In Files

  • matrix.rb
  • matrix/eigenvalue_decomposition.rb
  • matrix/lup_decomposition.rb

Matrix

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:

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:

frozen_string_literal: false

frozen_string_literal: false

Constants

SELECTORS

Attributes

column_count[R]

Returns the number of columns.

column_size[R]

Returns the number of columns.

rows[R]

instance creations

Public Class Methods

I(n) click to toggle source
Alias for: identity
[](*rows) click to toggle source

Creates a matrix where each argument is a row.

Matrix[ [25, 93], [-1, 66] ]
   =>  25 93
       -1 66
 
               # File matrix.rb, line 152
def Matrix.[](*rows)
  rows(rows, false)
end
            
build(row_count, column_count = row_count) click to toggle source

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
 
               # File matrix.rb, line 197
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) click to toggle source

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
 
               # File matrix.rb, line 283
def Matrix.column_vector(column)
  column = convert_to_array(column)
  new [column].transpose, 1
end
            
columns(columns) click to toggle source

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

Matrix.columns([[25, 93], [-1, 66]])
   =>  25 -1
       93 66
 
               # File matrix.rb, line 182
def Matrix.columns(columns)
  rows(columns, false).transpose
end
            
diagonal(*values) click to toggle source

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
 
               # File matrix.rb, line 217
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) click to toggle source

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]]
 
               # File matrix.rb, line 301
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) click to toggle source

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]]
 
               # File matrix.rb, line 336
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
            
identity(n) click to toggle source

Creates an n by n identity matrix.

Matrix.identity(2)
  => 1 0
     0 1
 
               # File matrix.rb, line 245
def Matrix.identity(n)
  scalar(n, 1)
end
            
Also aliased as: unit, I
new(rows, column_count = rows[0].size) click to toggle source

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

 
               # File matrix.rb, line 356
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
            
row_vector(row) click to toggle source

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
 
               # File matrix.rb, line 270
def Matrix.row_vector(row)
  row = convert_to_array(row)
  new [row]
end
            
rows(rows, copy = true) click to toggle source

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
 
               # File matrix.rb, line 164
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) click to toggle source

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

Matrix.scalar(2, 5)
  => 5 0
     0 5
 
               # File matrix.rb, line 235
def Matrix.scalar(n, value)
  diagonal(*Array.new(n, value))
end
            
unit(n) click to toggle source
Alias for: identity
vstack(x, *matrices) click to toggle source

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]]
 
               # File matrix.rb, line 315
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) click to toggle source

Creates a zero matrix.

Matrix.zero(2)
  => 0 0
     0 0
 
               # File matrix.rb, line 259
def Matrix.zero(row_count, column_count = row_count)
  rows = Array.new(row_count){Array.new(column_count, 0)}
  new rows, column_count
end
            

Public Instance Methods

*(m) click to toggle source

Matrix multiplication.

Matrix[[2,4], [6,8]] * Matrix.identity(2)
  => 2 4
     6 8
 
               # File matrix.rb, line 953
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) click to toggle source

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
 
               # File matrix.rb, line 1120
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) click to toggle source

Matrix addition.

Matrix.scalar(2,5) + Matrix[[1,0], [-4,7]]
  =>  6  0
     -4 12
 
               # File matrix.rb, line 986
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
            
+@() click to toggle source
 
               # File matrix.rb, line 1143
def +@
  self
end
            
-(m) click to toggle source

Matrix subtraction.

Matrix[[1,5], [4,2]] - Matrix[[9,3], [-4,1]]
  => -8  2
      8  1
 
               # File matrix.rb, line 1013
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
            
-@() click to toggle source
 
               # File matrix.rb, line 1147
def -@
  collect {|e| -e }
end
            
/(other) click to toggle source

Matrix division (multiplication by the inverse).

Matrix[[7,6], [3,9]] / Matrix[[2,9], [3,1]]
  => -7  1
     -3 -6
 
               # File matrix.rb, line 1040
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) click to toggle source

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

 
               # File matrix.rb, line 915
def ==(other)
  return false unless Matrix === other &&
                      column_count == other.column_count # necessary for empty matrices
  rows == other.rows
end
            
[](i, j) click to toggle source

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

 
               # File matrix.rb, line 372
def [](i, j)
  @rows.fetch(i){return nil}[j]
end
            
Also aliased as: element, component
adjugate() click to toggle source

Returns the adjugate of the matrix.

Matrix[ [7,6],[3,9] ].adjugate
  => 9 -6
     -3 7
 
               # File matrix.rb, line 701
def adjugate
  Matrix.Raise ErrDimensionMismatch unless square?
  Matrix.build(row_count, column_count) do |row, column|
    cofactor(column, row)
  end
end
            
clone() click to toggle source

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.

 
               # File matrix.rb, line 932
def clone
  new_matrix @rows.map(&:dup), column_count
end
            
coerce(other) click to toggle source

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.

 
               # File matrix.rb, line 1457
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) click to toggle source

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
 
               # File matrix.rb, line 686
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) click to toggle source
Alias for: laplace_expansion
collect() click to toggle source

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
 
               # File matrix.rb, line 440
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
            
Also aliased as: map
column(j) click to toggle source

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.

 
               # File matrix.rb, line 417
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() click to toggle source

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

 
               # File matrix.rb, line 1478
def column_vectors
  Array.new(column_count) {|i|
    column(i)
  }
end
            
component(i, j) click to toggle source
Alias for: []
conj() click to toggle source
Alias for: conjugate
conjugate() click to toggle source

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
 
               # File matrix.rb, line 1403
def conjugate
  collect(&:conjugate)
end
            
Also aliased as: conj
det() click to toggle source
Alias for: determinant
det_e() click to toggle source
Alias for: determinant_e
determinant() click to toggle source

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
 
               # File matrix.rb, line 1165
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
            
Also aliased as: det
determinant_e() click to toggle source

deprecated; use #determinant

 
               # File matrix.rb, line 1247
def determinant_e
  warn "#{caller(1)[0]}: warning: Matrix#determinant_e is deprecated; use #determinant"
  determinant
end
            
Also aliased as: det_e
diagonal?() click to toggle source

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

 
               # File matrix.rb, line 747
def diagonal?
  Matrix.Raise ErrDimensionMismatch unless square?
  each(:off_diagonal).all?(&:zero?)
end
            
each(which = :all) click to toggle source

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]

 
               # File matrix.rb, line 463
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) click to toggle source

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
 
               # File matrix.rb, line 524
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() click to toggle source
Alias for: eigensystem
eigensystem() click to toggle source

Returns the Eigensystem of the matrix; see 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
 
               # File matrix.rb, line 1370
def eigensystem
  EigenvalueDecomposition.new(self)
end
            
Also aliased as: eigen
element(i, j) click to toggle source
Alias for: []
elements_to_f() click to toggle source
 
               # File matrix.rb, line 1491
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() click to toggle source
 
               # File matrix.rb, line 1496
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() click to toggle source
 
               # File matrix.rb, line 1501
def elements_to_r
  warn "#{caller(1)[0]}: warning: Matrix#elements_to_r is deprecated, use map(&:to_r)"
  map(&:to_r)
end
            
empty?() click to toggle source

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

 
               # File matrix.rb, line 756
def empty?
  column_count == 0 || row_count == 0
end
            
eql?(other) click to toggle source
 
               # File matrix.rb, line 921
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) click to toggle source
Alias for: index
first_minor(row, column) click to toggle source

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
 
               # File matrix.rb, line 659
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() click to toggle source

Returns a hash-code for the matrix.

 
               # File matrix.rb, line 939
def hash
  @rows.hash
end
            
hermitian?() click to toggle source

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

 
               # File matrix.rb, line 764
def hermitian?
  Matrix.Raise ErrDimensionMismatch unless square?
  each_with_index(:upper).all? do |e, row, col|
    e == rows[col][row].conj
  end
end
            
hstack(*matrices) click to toggle source

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]]
 
               # File matrix.rb, line 1261
def hstack(*matrices)
  self.class.hstack(self, *matrices)
end
            
imag() click to toggle source
Alias for: imaginary
imaginary() click to toggle source

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
 
               # File matrix.rb, line 1417
def imaginary
  collect(&:imaginary)
end
            
Also aliased as: imag
index(value, selector = :all) → [row, column] click to toggle source
index(selector = :all){ block } → [row, column]
index(selector = :all) → an_enumerator

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]
 
               # File matrix.rb, line 587
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
            
Also aliased as: find_index
inspect() click to toggle source

Overrides Object#inspect

 
               # File matrix.rb, line 1526
def inspect
  if empty?
    "#{self.class}.empty(#{row_count}, #{column_count})"
  else
    "#{self.class}#{@rows.inspect}"
  end
end
            
inv() click to toggle source
Alias for: inverse
inverse() click to toggle source

Returns the inverse of the matrix.

Matrix[[-1, -1], [0, -1]].inverse
  => -1  1
      0 -1
 
               # File matrix.rb, line 1060
def inverse
  Matrix.Raise ErrDimensionMismatch unless square?
  self.class.I(row_count).send(:inverse_from, self)
end
            
Also aliased as: inv
laplace_expansion(row: nil, column: nil) click to toggle source

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]
 
               # File matrix.rb, line 718
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
            
Also aliased as: cofactor_expansion
lower_triangular?() click to toggle source

Returns true if this is a lower triangular matrix.

 
               # File matrix.rb, line 774
def lower_triangular?
  each(:strict_upper).all?(&:zero?)
end
            
lup() click to toggle source

Returns the LUP decomposition of the matrix; see 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)]
 
               # File matrix.rb, line 1385
def lup
  LUPDecomposition.new(self)
end
            
Also aliased as: lup_decomposition
lup_decomposition() click to toggle source
Alias for: lup
map() click to toggle source
Alias for: collect
minor(*param) click to toggle source

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.

 
               # File matrix.rb, line 618
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
            
normal?() click to toggle source

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

 
               # File matrix.rb, line 782
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?() click to toggle source

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

 
               # File matrix.rb, line 800
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?() click to toggle source

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

 
               # File matrix.rb, line 818
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
            
rank() click to toggle source

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
 
               # File matrix.rb, line 1274
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() click to toggle source

deprecated; use #rank

 
               # File matrix.rb, line 1305
def rank_e
  warn "#{caller(1)[0]}: warning: Matrix#rank_e is deprecated; use #rank"
  rank
end
            
real() click to toggle source

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
 
               # File matrix.rb, line 1431
def real
  collect(&:real)
end
            
real?() click to toggle source

Returns true if all entries of the matrix are real.

 
               # File matrix.rb, line 839
def real?
  all?(&:real?)
end
            
rect() click to toggle source

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

 
               # File matrix.rb, line 1441
def rect
  [real, imag]
end
            
Also aliased as: rectangular
rectangular() click to toggle source
Alias for: rect
regular?() click to toggle source

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

 
               # File matrix.rb, line 846
def regular?
  not singular?
end
            
round(ndigits=0) click to toggle source

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

 
               # File matrix.rb, line 1313
def round(ndigits=0)
  map{|e| e.round(ndigits)}
end
            
row(i) click to toggle source

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.

 
               # File matrix.rb, line 403
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() click to toggle source

Returns the number of rows.

 
               # File matrix.rb, line 388
def row_count
  @rows.size
end
            
Also aliased as: row_size
row_size() click to toggle source
Alias for: row_count
row_vectors() click to toggle source

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

 
               # File matrix.rb, line 1469
def row_vectors
  Array.new(row_count) {|i|
    row(i)
  }
end
            
singular?() click to toggle source

Returns true if this is a singular matrix.

 
               # File matrix.rb, line 853
def singular?
  determinant == 0
end
            
square?() click to toggle source

Returns true if this is a square matrix.

 
               # File matrix.rb, line 860
def square?
  column_count == row_count
end
            
symmetric?() click to toggle source

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

 
               # File matrix.rb, line 868
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
            
t() click to toggle source
Alias for: transpose
to_a() click to toggle source

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

 
               # File matrix.rb, line 1487
def to_a
  @rows.collect(&:dup)
end
            
to_s() click to toggle source

Overrides Object#to_s

 
               # File matrix.rb, line 1513
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() click to toggle source
Alias for: trace
trace() click to toggle source

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

Matrix[[7,6], [3,9]].trace
  => 16
 
               # File matrix.rb, line 1322
def trace
  Matrix.Raise ErrDimensionMismatch unless square?
  (0...column_count).inject(0) do |tr, i|
    tr + @rows[i][i]
  end
end
            
Also aliased as: tr
transpose() click to toggle source

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
 
               # File matrix.rb, line 1340
def transpose
  return self.class.empty(column_count, 0) if row_count.zero?
  new_matrix @rows.transpose, row_count
end
            
Also aliased as: t
unitary?() click to toggle source

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

 
               # File matrix.rb, line 880
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?() click to toggle source

Returns true if this is an upper triangular matrix.

 
               # File matrix.rb, line 897
def upper_triangular?
  each(:strict_lower).all?(&:zero?)
end
            
vstack(*matrices) click to toggle source

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]]
 
               # File matrix.rb, line 1354
def vstack(*matrices)
  self.class.vstack(self, *matrices)
end
            
zero?() click to toggle source

Returns true if this is a matrix with only zero elements

 
               # File matrix.rb, line 904
def zero?
  all?(&:zero?)
end