[index]
Algebra::MatrixAlgebra / Algebra::Vector / Algebra::Covector / Algebra::SquareMatrix / Algebra::GaussianElimination
(Class of Matrices)
This class expresses matrices. For creating an actual class, use the class method ::create or the function Algebra.MatrixAlgebra(), giving the ground ring and sizes.
That has Algebra::Vector(column vectorj, Algebra::Covector(row vector), Algebra::SquareMatrix(square matrix) as subclass.
Algebra.MatrixAlgebra(ring, m, n)
::create(ring, m, n)
Creates the class of matrix of type (m, n)
with
elements of the ring ring.
The return value of this method is a subclass of
Algebra::MatrixAlgebra.
The subclass has class methods:
ground, rsize, csize and sizes,
which returns the ground ring, the size of rows( m ),
the size of columns( n ) and the array of [m, n]
respectively.
To create the actual matrix, use the class methods: ::new, ::matrix or ::[].
::new(array)
Returns the matrix of the elements designated by the array of arrays array.
Example:
M = Algebra.MatrixAlgebra(Integer, 2, 3) a = M.new([[1, 2, 3], [4, 5, 6]]) a.display #=> [1, 2, 3] #=> [4, 5, 6]
::matrix{|i, j| ... }
Returns the matrix which has the i-j
-th elements
evaluating ..., where i and j are the row
and the column indices
Example:
M = Algebra.MatrixAlgebra(Integer, 2, 3) a = M.matrix{|i, j| 10*(i + 1) + j + 1} a.display #=> [11, 12, 13] #=> [21, 22, 23]
::[array1, array2, ..., array]
Returns the matrix which has array1, array2, ..., array
as rows.
Example:
M = Algebra.MatrixAlgebra(Integer, 2, 3) a = M[[1, 2, 3], [4, 5, 6]] a.display #=> [1, 2, 3] #=> [4, 5, 6]
::collect_ij{|i, j| ... }
::collect_row{|i| ... }
Returns the matrix whose i-th row is the array obtained by evaluating ....
Example:
M = Algebra.MatrixAlgebra(Integer, 2, 3) A = M.collect_row{|i| [i*10 + 11, i*10 + 12, i*10 + 13]} A.display #=> [11, 12, 13] #=> [21, 22, 23]
::collect_column{|j| ... }
Returns the matrix whose j-th column is the array obtained by evaluating ....
Example:
M = Algebra.MatrixAlgebra(Integer, 2, 3) A = M.collect_column{|j| [11 + j, 21 + j]} A.display #=> [11, 12, 13] #=> [21, 22, 23]
::*(otype)
Returns the class of matrix multiplicated by otype.
Example:
M = Algebra.MatrixAlgebra(Integer, 2, 3) N = Algebra.MatrixAlgebra(Integer, 3, 4) L = M * N p L.sizes #=> [3, 4]
::vector_type
::covector_type
::transpose
::zero
[i, j]
(i, j)
-th component.[i, j] = x
(i, j)
-th component with x.rsize
csize
sizes
rows
Returns the array of rows.
Example:
M = Algebra.MatrixAlgebra(Integer, 2, 3) a = M.new([[1, 2, 3], [4, 5, 6]]) p a.rows #=> [[1, 2, 3], [4, 5, 6]] p a.row(1) #=> [4, 5, 6] a.set_row(1, [40, 50, 60]) a.display #=> [1, 2, 3] #=> [40, 50, 60]
row(i)
set_row(i, array)
columns
Returns the array of columns.
á:
M = Algebra.MatrixAlgebra(Integer, 2, 3) a = M.new([[1, 2, 3], [4, 5, 6]]) p a.columns #=> [[1, 4], [2, 5], [3, 6]] p a.column(1) #=> [2, 5] a.set_column(1, [20, 50]) a.display #=> [1, 20, 3] #=> [4, 50, 6]
column(j)
set_column(j, array)
each{|row| ...}
each_index{|i, j| ...}
(i, j)
.each_i{|i| ...}
i
of rows.each_j{|j| ...}
j
of columns.each_row{|r| ... }
each_column{|c| ... }
matrix{|i, j| ... }
collect_ij{|i, j| ... }
collect_row{|i| ... }
collect_column{|j| ... }
minor(i, j)
cofactor(i, j)
minor(i, j) ** (i + j)
.cofactor_matrix
self.class.transpose.matrix{|i, j| cofactor(j, i)}
.adjoint
==(other)
+(other)
-(other)
*(other)
Returns the product of self and other.
Example:
M = Algebra.MatrixAlgebra(Integer, 2, 3) N = Algebra.MatrixAlgebra(Integer, 3, 4) L = M * N a = M[[1, 2, 3], [4, 5, 6]] b = N[[-3, -2, -1, 0], [1, 2, 3, 4], [5, 6, 7, 8]] c = a * b p c.type #=> L c.display #=> [14, 20, 26, 32] #=> [23, 38, 53, 68]
**(n)
/(other)
rank
dsum(other)
Returns the direct sum of self and other.
Example:
a = Algebra.MatrixAlgebra(Integer, 2, 3)[ [1, 2, 3], [4, 5, 6] ] b = Algebra.MatrixAlgebra(Integer, 3, 2)[ [-1, -2], [-3, -4], [-5, -6] ] (a.dsum b).display #=> 1, 2, 3, 0, 0 #=> 4, 5, 6, 0, 0 #=> 0, 0, 0, -1, -2 #=> 0, 0, 0, -3, -4 #=> 0, 0, 0, -5, -6
to_ary
flatten
diag
convert_to(ring)
Returns the conversion of self to ring's object.
Example:
require "matrix-algebra" require "residue-class-ring" Z3 = Algebra.ResidueClassRing(Integer, 3) a = Algebra.MatrixAlgebra(Integer, 2, 3)[ [1, 2, 3], [4, 5, 6] ] a.convert_to(Algebra.MatrixAlgebra(Z3, 2, 3)).display #=> 1, 2, 0 #=> 1, 2, 0
transpose
Returns the transposed matrix.
Example:
M = Algebra.MatrixAlgebra(Integer, 2, 3) a = M.new([[1, 2, 3], [4, 5, 6]]) Mt = M.transpose b = a.transpose p b.type #=> Mt b.display #=> [1, 4] #=> [2, 5] #=> [3, 6]
dup
Returns the duplication of self.
Example:
M = Algebra.MatrixAlgebra(Integer, 2, 3) a = M.new([[1, 2, 3], [4, 5, 6]]) b = a.dup b[1, 1] = 50 a.display #=> [1, 2, 3] #=> [4, 5, 6] b.display #=> [1, 2, 3] #=> [4, 50, 6]
display([out])
(Class of Vector)
The class of column vectors.
none.
Algebra.Vector(ring, n)
Algebra::Vector.create(ring, n)
Creates the class of the n-th dimensional (column) vector over the ring.
The return value of this is a subclass of Algebra::Vector. This subclass has the class methods: ground and size, which returns ring and the size n respectively.
To get actual vectors, use the class methods: new, matrix or [].
Algebra::Vector is identified with
Algebra::MatrixAlgebra of type [n, 1]
.
Algebra::Vector::new(array)
Returns the vector of the array.
Example:
V = Algebra.Vector(Integer, 3) a = V.new([1, 2, 3]) a.display #=> [1] #=> [2] #=> [3]
Algebra::Vector::vector{|i| ... }
Returns the vector of ... as the i-th element.
Example:
V = Algebra.Vector(Integer, 3) a = V.vector{|j| j + 1} a.display #=> [1] #=> [2] #=> [3]
Algebra::Vector::matrix{|i, j| ... }
size
to_a
transpose
inner_product(other)
inner_product_complex(other)
inner_product(other.conjugate)
.norm2
inner_product(self)
.norm2_complex
inner_product(self.conjugate)
.(Row Vector Class)
The class of row vectors.
none.
Algebra.Covector(ring, n)
Algebra::Covector::create(ring, n)
Creates the class of the n-th dimensional (row) vector over the ring.
The return value of this is a subclass of Algebra::MatrixAlgebra::CoVector. This subclass has the class methods: ground and size, whic h returns ring and the size n respectively.
To get actual vectors, use the class methods: new, matrix or [].
Algebra::Covector is identified with
[1, n]
-type
Algebra::MatrixAlgebra.
Algebra::Covector::new(array)
Returns the row vector of the array.
Example:
V = Algebra::Covector(Integer, 3) a = V.new([1, 2, 3]) a.display #=> [1, 2, 3]
Algebra::Covector::covector{|j| ... }
Returns the vector of ... as the j-th element.
Example:
V = Algebra.Covector(Integer, 3) a = V.covector{|j| j + 1} a.display #=> [1, 2, 3]
Algebra::Covector::matrix{|i, j| ... }
size
to_a
transpose
inner_product(other)
inner_product_complex(other)
inner_product(other.conjugate)
.norm2
inner_product(self)
.norm2_complex
inner_product(self.conjugate)
.(Class of SquareMatrix)
The Ring of Square Matrices over a ring.
none.
Algebra.SquareMatrix(ring, size)
Algebra::SquareMatrix::create(ring, n)
Creates the class of square matrices.
The return value of this is the subclass of Algebra::SquareMatrix. This subclass has the class methods ground and size which returns ring and the size n respectively.
Algebra::SquareMatrix is identified
with Algebra::MatrixAlgebra::MatrixAlgebra of type
[n, n]
.
To get the actual matrices, use the class methods Algebra::SquareMatrix::new, Algebra::SquareMatrix::matrix or Algebra::SquareMatrix::[].
Algebra::SquareMatrix.determinant(aa)
Algebra::SquareMatrix.det(aa)
Algebra::SquareMatrix::unity
Algebra::SquareMatrix::zero
Algebra::SquareMatrix::const(x)
size
const(x)
determinant
inverse
/(other)
self * other.inverse
. If other is a schalar,
divides each entries by other.char_polynomial(ring)
char_matrix(ring)
_char_matrix(poly_ring_matrix)
(Module of Gaussian Elimination)
Module of the elimination method of Gauss.
gaussian-elimination.rb
none.
none.
swap_r!(i, j)
swap_r(i, j)
swap_c!(i, j)
swap_c(i, j)
multiply_r!(i, c)
multiply_r(i, c)
multiply_c!(j, c)
multiply_c(j, c)
divide_r!(i, c)
divide_r(i, c)
divide_c!(j, c)
divide_c(j, c)
mix_r!(i, j, c)
mix_r(i, j, c)
mix_c!(i, j, c)
mix_c(i, j, c)
left_eliminate!
Transform to the step matrix by the left fundamental transformation.
The return value is the array of the square matrix which used to transform, its determinant and the rank.
Example:
require "matrix-algebra" require "mathn" class Rational < Numeric def inspect; to_s; end end M = Algebra.MatrixAlgebra(Rational, 4, 3) a = M.matrix{|i, j| i*10 + j} b = a.dup c, d, e = b.left_eliminate! b.display #=> [1, 0, -1] #=> [0, 1, 2] #=> [0, 0, 0] #=> [0, 0, 0] c.display #=> [-11/10, 1/10, 0, 0] #=> [1, 0, 0, 0] #=> [1, -2, 1, 0] #=> [2, -3, 0, 1] p c*a == b#=> true p d #=> 1/10 p e #=> 2
left_inverse
left_sweep
step_matrix?
kernel_basis
Returns the array of vector( Algebra::Vector ) such that the right multiplication of it is null.
Example:
require "matrix-algebra" require "mathn" M = Algebra.MatrixAlgebra(Rational, 5, 4) a = M.matrix{|i, j| i + j} a.display #=> #[0, 1, 2, 3] #[1, 2, 3, 4] #[2, 3, 4, 5] #[3, 4, 5, 6] #[4, 5, 6, 7] a.kernel_basis.each do |v| puts "a * #{v} = #{a * v}" #=> a * [1, -2, 1, 0] = [0, 0, 0, 0, 0] #=> a * [2, -3, 0, 1] = [0, 0, 0, 0, 0] end
determinant_by_elimination