Linear Algebra¶

Matrices¶

Operations over matrices

simula.linalg.matrices.In(n)¶

Returns an identity matrix of order n.

Parameters: n – an integer

simula : identity_matrix(2)
Matrix([
[1, 0],
[0, 1]])
simula : identity_matrix(3)
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])

class simula.linalg.matrices.Matrix(*args, **kwargs)¶

Bases: sympy.matrices.dense.MutableDenseMatrix, simula.structure.simula_object.SimulaObject

Base class for Matrices.

EXAMPLES:

simula : A = matrix([[-2, 1, 4], [-5, 2, 5], [-1, 1, 3]]); A
Matrix([
[-2, 1, 4],
[-5, 2, 5],
[-1, 1, 3]])
simula : A.det()
-4
simula : A.rank()
3
simula : A.trace()
3
simula : A.transpose()
Matrix([
[-2, -5, -1],
[ 1,  2,  1],
[ 4,  5,  3]])
simula : A^3
Matrix([
[-19, 12, 27],
[-15,  8, 15],
[-18, 12, 26]])
simula : A.charpoly('x')
PurePoly(x^3 - 3x^2 + 4, x, domain='ZZ')
simula : A.eigenvals()
{-1: 1, 2: 2}
simula : A.cofactor_matrix()
Matrix([
[ 1,  10, -3],
[ 1,  -2,  1],
[-3, -10,  1]])
simula : B = matrix([[1, 2], [3, 4]]); B
Matrix([
[1, 2],
[3, 4]])
simula : C = matrix([[5, 6, 7], [8, 9, 10]]); C
Matrix([
[5, 6,  7],
[8, 9, 10]])
simula : matrix([B, C])
Matrix([
[1, 2, 5, 6,  7],
[3, 4, 8, 9, 10]])
simula : matrix([[B, C], [B, C]])
[1, 2, 5, 6,  7],
[3, 4, 8, 9, 10],
[1, 2, 5, 6,  7],
[3, 4, 8, 9, 10]])

algebraic_multiplicity(lamda)¶

Returns the algebraic multiplicity of lamda.

EXAMPLES:

simula : A = matrix([[-2, 1, 4], [-5, 2, 5], [-1, 1, 3]]); A
Matrix([
[-2, 1, 4],
[-5, 2, 5],
[-1, 1, 3]])
simula : A.eigenvalues()
{-1: 1, 2: 2}
simula : A.algebraic_multiplicity(2)
2

static circulant(v, shift=None)¶

Returns a circulant matrix.

See also circulant_matrix

dunford_decomposition()¶

Returns the Dunford decomposition of self.

EXAMPLES:

simula : A = matrix([[-2, 1, 4], [-5, 2, 5], [-1, 1, 3]]); A
Matrix([
[-2, 1, 4],
[-5, 2, 5],
[-1, 1, 3]])
simula : B, N = A.dunford_decomposition(); B, N
(Matrix([
[-1/3, 0, 7/3],
[  -5, 2,   5],
[ 2/3, 0, 4/3]]), Matrix([
[-5/3, 1, 5/3],
[   0, 0,   0],
[-5/3, 1, 5/3]]))
simula : B*N == N*B
True
simula : B.is_diagonalizable()
True
simula : N.is_nilpotent()
True
simula : B+N
Matrix([
[-2, 1, 4],
[-5, 2, 5],
[-1, 1, 3]])

eigenvalues(*args, **kwargs)¶

Returns the eigenvalues of self.

EXAMPLES:

simula : A = matrix([[-2, 1, 4], [-5, 2, 5], [-1, 1, 3]]); A
Matrix([
[-2, 1, 4],
[-5, 2, 5],
[-1, 1, 3]])
simula : A.eigenvalues()
{-1: 1, 2: 2}

eigenvectors_left(**kwargs)¶

Returns the left eigenvectors of self.

EXAMPLES:

simula : A = matrix([[-2, 1, 4], [-5, 2, 5], [-1, 1, 3]]); A
Matrix([
[-2, 1, 4],
[-5, 2, 5],
[-1, 1, 3]])
simula : A.eigenvectors_left()
[(-1, 1, [Matrix([[-1, 0, 1]])]), (2, 2, [Matrix([[-1, 3/5, 1]])])]

eigenvectors_right(**kwargs)¶

Returns the right eigenvectors of self.

EXAMPLES:

simula : A = matrix([[-2, 1, 4], [-5, 2, 5], [-1, 1, 3]]); A
Matrix([
[-2, 1, 4],
[-5, 2, 5],
[-1, 1, 3]])
simula : A.eigenvectors_right()
[(-1, 1, [Matrix([
[ -7/2],
[-15/2],
[    1]])]), (2, 2, [Matrix([
[1],
[0],
[1]])])]

geometric_multiplicity(lamda)¶

Returns the geometric multiplicity of lamda.

EXAMPLES:

simula : A = matrix([[-2, 1, 4], [-5, 2, 5], [-1, 1, 3]]); A
Matrix([
[-2, 1, 4],
[-5, 2, 5],
[-1, 1, 3]])
simula : A.eigenvalues()
{-1: 1, 2: 2}
simula : A.geometric_multiplicity(2)
1

linear_map(domain=None, codomain=None)¶

Returns the linear map associated to self.

Parameters: domain – (optional) a field or vector space

:param codomain : (optional) a field or vector space

EXAMPLES:

simula : A = matrix([[-2, 1, 4], [-5, 2, 5], [-1, 1, 3]]); A
Matrix([
[-2, 1, 4],
[-5, 2, 5],
[-1, 1, 3]])
simula : f = A.linear_map(RR, RR)
simula : f
Linear map from RR^3 --> RR^3 defined by (x1, x2, x3) |--> (-2x1 + x2 + 4x3, -5x1 + 2x2 + 5x3, -x1 + x2 + 3x3)
simula : f(1, 0, 0)
(-2, -5, -1)

reverse_cols()¶

Returns a matrix when its columns are the reversed columns of self.

EXAMPLES:

simula : A = matrix([[-2, 1, 4], [-5, 2, 5], [-1, 1, 3]]); A
Matrix([
[-2, 1, 4],
[-5, 2, 5],
[-1, 1, 3]])
simula : A.reverse_cols()
Matrix([
[-1, 1, 3],
[-5, 2, 5],
[-2, 1, 4]])

reverse_rows()¶

Returns a matrix when its rows are the reversed rows of self.

EXAMPLES:

simula : A = matrix([[-2, 1, 4], [-5, 2, 5], [-1, 1, 3]]); A
Matrix([
[-2, 1, 4],
[-5, 2, 5],
[-1, 1, 3]])
simula : A.reverse_rows()
Matrix([
[4, 1, -2],
[5, 2, -5],
[3, 1, -1]])

rref_mod(gf)¶

Returns the row reduced echelon form in finite field GF(q).

Parameters: gf – a finite field GF(q)

EXAMPLES:

simula : A = matrix([[2, 1, 0, 0, 1, 1, -1], [1, 1, 0, 1, 0, -1, 0], [2, -1, 1, 1, 0, 1, -1]]); A
Matrix([
[2,  1, 0, 0, 1,  1, -1],
[1,  1, 0, 1, 0, -1,  0],
[2, -1, 1, 1, 0,  1, -1]])
simula : A.rref_mod(GF(3))
Matrix([
[1, 0, 0, 2, 1, 2, 2],
[0, 1, 0, 2, 2, 0, 1],
[0, 0, 1, 2, 0, 0, 2]])
simula : A.rref_mod(GF(5))
Matrix([
[1, 0, 0, 4, 1, 2, 4],
[0, 1, 0, 2, 4, 2, 1],
[0, 0, 1, 0, 2, 4, 2]])

spectral_radius()¶

Returns the spectral radius of self.

EXAMPLES:

simula : A = matrix([[-2, 1, 4], [-5, 2, 5], [-1, 1, 3]]); A
Matrix([
[-2, 1, 4],
[-5, 2, 5],
[-1, 1, 3]])
simula : A.spectral_radius()
2

spectrum()¶

Returns the spectrum of self.

EXAMPLES:

simula : A = matrix([[-2, 1, 4], [-5, 2, 5], [-1, 1, 3]]); A
Matrix([
[-2, 1, 4],
[-5, 2, 5],
[-1, 1, 3]])
simula : A.spectrum()
{2, -1}

simula.linalg.matrices.block_matrix¶: alias of simula.linalg.matrices.BlockMatrix

simula.linalg.matrices.circulant_matrix(v, shift=None)¶

Returns a circulant matrix.

Parameters

v – a vector
shift – (optional) the number of rows. If it is not None, it should be less or equal than the length of v

EXAMPLES:

simula : v = [1,2,3,4,5]
simula : circulant_matrix(v)
Matrix([
[1, 2, 3, 4, 5],
[5, 1, 2, 3, 4],
[4, 5, 1, 2, 3],
[3, 4, 5, 1, 2],
[2, 3, 4, 5, 1]])
simula : circulant_matrix(v, shift=3)
Matrix([
[1, 2, 3, 4, 5],
[5, 1, 2, 3, 4],
[4, 5, 1, 2, 3]])

simula.linalg.matrices.companion_matrix(poly, format='right')¶

Returns a companion matrix associated to a polynomial for a given format.

Parameters

poly – a polynomial
format – a string (“right”, “left”, “top”, “bottom”), default is “right”

EXAMPLES:

simula : p = x^5-x^4-3x^3-x^2+5x-6
simula : companion_matrix(p)
Matrix([
[0, 0, 0, 0,  6],
[1, 0, 0, 0, -5],
[0, 1, 0, 0,  1],
[0, 0, 1, 0,  3],
[0, 0, 0, 1,  1]])
simula : companion_matrix(p, format='left')
Matrix([
[ 1, 1, 0, 0, 0],
[ 3, 0, 1, 0, 0],
[ 1, 0, 0, 1, 0],
[-5, 0, 0, 0, 1],
[ 6, 0, 0, 0, 0]])
simula : companion_matrix(p, format='top')
Matrix([
[1, 3, 1, -5, 6],
[1, 0, 0,  0, 0],
[0, 1, 0,  0, 0],
[0, 0, 1,  0, 0],
[0, 0, 0,  1, 0]])
simula : companion_matrix(p, format='bottom')
Matrix([
[0,  1, 0, 0, 0],
[0,  0, 1, 0, 0],
[0,  0, 0, 1, 0],
[0,  0, 0, 0, 1],
[6, -5, 1, 3, 1]])

simula.linalg.matrices.diag(*args)¶

Returns a diagonal matrix.

EXAMPLES:

simula : diagonal_matrix(1, 2, 3)
Matrix([
[1, 0, 0],
[0, 2, 0],
[0, 0, 3]])
simula : diagonal_matrix(-2, 2, 1, 1)
Matrix([
[-2, 0, 0, 0],
[ 0, 2, 0, 0],
[ 0, 0, 1, 0],
[ 0, 0, 0, 1]])

simula.linalg.matrices.diagonal_matrix(*args)¶

Returns a diagonal matrix.

EXAMPLES:

simula : diagonal_matrix(1, 2, 3)
Matrix([
[1, 0, 0],
[0, 2, 0],
[0, 0, 3]])
simula : diagonal_matrix(-2, 2, 1, 1)
Matrix([
[-2, 0, 0, 0],
[ 0, 2, 0, 0],
[ 0, 0, 1, 0],
[ 0, 0, 0, 1]])

simula.linalg.matrices.hilbert_matrix(n)¶

Return a Hilbert matrix of the given dimension.

H_ij = 1/(i + j - 1) for i, j = 1, …, n

Parameters: n – an integer

EXAMPLES:

simula : hilbert_matrix(2)
Matrix([
[  1, 1/2],
[1/2, 1/3]])
simula : hilbert_matrix(3)
Matrix([
[  1, 1/2, 1/3],
[1/2, 1/3, 1/4],
[1/3, 1/4, 1/5]])
simula : hilbert_matrix(4)
Matrix([
[  1, 1/2, 1/3, 1/4],
[1/2, 1/3, 1/4, 1/5],
[1/3, 1/4, 1/5, 1/6],
[1/4, 1/5, 1/6, 1/7]])

simula.linalg.matrices.identity_matrix(n)¶

Returns an identity matrix of order n.

Parameters: n – an integer

simula : identity_matrix(2)
Matrix([
[1, 0],
[0, 1]])
simula : identity_matrix(3)
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])

simula.linalg.matrices.jordan_cell(lamda, dim)¶

Returns a jordan block of dimension dim associated to the eigenvalue lamda.

Param: an eigenvalue
Parameters: dim – (an integer) the number of rows and columns

EXAMPLES:

simula : jordan_cell(2, 3)
Matrix([
[2, 1, 0],
[0, 2, 1],
[0, 0, 2]])
simula : jordan_cell(-3, 4)
Matrix([
[-3,  1,  0,  0],
[ 0, -3,  1,  0],
[ 0,  0, -3,  1],
[ 0,  0,  0, -3]])

simula.linalg.matrices.linear_system_to_matrix(expr, symbols=None)¶

Returns the two matrices associated to a linear system.

Parameters

expr – an expression
symbols – list of symbols

Return type

(<class ‘simula.linalg.matrices.Matrix’>, <class ‘simula.linalg.matrices.Matrix’>)

EXAMPLES

simula : x, y , z = var('x,y,z')
simula : A, b = linear_system_to_matrix([x+y-z-1, 2x-3y-z-7, 2x+z-5], (x,y,z))
simula : A
Matrix([
[1,  1, -1],
[2, -3, -1],
[2,  0,  1]])
simula : b
Matrix([
[1],
[7],
[5]])

simula.linalg.matrices.matrix¶: alias of simula.linalg.matrices.Matrix

simula.linalg.matrices.ones(rows, cols=None)¶

Returns a matrix when all its coefficients are equal to one.

Parameters

rows – (an integer) the number or rows
cols – (optional) the number or columns

EXAMPLES:

simula : ones_matrix(2, 4)
Matrix([
[1, 1, 1, 1],
[1, 1, 1, 1]])
simula : ones_matrix(3)
Matrix([
[1, 1, 1],
[1, 1, 1],
[1, 1, 1]])

simula.linalg.matrices.ones_matrix(rows, cols=None)¶

Returns a matrix when all its coefficients are equal to one.

Parameters

rows – (an integer) the number or rows
cols – (optional) the number or columns

EXAMPLES:

simula : ones_matrix(2, 4)
Matrix([
[1, 1, 1, 1],
[1, 1, 1, 1]])
simula : ones_matrix(3)
Matrix([
[1, 1, 1],
[1, 1, 1],
[1, 1, 1]])

simula.linalg.matrices.zero_matrix(rows, cols=None)¶

Returns a zero matrix with rows rows and cols cols. If cols is not given it is equal to the rows.

Parameters

rows – (an integer) the number or rows
cols – (optional) the number or columns

EXAMPLES:

simula : zero_matrix(2, 3)
Matrix([
[0, 0, 0],
[0, 0, 0]])
simula : zero_matrix(3)
Matrix([
[0, 0, 0],
[0, 0, 0],
[0, 0, 0]])

simula.linalg.matrices.zeros(rows, cols=None)¶

Returns a zero matrix with rows rows and cols cols. If cols is not given it is equal to the rows.

Parameters

rows – (an integer) the number or rows
cols – (optional) the number or columns

EXAMPLES:

simula : zero_matrix(2, 3)
Matrix([
[0, 0, 0],
[0, 0, 0]])
simula : zero_matrix(3)
Matrix([
[0, 0, 0],
[0, 0, 0],
[0, 0, 0]])

Vector Spaces¶

Vectors Spaces

Classes

VectorSpace K^n
SubSpace
Vect
MatrixSpace
vector

class simula.linalg.vector_space.MatrixSpace(field, rows=None, cols=None)¶

Bases: simula.linalg.vector_space.VectorSpace, abc.ABC

Representation of Matrix spaces

Parameters

field – a field
rows – the number of rows
cols – the number of columns

EXAMPLES:

simula : M = MatrixSpace(QQ, 3, 2)
Matrix Space of dimension 3 x 2 over Rational Numbers
simula : M.canonical_basis()
[Matrix([
[1, 0],
[0, 0],
[0, 0]]), Matrix([
[0, 1],
[0, 0],
[0, 0]]), Matrix([
[0, 0],
[1, 0],
[0, 0]]), Matrix([
[0, 0],
[0, 1],
[0, 0]]), Matrix([
[0, 0],
[0, 0],
[1, 0]]), Matrix([
[0, 0],
[0, 0],
[0, 1]])]
simula : A = matrix([[1, 2.0], [0.5, 2.5], [0.3, 1]]) ; A
Matrix([
[  1, 2.0],
[0.5, 2.5],
[0.3,   1]])
simula : M(A)
Matrix([
[   1,   2],
[ 1/2, 5/2],
[3/10,   1]])

are_linearly_independent(family)¶

Tests if the vectors in``family`` are linearly independent in self.

Parameters: family (Union[Iterable, Sized]) –

canonical_basis()¶: Returns a canonical basis of self.

change_field(field)¶: Returns a matrix space of same dimension of self for the new field.

get_a_basis()¶: Returns a basis of self.

get_component_in_basis(v, family=None)¶: Returns the components of the vector v in family.

is_basis(family)¶

Tests if family is a basis of self.

Parameters: family (Union[Iterable, Sized]) –

linear_combination(family, coeffs)¶: Returns a linear combination of the vectors in family by the coefficients in self.

random_element(a=None, b=None)¶: Returns a random matrix in self.

class simula.linalg.vector_space.SubSpace(family=None, domain=None, name='')¶

Bases: simula.linalg.vector_space.VectorSpace, abc.ABC

Representation of a subspace

INPUT:

Parameters

family – a set of vectors
domain – a vector space (optional)
name – the name of self (optional)

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : F = SubSpace({(1, 2, 0), (1, 1, 2)}, V)
Subspace of QQ^3 generated by the family {(1, 2, 0), (1, 1, 2)}
simula : F == V
False
simula : W = SubSpace({(1, 2, 0), (1, 1, 2), (1, 0, 0)}, V)
Subspace of QQ^3 generated by the family {(1, 0, 0), (1, 2, 0), (1, 1, 2)}
simula : W == V

get_a_basis()¶: Returns a basis of self.

get_component_in_basis(v, family=None)¶: Returns the component of v (if it exists) for the family.

class simula.linalg.vector_space.Vect(family=None, domain=None, transpose=False)¶

Bases: simula.linalg.vector_space.SubSpace, abc.ABC

Representation of (an abstract) subspace.

simula : V = QQ^3; V
Vector Space of dimension 3 over Rational Numbers
simula : F = Vect({(1,0,0), (1, 1, 1)}, V); F
Vect({(1, 0, 0), (1, 1, 1)})
simula : W = Vect({(1, 0, 0), (1, 1, 1)}); W
Vect({(1, 0, 0), (1, 1, 1)})
simula : W.dimension()
2

class simula.linalg.vector_space.VectorSpace(field, dim=1)¶

Bases: sympy.polys.domains.domain.Domain, simula.structure.simula_object.SimulaObject, abc.ABC

are_linearly_dependent(family)¶

Tests if the vectors in family` are linearly dependent in ``self.

Parameters: family (Iterable) – a set of vectors
Returns: a boolean

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : V.are_linearly_dependent({(1, 1, 1), (2, 1, 0), (3, 2, 1)})
True

are_linearly_independent(family)¶

Tests if the vectors in family` are linearly independent in ``self.

Parameters: family (Union[Iterable, Sized]) – a set of vectors
Returns: a boolean

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : V.are_linearly_independent({(1, 1, 1), (2, 1, 0), (3, 2, 1)})
False

canonical_basis()¶

Returns the canonical basis of self.

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : V.canonical_basis()
{(1, 0, 0), (0, 1, 0), (0, 0, 1)}

cardinality()¶

Returns the cardinality of self

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : V.cardinality()
oo
simula : V2 = VectorSpace(GF(5), 3); V2
Vector Space of dimension 3 over GF(5)
simula : V2.cardinality()
125

change_field(field)¶

Change the base field of self.

Parameters: field – a field (QQ, RR, CC or GF(q))
Returns: a vector space

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : V.change_field(RR)
Vector Space of dimension 3 over Real Numbers with 53 bits of precision

contains(family)¶

Tests if family` is included in ``self.

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : V.include({(1, 1, 0), (2, 1, 0), (1/2, 0, 1)})
True

Parameters: family (Iterable) –

dim()¶

The dimension of self.

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : V.dimension()
3

dimension()¶

The dimension of self.

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : V.dimension()
3

get_a_basis()¶

Returns a basis of self.

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : V.get_a_basis()
{(1, 0, 0), (0, 1, 0), (0, 0, 1)}

get_component_in_basis(v, family=None)¶

Returns the component of v (if it exists) for the family.

Parameters

v – a vector
family – a list vectors

Returns

a tuple of scalars

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : V.get_component_in_basis((1,2,3), [(1, 1, -2), (2, 0, 1), (0, 0, 1)])
(2, -1/2, 15/2)

include(family)¶

Tests if family` is included in ``self.

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : V.include({(1, 1, 0), (2, 1, 0), (1/2, 0, 1)})
True

Parameters: family (Iterable) –

is_basis(family)¶

Tests if family` is a basis of ``self.

Parameters: family (Union[Iterable, Sized]) – a set of vectors
Returns: a boolean

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : V.is_basis({(1, 1, 1), (2, 1, 0), (3, 2, 1)})
False

is_finite()¶

Tests if self is finite.

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : V.is_finite()
False
simula : V2 = VectorSpace(GF(5), 3); V2
Vector Space of dimension 3 over GF(5)
simula : V2.is_finite()
True

is_generators(family)¶

Tests if family` generate ``self.

Parameters: family – a set of vectors
Returns: a boolean

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : V.is_generators({(0, 0, 1), (1, 1, 1), (1, 0, -1)})
True

is_infinite()¶

Tests if self is infinite.

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : V.is_infinite()
True
simula : V2 = VectorSpace(GF(5), 3); V2
Vector Space of dimension 3 over GF(5)
simula : V2.is_infinite()
False

is_subspace(domain)¶

Tests if domain is a subspace of self.

Parameters: domain – a vector space
Returns: a boolean True or False

linear_combination(family, coeffs)¶

Returns a linear combination of family by the coefficients coeffs.

Parameters

family – a list of vectors
coeffs – a list of scalars

Returns

a vector

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : V.linear_combination([(1, 1, -2), (2, 0, 1)], [-2, 3])
(4, -2, 7)

matrix_change_basis(basis1, basis2)¶

Returns subspace of self generated by family.

Parameters

basis1 (Iterable) – a basis of self
basis2 (Iterable) – a basis of self

Returns

a matrix

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : V.matrix_change_basis([(1, 0, 0), (1, -1, 0), (0, 2, 2)], [(1,0,0), (0,1,0), (0, 0, 1)])
Matrix([
[1,  1,  -1],
[0, -1,   1],
[0,  0, 1/2]])

subspace(family)¶

Returns subspace of self generated by family.

Parameters: family (Iterable) – a set of vectors
Returns: a vector space

EXAMPLES:

simula : V = VectorSpace(QQ, 3); V
Vector Space of dimension 3 over Rational Numbers
simula : V.subspace({(1, 1, -2), (2, 0, 1)})
Subspace of QQ^3 generated by the family {(1, 1, -2), (2, 0, 1)}

simula.linalg.vector_space.gramSchmidt(*args, orthonormal=False)¶: GramShmidt orthogonalisation

class simula.linalg.vector_space.vector(v, *, column=False)¶

Bases: object

Representation of a vector.

EXAMPLES:

simula : v = vector((1, -2, 3))
simula : v
(1, -2, 3)
simula : 5v
(5, -10, 15)
simula : v in QQ^3
True
simula : v.T
[ 1]
[-2]
[ 3]
simula : v * v.T
14
simula :  A = matrix([[1, 2, 0], [-1, 2, 3], [0, -1, 2]]); A
Matrix([
[ 1,  2, 0],
[-1,  2, 3],
[ 0, -1, 2]])
simula : v * A
(3, -5, 0)
simula : A * v.T
[-3]
[ 4]
[ 8]
simula : w = vector((2, 2, 5), column=True); w
[2]
[2]
[5]
simula : v - 3w.T
(-5, -8, -12)

Linear Maps¶

Operations over linear maps

class simula.linalg.linear_map.LinearMap(symbols=None, expr=None, *, domain=None, codomain=None, matrix=None, basis1=None, basis2=None, check=True)¶

Bases: simula.calculus.functions.Function

Representation of linear maps.

Parameters

symbols – a tuple of symbols
expr – an expression or a tuple of expression
domain – (optional) a field or a vector space
codomain – (optional) a field or a vector space
matrix – (optional) a matrix
basis1 – a basis of the domain
basis2 – a basis of the codomain
check – a boolean

One can define a linear map by two methods

First method : you know the expression of the linear map

simula : (x, y, z) = var("x,y,z")
simula : f = linear_map((x, y, z), (2x-y-z, x-y, -x+z)); f
Linear map from QQ^3 --> QQ^3 defined by (x, y, z) |--> (2x - y - z, x - y, -x + z)
simula : f(1, 2, -3)
(3, -1, -4)
simula : f.kernel()
Subspace of QQ^3 generated by the family {(1, 1, 1)}
simula : f.image()
Subspace of QQ^3 generated by the family {(2, 1, -1), (-1, -1, 0)}
simula : f.is_diagonalizable()
True
simula : f.eigenvals()
{1 - sqrt(2): 1, 1 + sqrt(2): 1, 0: 1}
simula : f.is_one_to_one()
False

Second methodyou know the matrix associated to the linear map in some bases (if no basis
is specified they are equal to canonical bases).

simula : A = matrix([[-1, 1, 1], [1, -1, 1], [1, 1, -1]]); A
Matrix([
[-1,  1,  1],
[ 1, -1,  1],
[ 1,  1, -1]])
simula : g = linear_map(matrix=A, domain=RR^3, codomain=RR^3); g
g = linear_map(matrix=A, domain=RR^3, codomain=RR^3); g
Linear map from RR^3 --> RR^3 defined by (x1, x2, x3) |--> (-x1 + x2 + x3, x1 - x2 + x3, x1 + x2 - x3)
simula : g(0, 1, 1)
(2, 0, 0)
simula : g.spectrum()
{1, -2}
simula : g.image()
Subspace of RR^3 generated by the family {(1, 1, -1), (1, -1, 1), (-1, 1, 1)}
simula : g.is_endomorphism()
True
simula : g.is_surjective()
True

det()¶: Returns the determinant of the linear map.

eigenvals()¶: Returns the eigen values of self.

eigenvects()¶: Returns the eigen vectors of self.

get_matrix(basis1=None, basis2=None)¶: Return the matrix associated to the linear map with respect to basis1 and basis2.

im()¶: Returns the image of self.

image()¶: Returns the image of self.

is_diagonalizable()¶: Tests if self is diagonalizable.

is_endomorphism()¶: Tests if self is an endomorphism.

is_idempotent()¶: Tests if self is idempotent.

is_injective()¶: Tests if self is an one-to-one.

is_isomorphism()¶: Tests if self is an isomorphism.

is_nilpotent()¶: Tests if self is nilpotent.

is_one_to_one()¶: Tests if self is an one-to-one.

is_surjective()¶: Tests if self is an surjective.

is_zero()¶: Tests if self is null.

ker()¶: Returns the kernel of self.

kernel()¶: Returns the kernel of self.

nullity()¶: Returns the nullity of the kernel of the linear map.

rank()¶: Returns the rank of the linear map.

spectrum()¶: Returns the spectrum of self i.e the set of eigen values of self.

trace()¶: Returns the trace of the linear map.

simula.linalg.linear_map.image(expr)¶: Image of a linear map or a matrix.

simula.linalg.linear_map.ker(expr)¶: Kernel of a linear map or a matrix.

simula.linalg.linear_map.kernel(expr)¶: Kernel of a linear map or a matrix.

simula.linalg.linear_map.linear_map¶: alias of simula.linalg.linear_map.LinearMap

simula.linalg.linear_map.linear_transformation¶: alias of simula.linalg.linear_map.LinearMap