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Class: Matrix

Relationships & Source Files
Namespace Children
Classes:
EigenvalueDecomposition, LUPDecomposition
Super Chains via Extension / Inclusion / Inheritance
Instance Chain:
self, Enumerable
Inherits: Object
Defined in: lib/matrix.rb,
lib/matrix/eigenvalue_decomposition.rb,
lib/matrix/lup_decomposition.rb

Overview

The Matrix class represents a mathematical matrix. It provides methods for creating matrices, operating on them arithmetically and algebraically, and determining their mathematical properties (trace, rank, inverse, determinant).

Method Catalogue

To create a matrix:

  • Matrix

  • Matrix.[](*rows)

  • Matrix.rows(rows, copy = true)

  • Matrix.columns(columns)

  • Matrix.build(row_count, column_count, &block)

  • Matrix.diagonal(*values)

  • Matrix.scalar(n, value)

  • Matrix.identity(n)

  • Matrix.unit(n)

  • Matrix.I(n)

  • Matrix.zero(n)

  • Matrix.row_vector(row)

  • Matrix.column_vector(column)

  • Matrix.hstack(*matrices)

  • Matrix.vstack(*matrices)

To access Matrix elements/columns/rows/submatrices/properties:

Properties of a matrix:

Matrix arithmetic:

Matrix functions:

Matrix decompositions:

Complex arithmetic:

  • conj

  • conjugate

  • imag

  • imaginary

  • real

  • rect

  • rectangular

Conversion to other data types:

String representations:

Constant Summary

  • SELECTORS =
    # File 'lib/matrix.rb', line 573 
    {all: true, diagonal: true, off_diagonal: true, lower: true, strict_lower: true, strict_upper: true, upper: true}.freeze

Class Method Summary

Instance Attribute Summary

Instance Method Summary

Constructor Details

.new(rows, column_count = rows[0].size) ⇒ Matrix

new is private; use .rows, columns, [], etc… to create.

[ GitHub ] 

  


# File 'lib/matrix.rb', line 355



def initialize(rows, column_count = rows[0].size)
  # No checking is done at this point. rows must be an Array of Arrays.
  # column_count must be the size of the first row, if there is one,
  # otherwise it *must* be specified and can be any integer >= 0
  @rows = rows
  @column_count = column_count
end


  







Class Method Details



  

    .[](*rows)  
  



  

    

Creates a matrix where each argument is a row.



Matrix[ [25, 93], [-1, 66] ]
   #=>  25 93
       -1 66


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 151



def Matrix.[](*rows)
  rows(rows, false)
end


  








  

    .build(row_count, column_count = row_count)  
  



  

    

Creates a matrix of size #row_count x #column_count. It fills the values by calling the given block, passing the current row and column. Returns an enumerator if no block is given.



m = Matrix.build(2, 4) {|row, col| col - row }
  #=> Matrix[[0, 1, 2, 3], [-1, 0, 1, 2]]
m = Matrix.build(3) { rand }
  #=> a 3x3 matrix with random elements


  





  
Raises:


    
  
  • 
    (ArgumentError)
  
    • 





  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 196



def Matrix.build(row_count, column_count = row_count)
  row_count = CoercionHelper.coerce_to_int(row_count)
  column_count = CoercionHelper.coerce_to_int(column_count)
  raise ArgumentError if row_count < 0 || column_count < 0
  return to_enum :build, row_count, column_count unless block_given?
  rows = Array.new(row_count) do |i|
    Array.new(column_count) do |j|
      yield i, j
    end
  end
  new rows, column_count
end


  








  

    .column_vector(column)  
  



  

    

Creates a single-column matrix where the values of that column are as given in #column.



Matrix.column_vector([4,5,6])
  #=> 4
     5
     6


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 282



def Matrix.column_vector(column)
  column = convert_to_array(column)
  new [column].transpose, 1
end


  








  

    .columns(columns)  
  



  

    

Creates a matrix using columns as an array of column vectors.



Matrix.columns([[25, 93], [-1, 66]])
   #=>  25 -1
       93 66


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 181



def Matrix.columns(columns)
  rows(columns, false).transpose
end


  








  

    .diagonal(*values)  
  



  

    

Creates a matrix where the diagonal elements are composed of values.



Matrix.diagonal(9, 5, -3)
  #=>  9  0  0
      0  5  0
      0  0 -3


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 216



def Matrix.diagonal(*values)
  size = values.size
  return Matrix.empty if size == 0
  rows = Array.new(size) {|j|
    row = Array.new(size, 0)
    row[j] = values[j]
    row
  }
  new rows
end


  








  

    .empty(row_count = 0, column_count = 0)  
  



  

    

Creates a empty matrix of #row_count x #column_count. At least one of #row_count or #column_count must be 0.



m = Matrix.empty(2, 0)
m == Matrix[ [], [] ]
  #=> true
n = Matrix.empty(0, 3)
n == Matrix.columns([ [], [], [] ])
  #=> true
m * n
  #=> Matrix[[0, 0, 0], [0, 0, 0]]


  





  
Raises:


    
  
  • 
    (ArgumentError)
  
    • 





  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 300



def Matrix.empty(row_count = 0, column_count = 0)
  raise ArgumentError, "One size must be 0" if column_count != 0 && row_count != 0
  raise ArgumentError, "Negative size" if column_count < 0 || row_count < 0

  new([[]]*row_count, column_count)
end


  








  

    .hstack(x, *matrices)  
  



  

    

Create a matrix by stacking matrices horizontally



x = Matrix[[1, 2], [3, 4]]
y = Matrix[[5, 6], [7, 8]]
Matrix.hstack(x, y) # => Matrix[[1, 2, 5, 6], [3, 4, 7, 8]]


  





  
Raises:


    
  
  • 
    (TypeError)
  
    • 





  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 335



def Matrix.hstack(x, *matrices)
  raise TypeError, "Expected a Matrix, got a #{x.class}" unless x.is_a?(Matrix)
  result = x.send(:rows).map(&:dup)
  total_column_count = x.column_count
  matrices.each do |m|
    raise TypeError, "Expected a Matrix, got a #{m.class}" unless m.is_a?(Matrix)
    if m.row_count != x.row_count
      raise ErrDimensionMismatch, "The given matrices must have #{x.row_count} rows, but one has #{m.row_count}"
    end
    result.each_with_index do |row, i|
      row.concat m.send(:rows)[i]
    end
    total_column_count += m.column_count
  end
  new result, total_column_count
end


  








  

    I(n)  
  



  

    

Alias for .identity.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 249



alias I identity


  








  

    .identity(n)  
    Also known as: .unit, .I
  



  

    

Creates an n by n identity matrix.



Matrix.identity(2)
  #=> 1 0
     0 1


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 244



def Matrix.identity(n)
  scalar(n, 1)
end


  








  

    .row_vector(row)  
  



  

    

Creates a single-row matrix where the values of that row are as given in #row.



Matrix.row_vector([4,5,6])
  #=> 4 5 6


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 269



def Matrix.row_vector(row)
  row = convert_to_array(row)
  new [row]
end


  








  

    .rows(rows, copy = true)  
  



  

    

Creates a matrix where rows is an array of arrays, each of which is a row of the matrix.  If the optional argument copy is false, use the given arrays as the internal structure of the matrix without copying.



Matrix.rows([[25, 93], [-1, 66]])
   #=>  25 93
       -1 66


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 163



def Matrix.rows(rows, copy = true)
  rows = convert_to_array(rows, copy)
  rows.map! do |row|
    convert_to_array(row, copy)
  end
  size = (rows[0] || []).size
  rows.each do |row|
    raise ErrDimensionMismatch, "row size differs (#{row.size} should be #{size})" unless row.size == size
  end
  new rows, size
end


  








  

    .scalar(n, value)  
  



  

    

Creates an n by n diagonal matrix where each diagonal element is value.



Matrix.scalar(2, 5)
  #=> 5 0
     0 5


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 234



def Matrix.scalar(n, value)
  diagonal(*Array.new(n, value))
end


  








  

    .unit(n)  
  



  

    

Alias for .identity.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 248



alias unit identity


  








  

    .vstack(x, *matrices)  
  



  

    

Create a matrix by stacking matrices vertically



x = Matrix[[1, 2], [3, 4]]
y = Matrix[[5, 6], [7, 8]]
Matrix.vstack(x, y) # => Matrix[[1, 2], [3, 4], [5, 6], [7, 8]]


  





  
Raises:


    
  
  • 
    (TypeError)
  
    • 





  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 314



def Matrix.vstack(x, *matrices)
  raise TypeError, "Expected a Matrix, got a #{x.class}" unless x.is_a?(Matrix)
  result = x.send(:rows).map(&:dup)
  matrices.each do |m|
    raise TypeError, "Expected a Matrix, got a #{m.class}" unless m.is_a?(Matrix)
    if m.column_count != x.column_count
      raise ErrDimensionMismatch, "The given matrices must have #{x.column_count} columns, but one has #{m.column_count}"
    end
    result.concat(m.send(:rows))
  end
  new result, x.column_count
end


  








  

    .zero(row_count, column_count = row_count)  
  



  

    

Creates a zero matrix.



Matrix.zero(2)
  #=> 0 0
     0 0


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 258



def Matrix.zero(row_count, column_count = row_count)
  rows = Array.new(row_count){Array.new(column_count, 0)}
  new rows, column_count
end


  







Instance Attribute Details



  

    #column_count   (readonly)
    Also known as: #column_size
  



  

    

Returns the number of columns.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 395



attr_reader :column_count


  








  

    #column_size   (readonly)
  



  

    

Alias for #column_count.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 396



alias_method :column_size, :column_count


  








  

    #diagonal?  ⇒ Boolean  (readonly)
  



  

    

Returns true if this is a diagonal matrix. Raises an error if matrix is not square.


  





  


  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 746



def diagonal?
  Matrix.Raise ErrDimensionMismatch unless square?
  each(:off_diagonal).all?(&:zero?)
end


  








  

    #empty?  ⇒ Boolean  (readonly)
  



  

    

Returns true if this is an empty matrix, i.e. if the number of rows or the number of columns is 0.


  





  


  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 755



def empty?
  column_count == 0 || row_count == 0
end


  








  

    #hermitian?  ⇒ Boolean  (readonly)
  



  

    

Returns true if this is an hermitian matrix. Raises an error if matrix is not square.


  





  


  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 763



def hermitian?
  Matrix.Raise ErrDimensionMismatch unless square?
  each_with_index(:upper).all? do |e, row, col|
    e == rows[col][row].conj
  end
end


  








  

    #lower_triangular?  ⇒ Boolean  (readonly)
  



  

    

Returns true if this is a lower triangular matrix.


  





  


  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 773



def lower_triangular?
  each(:strict_upper).all?(&:zero?)
end


  








  

    #normal?  ⇒ Boolean  (readonly)
  



  

    

Returns true if this is a normal matrix. Raises an error if matrix is not square.


  





  


  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 781



def normal?
  Matrix.Raise ErrDimensionMismatch unless square?
  rows.each_with_index do |row_i, i|
    rows.each_with_index do |row_j, j|
      s = 0
      rows.each_with_index do |row_k, k|
        s += row_i[k] * row_j[k].conj - row_k[i].conj * row_k[j]
      end
      return false unless s == 0
    end
  end
  true
end


  








  

    #orthogonal?  ⇒ Boolean  (readonly)
  



  

    

Returns true if this is an orthogonal matrix Raises an error if matrix is not square.


  





  


  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 799



def orthogonal?
  Matrix.Raise ErrDimensionMismatch unless square?
  rows.each_with_index do |row, i|
    column_count.times do |j|
      s = 0
      row_count.times do |k|
        s += row[k] * rows[k][j]
      end
      return false unless s == (i == j ? 1 : 0)
    end
  end
  true
end


  








  

    #permutation?  ⇒ Boolean  (readonly)
  



  

    

Returns true if this is a permutation matrix Raises an error if matrix is not square.


  





  


  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 817



def permutation?
  Matrix.Raise ErrDimensionMismatch unless square?
  cols = Array.new(column_count)
  rows.each_with_index do |row, i|
    found = false
    row.each_with_index do |e, j|
      if e == 1
        return false if found || cols[j]
        found = cols[j] = true
      elsif e != 0
        return false
      end
    end
    return false unless found
  end
  true
end


  








  

    #real?  ⇒ Boolean  (readonly)
  



  

    

Returns true if all entries of the matrix are real.


  





  


  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 838



def real?
  all?(&:real?)
end


  








  

    #regular?  ⇒ Boolean  (readonly)
  



  

    

Returns true if this is a regular (i.e. non-singular) matrix.


  





  


  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 845



def regular?
  not singular?
end


  








  

    #rows   (readonly, protected)
  

  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 142



attr_reader :rows


  








  

    #singular?  ⇒ Boolean  (readonly)
  



  

    

Returns true if this is a singular matrix.


  





  


  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 852



def singular?
  determinant == 0
end


  








  

    #square?  ⇒ Boolean  (readonly)
  



  

    

Returns true if this is a square matrix.


  





  


  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 859



def square?
  column_count == row_count
end


  








  

    #symmetric?  ⇒ Boolean  (readonly)
  



  

    

Returns true if this is a symmetric matrix. Raises an error if matrix is not square.


  





  


  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 867



def symmetric?
  Matrix.Raise ErrDimensionMismatch unless square?
  each_with_index(:strict_upper) do |e, row, col|
    return false if e != rows[col][row]
  end
  true
end


  








  

    #unitary?  ⇒ Boolean  (readonly)
  



  

    

Returns true if this is a unitary matrix Raises an error if matrix is not square.


  





  


  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 879



def unitary?
  Matrix.Raise ErrDimensionMismatch unless square?
  rows.each_with_index do |row, i|
    column_count.times do |j|
      s = 0
      row_count.times do |k|
        s += row[k].conj * rows[k][j]
      end
      return false unless s == (i == j ? 1 : 0)
    end
  end
  true
end


  








  

    #upper_triangular?  ⇒ Boolean  (readonly)
  



  

    

Returns true if this is an upper triangular matrix.


  





  


  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 896



def upper_triangular?
  each(:strict_lower).all?(&:zero?)
end


  








  

    #zero?  ⇒ Boolean  (readonly)
  



  

    

Returns true if this is a matrix with only zero elements


  





  


  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 903



def zero?
  all?(&:zero?)
end


  







Instance Method Details



  

    #*(m)  
  



  

    

Matrix multiplication.



Matrix[[2,4], [6,8]] * Matrix.identity(2)
  #=> 2 4
     6 8


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 952



def *(m) # m is matrix or vector or number
  case(m)
  when Numeric
    rows = @rows.collect {|row|
      row.collect {|e| e * m }
    }
    return new_matrix rows, column_count
  when Vector
    m = self.class.column_vector(m)
    r = self * m
    return r.column(0)
  when Matrix
    Matrix.Raise ErrDimensionMismatch if column_count != m.row_count

    rows = Array.new(row_count) {|i|
      Array.new(m.column_count) {|j|
        (0 ... column_count).inject(0) do |vij, k|
          vij + self[i, k] * m[k, j]
        end
      }
    }
    return new_matrix rows, m.column_count
  else
    return apply_through_coercion(m, __method__)
  end
end


  








  

    #**(other)  
  



  

    

Matrix exponentiation. Equivalent to multiplying the matrix by itself N times. Non integer exponents will be handled by diagonalizing the matrix.



Matrix[[7,6], [3,9]] ** 2
  #=> 67 96
     48 99


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1119



def ** (other)
  case other
  when Integer
    x = self
    if other <= 0
      x = self.inverse
      return self.class.identity(self.column_count) if other == 0
      other = -other
    end
    z = nil
    loop do
      z = z ? z * x : x if other[0] == 1
      return z if (other >>= 1).zero?
      x *= x
    end
  when Numeric
    v, d, v_inv = eigensystem
    v * self.class.diagonal(*d.each(:diagonal).map{|e| e ** other}) * v_inv
  else
    Matrix.Raise ErrOperationNotDefined, "**", self.class, other.class
  end
end


  








  

    #+(m)  
  



  

    

Matrix addition.



Matrix.scalar(2,5) + Matrix[[1,0], [-4,7]]
  #=>  6  0
     -4 12


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 985



def +(m)
  case m
  when Numeric
    Matrix.Raise ErrOperationNotDefined, "+", self.class, m.class
  when Vector
    m = self.class.column_vector(m)
  when Matrix
  else
    return apply_through_coercion(m, __method__)
  end

  Matrix.Raise ErrDimensionMismatch unless row_count == m.row_count && column_count == m.column_count

  rows = Array.new(row_count) {|i|
    Array.new(column_count) {|j|
      self[i, j] + m[i, j]
    }
  }
  new_matrix rows, column_count
end


  








  

    #+@  
  

  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1142



def +@
  self
end


  








  

    #-(m)  
  



  

    

Matrix subtraction.



Matrix[[1,5], [4,2]] - Matrix[[9,3], [-4,1]]
  #=> -8  2
      8  1


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1012



def -(m)
  case m
  when Numeric
    Matrix.Raise ErrOperationNotDefined, "-", self.class, m.class
  when Vector
    m = self.class.column_vector(m)
  when Matrix
  else
    return apply_through_coercion(m, __method__)
  end

  Matrix.Raise ErrDimensionMismatch unless row_count == m.row_count && column_count == m.column_count

  rows = Array.new(row_count) {|i|
    Array.new(column_count) {|j|
      self[i, j] - m[i, j]
    }
  }
  new_matrix rows, column_count
end


  








  

    #-@  
  

  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1146



def -@
  collect {|e| -e }
end


  








  

    #/(other)  
  



  

    

Matrix division (multiplication by the inverse).



Matrix[[7,6], [3,9]] / Matrix[[2,9], [3,1]]
  #=> -7  1
     -3 -6


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1039



def /(other)
  case other
  when Numeric
    rows = @rows.collect {|row|
      row.collect {|e| e / other }
    }
    return new_matrix rows, column_count
  when Matrix
    return self * other.inverse
  else
    return apply_through_coercion(other, __method__)
  end
end


  








  

    #==(other)  
  



  

    

Returns true if and only if the two matrices contain equal elements.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 914



def ==(other)
  return false unless Matrix === other &&
                      column_count == other.column_count # necessary for empty matrices
  rows == other.rows
end


  








  

    #[](i, j)  
    Also known as: #element, #component
  



  

    

Returns element (i,j) of the matrix.  That is: row i, column j.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 371



def [](i, j)
  @rows.fetch(i){return nil}[j]
end


  








  

    #[]=(i, j, v)  
    Also known as: #set_element, #set_component
  

  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 377



def []=(i, j, v)
  @rows[i][j] = v
end


  








  

    #adjugate  
  



  

    

Returns the adjugate of the matrix.



Matrix[ [7,6],[3,9] ].adjugate
  #=> 9 -6
     -3 7


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 700



def adjugate
  Matrix.Raise ErrDimensionMismatch unless square?
  Matrix.build(row_count, column_count) do |row, column|
    cofactor(column, row)
  end
end


  








  

    #clone  
  



  

    

Returns a clone of the matrix, so that the contents of each do not reference identical objects. There should be no good reason to do this since Matrices are immutable.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 931



def clone
  new_matrix @rows.map(&:dup), column_count
end


  








  

    #coerce(other)  
  



  

    

The coerce method provides support for Ruby type coercion. This coercion mechanism is used by Ruby to handle mixed-type numeric operations: it is intended to find a compatible common type between the two operands of the operator. See also Numeric#coerce.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1456



def coerce(other)
  case other
  when Numeric
    return Scalar.new(other), self
  else
    raise TypeError, "#{self.class} can't be coerced into #{other.class}"
  end
end


  








  

    #cofactor(row, column)  
  



  

    

Returns the (row, column) cofactor which is obtained by multiplying the first minor by (-1)**(row + column).



Matrix.diagonal(9, 5, -3, 4).cofactor(1, 1)
  #=> -108


  





  
Raises:


    
  
  • 
    (RuntimeError)
  
    • 





  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 685



def cofactor(row, column)
  raise RuntimeError, "cofactor of empty matrix is not defined" if empty?
  Matrix.Raise ErrDimensionMismatch unless square?

  det_of_minor = first_minor(row, column).determinant
  det_of_minor * (-1) ** (row + column)
end


  








  

    #cofactor_expansion(row: nil, column: nil)  
  



  

    

Alias for #laplace_expansion.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 735



alias_method :cofactor_expansion, :laplace_expansion


  








  

    #collect(&block)  
    Also known as: #map
  



  

    

Returns a matrix that is the result of iteration of the given block over all elements of the matrix.



Matrix[ [1,2], [3,4] ].collect { |e| e**2 }
  #=> 1  4
     9 16


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 439



def collect(&block) # :yield: e
  return to_enum(:collect) unless block_given?
  rows = @rows.collect{|row| row.collect(&block)}
  new_matrix rows, column_count
end


  








  

    #column(j)  
  



  

    

Returns column vector number j of the matrix as a ::Vector (starting at 0 like an array).  When a block is given, the elements of that vector are iterated.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 416



def column(j) # :yield: e
  if block_given?
    return self if j >= column_count || j < -column_count
    row_count.times do |i|
      yield @rows[i][j]
    end
    self
  else
    return nil if j >= column_count || j < -column_count
    col = Array.new(row_count) {|i|
      @rows[i][j]
    }
    Vector.elements(col, false)
  end
end


  








  

    #column_vectors  
  



  

    

Returns an array of the column vectors of the matrix.  See Vector.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1477



def column_vectors
  Array.new(column_count) {|i|
    column(i)
  }
end


  








  

    #component(i, j)  
  



  

    

Alias for #[].


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 375



alias component []


  








  

    #conj  
  



  

    

Alias for #conjugate.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1405



alias conj conjugate


  








  

    #conjugate  
    Also known as: #conj
  



  

    

Returns the conjugate of the matrix.



Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]]
  #=> 1+2i   i  0
        1   2  3
Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]].conjugate
  #=> 1-2i  -i  0
        1   2  3


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1402



def conjugate
  collect(&:conjugate)
end


  








  

    #det  
  



  

    

Alias for #determinant.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1202



alias_method :det, :determinant


  








  

    #det_e  
  



  

    

Alias for #determinant_e.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1250



alias det_e determinant_e


  








  

    #determinant  
    Also known as: #det
  



  

    

Returns the determinant of the matrix.



Beware that using Float values can yield erroneous results because of their lack of precision. Consider using exact types like Rational or BigDecimal instead.



Matrix[[7,6], [3,9]].determinant
  #=> 45


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1164



def determinant
  Matrix.Raise ErrDimensionMismatch unless square?
  m = @rows
  case row_count
    # Up to 4x4, give result using Laplacian expansion by minors.
    # This will typically be faster, as well as giving good results
    # in case of Floats
  when 0
    +1
  when 1
    + m[0][0]
  when 2
    + m[0][0] * m[1][1] - m[0][1] * m[1][0]
  when 3
    m0, m1, m2 = m
    + m0[0] * m1[1] * m2[2] - m0[0] * m1[2] * m2[1] \
    - m0[1] * m1[0] * m2[2] + m0[1] * m1[2] * m2[0] \
    + m0[2] * m1[0] * m2[1] - m0[2] * m1[1] * m2[0]
  when 4
    m0, m1, m2, m3 = m
    + m0[0] * m1[1] * m2[2] * m3[3] - m0[0] * m1[1] * m2[3] * m3[2] \
    - m0[0] * m1[2] * m2[1] * m3[3] + m0[0] * m1[2] * m2[3] * m3[1] \
    + m0[0] * m1[3] * m2[1] * m3[2] - m0[0] * m1[3] * m2[2] * m3[1] \
    - m0[1] * m1[0] * m2[2] * m3[3] + m0[1] * m1[0] * m2[3] * m3[2] \
    + m0[1] * m1[2] * m2[0] * m3[3] - m0[1] * m1[2] * m2[3] * m3[0] \
    - m0[1] * m1[3] * m2[0] * m3[2] + m0[1] * m1[3] * m2[2] * m3[0] \
    + m0[2] * m1[0] * m2[1] * m3[3] - m0[2] * m1[0] * m2[3] * m3[1] \
    - m0[2] * m1[1] * m2[0] * m3[3] + m0[2] * m1[1] * m2[3] * m3[0] \
    + m0[2] * m1[3] * m2[0] * m3[1] - m0[2] * m1[3] * m2[1] * m3[0] \
    - m0[3] * m1[0] * m2[1] * m3[2] + m0[3] * m1[0] * m2[2] * m3[1] \
    + m0[3] * m1[1] * m2[0] * m3[2] - m0[3] * m1[1] * m2[2] * m3[0] \
    - m0[3] * m1[2] * m2[0] * m3[1] + m0[3] * m1[2] * m2[1] * m3[0]
  else
    # For bigger matrices, use an efficient and general algorithm.
    # Currently, we use the Gauss-Bareiss algorithm
    determinant_bareiss
  end
end


  








  

    #determinant_bareiss   (private)
  



  

    

Private. Use #determinant



Returns the determinant of the matrix, using Bareiss' multistep integer-preserving gaussian elimination. It has the same computational cost order O(n^3) as standard Gaussian elimination. Intermediate results are fraction free and of lower complexity. A matrix of Integers will have thus intermediate results that are also Integers, with smaller bignums (if any), while a matrix of Float will usually have intermediate results with better precision.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1215



def determinant_bareiss
  size = row_count
  last = size - 1
  a = to_a
  no_pivot = Proc.new{ return 0 }
  sign = +1
  pivot = 1
  size.times do |k|
    previous_pivot = pivot
    if (pivot = a[k][k]) == 0
      switch = (k+1 ... size).find(no_pivot) {|row|
        a[row][k] != 0
      }
      a[switch], a[k] = a[k], a[switch]
      pivot = a[k][k]
      sign = -sign
    end
    (k+1).upto(last) do |i|
      ai = a[i]
      (k+1).upto(last) do |j|
        ai[j] =  (pivot * ai[j] - ai[k] * a[k][j]) / previous_pivot
      end
    end
  end
  sign * pivot
end


  








  

    #determinant_e  
    Also known as: #det_e
  



  

    

deprecated; use #determinant


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1246



def determinant_e
  warn "#{caller(1)[0]}: warning: Matrix#determinant_e is deprecated; use #determinant"
  determinant
end


  








  

    #each(which = :all)  
  



  

    

Yields all elements of the matrix, starting with those of the first row, or returns an Enumerator if no block given. Elements can be restricted by passing an argument:


  • 

    :all (default): yields all elements
    

  • 

    :diagonal: yields only elements on the diagonal
    

  • 

    :off_diagonal: yields all elements except on the diagonal
    

  • 

    :lower: yields only elements on or below the diagonal
    

  • 

    :strict_lower: yields only elements below the diagonal
    

  • 

    :strict_upper: yields only elements above the diagonal
    

  • 

    :upper: yields only elements on or above the diagonal
    


    Matrix[ [1,2], [3,4] ].each { |e| puts e }
    


    # => prints the numbers 1 to 4
    


    Matrix[ [1,2], [3,4] ].each(:strict_lower).to_a # => [3]
    



  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 462



def each(which = :all) # :yield: e
  return to_enum :each, which unless block_given?
  last = column_count - 1
  case which
  when :all
    block = Proc.new
    @rows.each do |row|
      row.each(&block)
    end
  when :diagonal
    @rows.each_with_index do |row, row_index|
      yield row.fetch(row_index){return self}
    end
  when :off_diagonal
    @rows.each_with_index do |row, row_index|
      column_count.times do |col_index|
        yield row[col_index] unless row_index == col_index
      end
    end
  when :lower
    @rows.each_with_index do |row, row_index|
      0.upto([row_index, last].min) do |col_index|
        yield row[col_index]
      end
    end
  when :strict_lower
    @rows.each_with_index do |row, row_index|
      [row_index, column_count].min.times do |col_index|
        yield row[col_index]
      end
    end
  when :strict_upper
    @rows.each_with_index do |row, row_index|
      (row_index+1).upto(last) do |col_index|
        yield row[col_index]
      end
    end
  when :upper
    @rows.each_with_index do |row, row_index|
      row_index.upto(last) do |col_index|
        yield row[col_index]
      end
    end
  else
    raise ArgumentError, "expected #{which.inspect} to be one of :all, :diagonal, :off_diagonal, :lower, :strict_lower, :strict_upper or :upper"
  end
  self
end


  








  

    #each_with_index(which = :all)  
  



  

    

Same as #each, but the row index and column index in addition to the element



Matrix[ [1,2], [3,4] ].each_with_index do |e, row, col|
  puts "#{e} at #{row}, #{col}"
end
  # => Prints:
  #    1 at 0, 0
  #    2 at 0, 1
  #    3 at 1, 0
  #    4 at 1, 1


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 523



def each_with_index(which = :all) # :yield: e, row, column
  return to_enum :each_with_index, which unless block_given?
  last = column_count - 1
  case which
  when :all
    @rows.each_with_index do |row, row_index|
      row.each_with_index do |e, col_index|
        yield e, row_index, col_index
      end
    end
  when :diagonal
    @rows.each_with_index do |row, row_index|
      yield row.fetch(row_index){return self}, row_index, row_index
    end
  when :off_diagonal
    @rows.each_with_index do |row, row_index|
      column_count.times do |col_index|
        yield row[col_index], row_index, col_index unless row_index == col_index
      end
    end
  when :lower
    @rows.each_with_index do |row, row_index|
      0.upto([row_index, last].min) do |col_index|
        yield row[col_index], row_index, col_index
      end
    end
  when :strict_lower
    @rows.each_with_index do |row, row_index|
      [row_index, column_count].min.times do |col_index|
        yield row[col_index], row_index, col_index
      end
    end
  when :strict_upper
    @rows.each_with_index do |row, row_index|
      (row_index+1).upto(last) do |col_index|
        yield row[col_index], row_index, col_index
      end
    end
  when :upper
    @rows.each_with_index do |row, row_index|
      row_index.upto(last) do |col_index|
        yield row[col_index], row_index, col_index
      end
    end
  else
    raise ArgumentError, "expected #{which.inspect} to be one of :all, :diagonal, :off_diagonal, :lower, :strict_lower, :strict_upper or :upper"
  end
  self
end


  








  

    #eigen  
  



  

    

Alias for #eigensystem.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1372



alias eigen eigensystem


  








  

    #eigensystem  
    Also known as: #eigen
  



  

    

Returns the Eigensystem of the matrix; see ::Matrix::EigenvalueDecomposition.



m = Matrix[[1, 2], [3, 4]]
v, d, v_inv = m.eigensystem
d.diagonal? # => true
v.inv == v_inv # => true
(v * d * v_inv).round(5) == m # => true


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1369



def eigensystem
  EigenvalueDecomposition.new(self)
end


  








  

    #element(i, j)  
  



  

    

Alias for #[].


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 374



alias element []


  








  

    #elements_to_f  
  

  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1490



def elements_to_f
  warn "#{caller(1)[0]}: warning: Matrix#elements_to_f is deprecated, use map(&:to_f)"
  map(&:to_f)
end


  








  

    #elements_to_i  
  

  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1495



def elements_to_i
  warn "#{caller(1)[0]}: warning: Matrix#elements_to_i is deprecated, use map(&:to_i)"
  map(&:to_i)
end


  








  

    #elements_to_r  
  

  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1500



def elements_to_r
  warn "#{caller(1)[0]}: warning: Matrix#elements_to_r is deprecated, use map(&:to_r)"
  map(&:to_r)
end


  








  

    #eql?(other)  ⇒ Boolean 
  



  

    
  





  


  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 920



def eql?(other)
  return false unless Matrix === other &&
                      column_count == other.column_count # necessary for empty matrices
  rows.eql? other.rows
end


  








  

    #find_index(*args)  
  



  

    

Alias for #index.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 602



alias_method :find_index, :index


  








  

    #first_minor(row, column)  
  



  

    

Returns the submatrix obtained by deleting the specified row and column.



Matrix.diagonal(9, 5, -3, 4).first_minor(1, 2)
  #=> 9 0 0
     0 0 0
     0 0 4


  





  
Raises:


    
  
  • 
    (RuntimeError)
  
    • 





  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 658



def first_minor(row, column)
  raise RuntimeError, "first_minor of empty matrix is not defined" if empty?

  unless 0 <= row && row < row_count
    raise ArgumentError, "invalid row (#{row.inspect} for 0..#{row_count - 1})"
  end

  unless 0 <= column && column < column_count
    raise ArgumentError, "invalid column (#{column.inspect} for 0..#{column_count - 1})"
  end

  arrays = to_a
  arrays.delete_at(row)
  arrays.each do |array|
    array.delete_at(column)
  end

  new_matrix arrays, column_count - 1
end


  








  

    #hash  
  



  

    

Returns a hash-code for the matrix.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 938



def hash
  @rows.hash
end


  








  

    #hstack(*matrices)  
  



  

    

Returns a new matrix resulting by stacking horizontally the receiver with the given matrices



x = Matrix[[1, 2], [3, 4]]
y = Matrix[[5, 6], [7, 8]]
x.hstack(y) # => Matrix[[1, 2, 5, 6], [3, 4, 7, 8]]


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1260



def hstack(*matrices)
  self.class.hstack(self, *matrices)
end


  








  

    #imag  
  



  

    

Alias for #imaginary.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1419



alias imag imaginary


  








  

    #imaginary  
    Also known as: #imag
  



  

    

Returns the imaginary part of the matrix.



Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]]
  #=> 1+2i  i  0
        1  2  3
Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]].imaginary
  #=>   2i  i  0
        0  0  0


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1416



def imaginary
  collect(&:imaginary)
end


  








  

    

      #index(value, selector = :all)  ⇒ Array, column 
      #index(selector = :all)  ⇒ Array, column 
      #index(selector = :all)  ⇒ Enumerator 
    

    Also known as: #find_index
  



  

    

The index method is specialized to return the index as [row, column] It also accepts an optional selector argument, see #each for details.



Matrix[ [1,2], [3,4] ].index(&:even?) # => [0, 1]
Matrix[ [1,1], [1,1] ].index(1, :strict_lower) # => [1, 0]


  





  
Raises:


    
  
  • 
    (ArgumentError)
  
    • 





  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 586



def index(*args)
  raise ArgumentError, "wrong number of arguments(#{args.size} for 0-2)" if args.size > 2
  which = (args.size == 2 || SELECTORS.include?(args.last)) ? args.pop : :all
  return to_enum :find_index, which, *args unless block_given? || args.size == 1
  if args.size == 1
    value = args.first
    each_with_index(which) do |e, row_index, col_index|
      return row_index, col_index if e == value
    end
  else
    each_with_index(which) do |e, row_index, col_index|
      return row_index, col_index if yield e
    end
  end
  nil
end


  








  

    #inspect  
  



  

    

Overrides Object#inspect


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1525



def inspect
  if empty?
    "#{self.class}.empty(#{row_count}, #{column_count})"
  else
    "#{self.class}#{@rows.inspect}"
  end
end


  








  

    #inv  
  



  

    

Alias for #inverse.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1063



alias inv inverse


  








  

    #inverse  
    Also known as: #inv
  



  

    

Returns the inverse of the matrix.



Matrix[[-1, -1], [0, -1]].inverse
  #=> -1  1
      0 -1


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1059



def inverse
  Matrix.Raise ErrDimensionMismatch unless square?
  self.class.I(row_count).send(:inverse_from, self)
end


  








  

    #laplace_expansion(row: nil, column: nil)  
    Also known as: #cofactor_expansion
  



  

    

Returns the Laplace expansion along given row or column.



Matrix[[7,6], [3,9]].laplace_expansion(column: 1)
 #=> 45

Matrix[[Vector[1, 0], Vector[0, 1]], [2, 3]].laplace_expansion(row: 0)
 #=> Vector[3, -2]


  





  
Raises:


    
  
  • 
    (RuntimeError)
  
    • 





  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 717



def laplace_expansion(row: nil, column: nil)
  num = row || column

  if !num || (row && column)
    raise ArgumentError, "exactly one the row or column arguments must be specified"
  end

  Matrix.Raise ErrDimensionMismatch unless square?
  raise RuntimeError, "laplace_expansion of empty matrix is not defined" if empty?

  unless 0 <= num && num < row_count
    raise ArgumentError, "invalid num (#{num.inspect} for 0..#{row_count - 1})"
  end

  send(row ? :row : :column, num).map.with_index { |e, k|
    e * cofactor(*(row ? [num, k] : [k,num]))
  }.inject(:+)
end


  








  

    #lup  
    Also known as: #lup_decomposition
  



  

    

Returns the LUP decomposition of the matrix; see ::Matrix::LUPDecomposition.



a = Matrix[[1, 2], [3, 4]]
l, u, p = a.lup
l.lower_triangular? # => true
u.upper_triangular? # => true
p.permutation?      # => true
l * u == p * a      # => true
a.lup.solve([2, 5]) # => Vector[(1/1), (1/2)]


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1384



def lup
  LUPDecomposition.new(self)
end


  








  

    #lup_decomposition  
  



  

    

Alias for #lup.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1387



alias lup_decomposition lup


  








  

    #map(&block)  
  



  

    

Alias for #collect.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 444



alias map collect


  








  

    #minor(*param)  
  



  

    

Returns a section of the matrix.  The parameters are either:


  • 

    start_row, nrows, start_col, ncols; OR
    

  • 

    row_range, col_range
    




Matrix.diagonal(9, 5, -3).minor(0..1, 0..2)
  #=> 9 0 0
     0 5 0



Like Array#[], negative indices count backward from the end of the row or column (-1 is the last element). Returns nil if the starting row or column is greater than row_count or column_count respectively.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 617



def minor(*param)
  case param.size
  when 2
    row_range, col_range = param
    from_row = row_range.first
    from_row += row_count if from_row < 0
    to_row = row_range.end
    to_row += row_count if to_row < 0
    to_row += 1 unless row_range.exclude_end?
    size_row = to_row - from_row

    from_col = col_range.first
    from_col += column_count if from_col < 0
    to_col = col_range.end
    to_col += column_count if to_col < 0
    to_col += 1 unless col_range.exclude_end?
    size_col = to_col - from_col
  when 4
    from_row, size_row, from_col, size_col = param
    return nil if size_row < 0 || size_col < 0
    from_row += row_count if from_row < 0
    from_col += column_count if from_col < 0
  else
    raise ArgumentError, param.inspect
  end

  return nil if from_row > row_count || from_col > column_count || from_row < 0 || from_col < 0
  rows = @rows[from_row, size_row].collect{|row|
    row[from_col, size_col]
  }
  new_matrix rows, [column_count - from_col, size_col].min
end


  








  

    #rank  
  



  

    

Returns the rank of the matrix. Beware that using Float values can yield erroneous results because of their lack of precision. Consider using exact types like Rational or BigDecimal instead.



Matrix[[7,6], [3,9]].rank
  #=> 2


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1273



def rank
  # We currently use Bareiss' multistep integer-preserving gaussian elimination
  # (see comments on determinant)
  a = to_a
  last_column = column_count - 1
  last_row = row_count - 1
  pivot_row = 0
  previous_pivot = 1
  0.upto(last_column) do |k|
    switch_row = (pivot_row .. last_row).find {|row|
      a[row][k] != 0
    }
    if switch_row
      a[switch_row], a[pivot_row] = a[pivot_row], a[switch_row] unless pivot_row == switch_row
      pivot = a[pivot_row][k]
      (pivot_row+1).upto(last_row) do |i|
         ai = a[i]
         (k+1).upto(last_column) do |j|
           ai[j] =  (pivot * ai[j] - ai[k] * a[pivot_row][j]) / previous_pivot
         end
       end
      pivot_row += 1
      previous_pivot = pivot
    end
  end
  pivot_row
end


  








  

    #rank_e  
  



  

    

deprecated; use #rank


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1304



def rank_e
  warn "#{caller(1)[0]}: warning: Matrix#rank_e is deprecated; use #rank"
  rank
end


  








  

    #real   (readonly)
  



  

    

Returns the real part of the matrix.



Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]]
  #=> 1+2i  i  0
        1  2  3
Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]].real
  #=>    1  0  0
        1  2  3


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1430



def real
  collect(&:real)
end


  








  

    #rect  
    Also known as: #rectangular
  



  

    

Returns an array containing matrices corresponding to the real and imaginary parts of the matrix



m.rect == [m.real, m.imag]  # ==> true for all matrices m


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1440



def rect
  [real, imag]
end


  








  

    #rectangular  
  



  

    

Alias for #rect.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1443



alias rectangular rect


  








  

    #round(ndigits = 0)  
  



  

    

Returns a matrix with entries rounded to the given precision (see Float#round)


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1312



def round(ndigits=0)
  map{|e| e.round(ndigits)}
end


  








  

    #row(i, &block)  
  



  

    

Returns row vector number i of the matrix as a ::Vector (starting at 0 like an array).  When a block is given, the elements of that vector are iterated.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 402



def row(i, &block) # :yield: e
  if block_given?
    @rows.fetch(i){return self}.each(&block)
    self
  else
    Vector.elements(@rows.fetch(i){return nil})
  end
end


  








  

    #row_count  
    Also known as: #row_size
  



  

    

Returns the number of rows.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 387



def row_count
  @rows.size
end


  








  

    #row_size  
  



  

    

Alias for #row_count.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 391



alias_method :row_size, :row_count


  








  

    #row_vectors  
  



  

    

Returns an array of the row vectors of the matrix.  See Vector.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1468



def row_vectors
  Array.new(row_count) {|i|
    row(i)
  }
end


  








  

    #set_component(i, j, v)   (private)
  



  

    

Alias for #[]=.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 381



alias set_component []=


  








  

    #set_element(i, j, v)   (private)
  



  

    

Alias for #[]=.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 380



alias set_element []=


  








  

    #t  
  



  

    

Alias for #transpose.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1343



alias t transpose


  








  

    #to_a  
  



  

    

Returns an array of arrays that describe the rows of the matrix.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1486



def to_a
  @rows.collect(&:dup)
end


  








  

    #to_s  
  



  

    

Overrides Object#to_s


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1512



def to_s
  if empty?
    "#{self.class}.empty(#{row_count}, #{column_count})"
  else
    "#{self.class}[" + @rows.collect{|row|
      "[" + row.collect{|e| e.to_s}.join(", ") + "]"
    }.join(", ")+"]"
  end
end


  








  

    #tr  
  



  

    

Alias for #trace.


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1327



alias tr trace


  








  

    #trace  
    Also known as: #tr
  



  

    

Returns the trace (sum of diagonal elements) of the matrix.



Matrix[[7,6], [3,9]].trace
  #=> 16


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1321



def trace
  Matrix.Raise ErrDimensionMismatch unless square?
  (0...column_count).inject(0) do |tr, i|
    tr + @rows[i][i]
  end
end


  








  

    #transpose  
    Also known as: #t
  



  

    

Returns the transpose of the matrix.



Matrix[[1,2], [3,4], [5,6]]
  #=> 1 2
     3 4
     5 6
Matrix[[1,2], [3,4], [5,6]].transpose
  #=> 1 3 5
     2 4 6


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1339



def transpose
  return self.class.empty(column_count, 0) if row_count.zero?
  new_matrix @rows.transpose, row_count
end


  








  

    #vstack(*matrices)  
  



  

    

Returns a new matrix resulting by stacking vertically the receiver with the given matrices



x = Matrix[[1, 2], [3, 4]]
y = Matrix[[5, 6], [7, 8]]
x.vstack(y) # => Matrix[[1, 2], [3, 4], [5, 6], [7, 8]]


  



  [ GitHub ]


  

  


# File 'lib/matrix.rb', line 1353



def vstack(*matrices)
  self.class.vstack(self, *matrices)
end