swap(i, j)[source]

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

Examples

>>> M = Matrix([[0, 1], [1, 0]])
>>> M
Matrix([
[0, 1],
[1, 0]])
>>> M.row_swap(0, 1)
>>> M
Matrix([
[1, 0],
[0, 1]])

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

Applies simplify to the elements of a matrix in place.

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

zip_row_op(i, k, f)[source]

In-place operation on row i using two-arg functor whose args are interpreted as (self[i, j], self[k, j]).

Examples

>>> M = eye(3)
>>> M.zip_row_op(1, 0, lambda v, u: v + 2*u); M
Matrix([
[1, 0, 0],
[2, 1, 0],
[0, 0, 1]])

class diofant.matrices.dense.MutableMatrix

alias of MutableDenseMatrix

## ImmutableMatrix Class Reference¶

class diofant.matrices.immutable.ImmutableMatrix[source]

Create an immutable version of a matrix.

Examples

>>> ImmutableMatrix(eye(3))
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
>>> _[0, 0] = 42
Traceback (most recent call last):
...
TypeError: Cannot set values of ImmutableDenseMatrix

C

By-element conjugation.

adjoint()

Conjugate transpose or Hermitian conjugation.

as_mutable()

Returns a mutable version of this matrix

Examples

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

conjugate()

By-element conjugation.

equals(other, failing_expression=False)

Applies equals to corresponding elements of the matrices, trying to prove that the elements are equivalent, returning True if they are, False if any pair is not, and None (or the first failing expression if failing_expression is True) if it cannot be decided if the expressions are equivalent or not. This is, in general, an expensive operation.

Examples

>>> A = Matrix([x*(x - 1), 0])
>>> B = Matrix([x**2 - x, 0])
>>> A == B
False
>>> A.simplify() == B.simplify()
True
>>> A.equals(B)
True
>>> A.equals(2)
False

is_zero

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

>>> 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
True
>>> b.is_zero
True
>>> c.is_zero
False
>>> d.is_zero
True
>>> e.is_zero