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Algebra::MatrixAlgebra / Algebra::Vector / Algebra::Covector / Algebra::SquareMatrix / Algebra::GaussianElimination

# Algebra::MatrixAlgebra

(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 vectorÅj, Algebra::Covector(row vector), Algebra::SquareMatrix(square matrix) as subclass.

## File Name:

• matrix-algebra.rb

• Object

## Included Module:

• Algebra::GaussianElimination

## Associated Function:

`Algebra.MatrixAlgebra(ring, m, n)`

Same as ::create(ring, m, n).

## Class Methods:

`::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| ... }`

Returns the array of arrays with the value ... as the j-th element of the i-th array.

`::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`

Returns the class of column-vector(Vector) which has the same size of rsize.

`::covector`

Returns the class of row-vector(CoVector) which has the same size of csize.

`::transpose`

Returns the transposed matrix

`::zero`

Returns the zero matrix.

## Methods:

`[i, j]`

Returns the `(i, j)`-th component.

`[i, j] = x`

Replaces the `(i, j)`-th component with x.

`rsize`

Same as ::rsize.

`csize`

Same as ::csize.

`sizes`

Same as ::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)`

Returns the i-th row as an array.

`set_row(i, array)`

Replaces the i-th row with array.

`columns`

Returns the array of columns.

ó·:

```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)`

Returns the j-th column as an array.

`set_column(j, array)`

Replaces the i-th column with array.

`each{|row| ...}`

Iterates with row.

`each_index{|i, j| ...}`

Iterates with indices ` (i, j) `.

`each_i{|i| ...}`

Iterates with the index `i` of rows.

`each_j{|j| ...}`

Iterates with the index `j` of columns.

`each_row{|r| ... }`

Iterates with the row r. Same as each.

`each_column{|c| ... }`

Iterates with the column c.

`matrix{|i, j| ... }`

Same as ::matrix.

`collect_ij{|i, j| ... }`

Same as ::collect_ij.

`collect_row{|i| ... }`

Same as ::collect_row.

`collect_column{|j| ... }`

Same as ::collect_column.

`==(other)`

Returns true if self is equal to other.

`+(other)`

Returns the sum of self and other.

`-(other)`

Returns the difference of self from 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)`

Returns the n-th power of self.

`/(other)`

Returns the quotient self by other.

`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
```
`diag`

Returns the array of the diagonal compotents.

`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])`

Displays self to out. If out is omitted, out is \$stdout.

# Algebra::Vector

(Class of Vector)

The class of column vectors.

## SuperClass:

• MatrixAlgebra

none.

## Associated Functions:

`Algebra.Vector(ring, n)`

Same as Algebra::Matrix::Vector.create(ring, n).

## Class Methods:

`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
#=> 
#=> 
#=> 
```
`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
#=> 
#=> 
#=> 
```
`Algebra::Vector::matrix{|i, j| ... }`

Returns the vector of ... as the i-th element. j is always 0.

## Methods

`size`

Returns the dimension.

`to_a`

Returns the array of elements.

`transpose`

Transpose to the row vector Algebra::Covector.

# Algebra::Covector

(Row Vector Class)

The class of row vectors.

## SuperClass:

• MatrixAlgebra

none.

## Associated Functions:

`Algebra.Covector(ring, n)`

Same as Algebra::Covector::create(ring, n).

## Class Methods:

`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| ... }`

Returns the vector of ... as the j-th element. i is always 0.

## Methods:

`size`

Returns the dimension.

`to_a`

Returns the array of elements.

`transpose`

Transpose to the column vector Algebra::Vector.

# Algebra::SquareMatrix

(Class of SquareMatrix)

The Ring of Square Matrices over a ring.

## SuperClass:

• Algebra::MatrixAlgebra

none.

## Associated Functions:

`Algebra.SquareMatrix(ring, size)`

Same as Algebra::SquareMatrix.create(ring, n).

## Class Methods:

`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::unity`

Returns the unity.

`Algebra::SquareMatrix::zero`

Returns the zero.

`Algebra::SquareMatrix::const(x)`

Returns the scalar matrix with by the diagonal components x.

## Methods:

`size`

Returns the dimension.

`const(x)`

Returns the scalar matrix with the diagonal components x.

`determinant`

Returns the determinant.

`char_polynomial(ring)`

Returns the characteristic polynomial over ring.

# Algebra::GaussianElimination

(Module of Gaussian Elimination)

Module of the elimination method of Gauss.

## FileName:

gaussian-elimination.rb

none.

none.

## Methods:

`swap_r!(i, j)`

Swaps i-th row and j-th row.

`swap_r(i, j)`

Returns the new matrix with i-th row and j-th row swapped.

`swap_c!(i, j)`

Swaps i-th column and j-th column.

`swap_c(i, j)`

Returns the new matrix with i-th column and j-th column swapped.

`multiply_r!(i, c)`

Multiplys the i-th row by c.

`multiply_r(i, c)`

Returns the new Matrix with the i-th row multiplied by c.

`multiply_c!(j, c)`

Multiplys the j-th column by c.

`multiply_c(j, c)`

Returns the new Matrix with the j-th column multiplied by c.

`divide_r!(i, c)`

Divides the i-th row by c.

`divide_r(i, c)`

Returns the new Matrix with the i-th row divided by c.

`divide_c!(j, c)`

Divides the j-th column by c.

`divide_c(j, c)`

Returns the new Matrix with the j-th column divided by c.

`mix_r!(i, j, c)`

Adds the j-th row multiplied by c to the i-th row.

`mix_r(i, j, c)`

Returns the new matrix such that the j-th row multiplied by c is added to the i-th row.

`mix_c!(i, j, c)`

Adds the j-th column multiplied by c to the i-th column.

`mix_c(i, j, c)`

Returns the new matrix such that the j-th column multiplied by c is added to the i-th column.

`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 and its determinant.

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 = 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
```
`left_inverse`

The general inverse matrix obtained by the left fundamental transformation.

`left_sweep`

Returns the step matrix by the left fundamental transformation.

`step_matrix?`

Returns the array of pivots if self is a step matrix, otherwise returns nil.

`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`

Calculate the determinant by elimination.