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/Ruby 3

class Matrix

Parent:
Object
Included modules:
Enumerable, ExceptionForMatrix

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

Constants

SELECTORS
VERSION

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)
Alias for: identity
[](*rows) Show source
# File lib/matrix.rb, line 78
def Matrix.[](*rows)
  rows(rows, false)
end

Creates a matrix where each argument is a row.

Matrix[ [25, 93], [-1, 66] ]
#   =>  25 93
#       -1 66
build(row_count, column_count = row_count) { |i, j| ... } Show source
# File lib/matrix.rb, line 123
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

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
column_vector(column) Show source
# File lib/matrix.rb, line 209
def Matrix.column_vector(column)
  column = convert_to_array(column)
  new [column].transpose, 1
end

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
columns(columns) Show source
# File lib/matrix.rb, line 108
def Matrix.columns(columns)
  rows(columns, false).transpose
end

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

Matrix.columns([[25, 93], [-1, 66]])
#   =>  25 -1
#       93 66
combine(*matrices) { |*elements| ... } Show source
# File lib/matrix.rb, line 288
def Matrix.combine(*matrices)
  return to_enum(__method__, *matrices) unless block_given?

  return Matrix.empty if matrices.empty?
  matrices.map!(&CoercionHelper.method(:coerce_to_matrix))
  x = matrices.first
  matrices.each do |m|
    raise ErrDimensionMismatch unless x.row_count == m.row_count && x.column_count == m.column_count
  end

  rows = Array.new(x.row_count) do |i|
    Array.new(x.column_count) do |j|
      yield matrices.map{|m| m[i,j]}
    end
  end
  new rows, x.column_count
end

Create a matrix by combining matrices entrywise, using the given block

x = Matrix[[6, 6], [4, 4]]
y = Matrix[[1, 2], [3, 4]]
Matrix.combine(x, y) {|a, b| a - b} # => Matrix[[5, 4], [1, 0]]
diagonal(*values) Show source
# File lib/matrix.rb, line 143
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

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
empty(row_count = 0, column_count = 0) Show source
# File lib/matrix.rb, line 227
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

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]]
hstack(x, *matrices) Show source
# File lib/matrix.rb, line 262
def Matrix.hstack(x, *matrices)
  x = CoercionHelper.coerce_to_matrix(x)
  result = x.send(:rows).map(&:dup)
  total_column_count = x.column_count
  matrices.each do |m|
    m = CoercionHelper.coerce_to_matrix(m)
    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

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]]
identity(n) Show source
# File lib/matrix.rb, line 171
def Matrix.identity(n)
  scalar(n, 1)
end

Creates an n by n identity matrix.

Matrix.identity(2)
#  => 1 0
#     0 1
Also aliased as: unit, I
new(rows, column_count = rows[0].size) Show source
# File lib/matrix.rb, line 322
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

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

row_vector(row) Show source
# File lib/matrix.rb, line 196
def Matrix.row_vector(row)
  row = convert_to_array(row)
  new [row]
end

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
rows(rows, copy = true) Show source
# File lib/matrix.rb, line 90
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

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
scalar(n, value) Show source
# File lib/matrix.rb, line 161
def Matrix.scalar(n, value)
  diagonal(*Array.new(n, value))
end

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

Matrix.scalar(2, 5)
#  => 5 0
#     0 5
unit(n)
Alias for: identity
vstack(x, *matrices) Show source
# File lib/matrix.rb, line 241
def Matrix.vstack(x, *matrices)
  x = CoercionHelper.coerce_to_matrix(x)
  result = x.send(:rows).map(&:dup)
  matrices.each do |m|
    m = CoercionHelper.coerce_to_matrix(m)
    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

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]]
zero(row_count, column_count = row_count) Show source
# File lib/matrix.rb, line 185
def Matrix.zero(row_count, column_count = row_count)
  rows = Array.new(row_count){Array.new(column_count, 0)}
  new rows, column_count
end

Creates a zero matrix.

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

Public Instance Methods

# File lib/matrix.rb, line 1058
def *(m) # m is matrix or vector or number
  case(m)
  when Numeric
    new_rows = @rows.collect {|row|
      row.collect {|e| e * m }
    }
    return new_matrix new_rows, column_count
  when Vector
    m = self.class.column_vector(m)
    r = self * m
    return r.column(0)
  when Matrix
    raise ErrDimensionMismatch if column_count != m.row_count
    m_rows = m.rows
    new_rows = rows.map do |row_i|
      Array.new(m.column_count) do |j|
        vij = 0
        column_count.times do |k|
          vij += row_i[k] * m_rows[k][j]
        end
        vij
      end
    end
    return new_matrix new_rows, m.column_count
  else
    return apply_through_coercion(m, __method__)
  end
end

Matrix multiplication.

Matrix[[2,4], [6,8]] * Matrix.identity(2)
#  => 2 4
#     6 8
**(exp) Show source
# File lib/matrix.rb, line 1237
def **(exp)
  case exp
  when Integer
    case
    when exp == 0
      _make_sure_it_is_invertible = inverse
      self.class.identity(column_count)
    when exp < 0
      inverse.power_int(-exp)
    else
      power_int(exp)
    end
  when Numeric
    v, d, v_inv = eigensystem
    v * self.class.diagonal(*d.each(:diagonal).map{|e| e ** exp}) * v_inv
  else
    raise ErrOperationNotDefined, ["**", self.class, exp.class]
  end
end

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 lib/matrix.rb, line 1093
def +(m)
  case m
  when Numeric
    raise ErrOperationNotDefined, ["+", self.class, m.class]
  when Vector
    m = self.class.column_vector(m)
  when Matrix
  else
    return apply_through_coercion(m, __method__)
  end

  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

Matrix addition.

Matrix.scalar(2,5) + Matrix[[1,0], [-4,7]]
#  =>  6  0
#     -4 12
# File lib/matrix.rb, line 1283
def [email protected]
  self
end
# File lib/matrix.rb, line 1120
def -(m)
  case m
  when Numeric
    raise ErrOperationNotDefined, ["-", self.class, m.class]
  when Vector
    m = self.class.column_vector(m)
  when Matrix
  else
    return apply_through_coercion(m, __method__)
  end

  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

Matrix subtraction.

Matrix[[1,5], [4,2]] - Matrix[[9,3], [-4,1]]
#  => -8  2
#      8  1
# File lib/matrix.rb, line 1292
def [email protected]
  collect {|e| -e }
end

Unary matrix negation.

-Matrix[[1,5], [4,2]]
# => -1 -5
#    -4 -2
/(other) Show source
# File lib/matrix.rb, line 1147
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

Matrix division (multiplication by the inverse).

Matrix[[7,6], [3,9]] / Matrix[[2,9], [3,1]]
#  => -7  1
#     -3 -6
==(other) Show source
# File lib/matrix.rb, line 1021
def ==(other)
  return false unless Matrix === other &&
                      column_count == other.column_count # necessary for empty matrices
  rows == other.rows
end

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

[](i, j) Show source
# File lib/matrix.rb, line 337
def [](i, j)
  @rows.fetch(i){return nil}[j]
end

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

Also aliased as: element, component
matrix[range, range] = matrix/element Show source
matrix[range, integer] = vector/column_matrix/element
matrix[integer, range] = vector/row_matrix/element
matrix[integer, integer] = element
# File lib/matrix.rb, line 351
def []=(i, j, v)
  raise FrozenError, "can't modify frozen Matrix" if frozen?
  rows = check_range(i, :row) or row = check_int(i, :row)
  columns = check_range(j, :column) or column = check_int(j, :column)
  if rows && columns
    set_row_and_col_range(rows, columns, v)
  elsif rows
    set_row_range(rows, column, v)
  elsif columns
    set_col_range(row, columns, v)
  else
    set_value(row, column, v)
  end
end

Set element or elements of matrix.

Also aliased as: set_element, set_component
abs() Show source
# File lib/matrix.rb, line 1299
def abs
  collect(&:abs)
end

Returns the absolute value elementwise

adjoint() Show source
# File lib/matrix.rb, line 1566
def adjoint
  conjugate.transpose
end

Returns the adjoint of the matrix.

Matrix[ [i,1],[2,-i] ].adjoint
#  => -i 2
#      1 i
adjugate() Show source
# File lib/matrix.rb, line 793
def adjugate
  raise ErrDimensionMismatch unless square?
  Matrix.build(row_count, column_count) do |row, column|
    cofactor(column, row)
  end
end

Returns the adjugate of the matrix.

Matrix[ [7,6],[3,9] ].adjugate
#  => 9 -6
#     -3 7
antisymmetric?() Show source
# File lib/matrix.rb, line 973
def antisymmetric?
  raise ErrDimensionMismatch unless square?
  each_with_index(:upper) do |e, row, col|
    return false unless e == -rows[col][row]
  end
  true
end

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

Also aliased as: skew_symmetric?
coerce(other) Show source
# File lib/matrix.rb, line 1619
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

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.

cofactor(row, column) Show source
# File lib/matrix.rb, line 778
def cofactor(row, column)
  raise RuntimeError, "cofactor of empty matrix is not defined" if empty?
  raise ErrDimensionMismatch unless square?

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

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
cofactor_expansion(row: nil, column: nil)
Alias for: laplace_expansion
collect(which = :all) { |e| ... } Show source
# File lib/matrix.rb, line 508
def collect(which = :all, &block) # :yield: e
  return to_enum(:collect, which) unless block_given?
  dup.collect!(which, &block)
end

Returns a matrix that is the result of iteration of the given block over all elements of the matrix. 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] ].collect { |e| e**2 } # => 1 4 # 9 16

Also aliased as: map
collect!(which = :all) { |e| ... } Show source
# File lib/matrix.rb, line 526
def collect!(which = :all)
  return to_enum(:collect!, which) unless block_given?
  raise FrozenError, "can't modify frozen Matrix" if frozen?
  each_with_index(which){ |e, row_index, col_index| @rows[row_index][col_index] = yield e }
end

Invokes the given block for each element of matrix, replacing the element with the value returned by the block. 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

Also aliased as: map!
column(j) { |e| ... } Show source
# File lib/matrix.rb, line 477
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

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.

column_vectors() Show source
# File lib/matrix.rb, line 1640
def column_vectors
  Array.new(column_count) {|i|
    column(i)
  }
end

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

combine(*other_matrices) { |*elements| ... } Show source
# File lib/matrix.rb, line 315
def combine(*matrices, &block)
  Matrix.combine(self, *matrices, &block)
end

Creates new matrix by combining with other_matrices entrywise, using the given block.

x = Matrix[[6, 6], [4, 4]]
y = Matrix[[1, 2], [3, 4]]
x.combine(y) {|a, b| a - b} # => Matrix[[5, 4], [1, 0]]
component(i, j)
Alias for: []
conj()
Alias for: conjugate
conjugate() Show source
# File lib/matrix.rb, line 1554
def conjugate
  collect(&:conjugate)
end

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
Also aliased as: conj
det()
Alias for: determinant
det_e()
Alias for: determinant_e
determinant() Show source
# File lib/matrix.rb, line 1317
def determinant
  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

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
Also aliased as: det
determinant_e() Show source
# File lib/matrix.rb, line 1398
def determinant_e
  warn "Matrix#determinant_e is deprecated; use #determinant", uplevel: 1
  determinant
end

deprecated; use Matrix#determinant

Also aliased as: det_e
diagonal?() Show source
# File lib/matrix.rb, line 839
def diagonal?
  raise ErrDimensionMismatch unless square?
  each(:off_diagonal).all?(&:zero?)
end

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

each(which = :all) { |e| ... } Show source
# File lib/matrix.rb, line 556
def each(which = :all, &block) # :yield: e
  return to_enum :each, which unless block_given?
  last = column_count - 1
  case which
  when :all
    @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

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]
    
each_with_index(which = :all) { |e, row, column| ... } Show source
# File lib/matrix.rb, line 616
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

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
eigen()
Alias for: eigensystem
eigensystem() Show source
# File lib/matrix.rb, line 1521
def eigensystem
  EigenvalueDecomposition.new(self)
end

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
Also aliased as: eigen
element(i, j)
Alias for: []
elements_to_f() Show source
# File lib/matrix.rb, line 1663
def elements_to_f
  warn "Matrix#elements_to_f is deprecated, use map(&:to_f)", uplevel: 1
  map(&:to_f)
end

Deprecated.

Use map(&:to_f)

elements_to_i() Show source
# File lib/matrix.rb, line 1671
def elements_to_i
  warn "Matrix#elements_to_i is deprecated, use map(&:to_i)", uplevel: 1
  map(&:to_i)
end

Deprecated.

Use map(&:to_i)

elements_to_r() Show source
# File lib/matrix.rb, line 1679
def elements_to_r
  warn "Matrix#elements_to_r is deprecated, use map(&:to_r)", uplevel: 1
  map(&:to_r)
end

Deprecated.

Use map(&:to_r)

empty?() Show source
# File lib/matrix.rb, line 848
def empty?
  column_count == 0 || row_count == 0
end

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

entrywise_product(m)
Alias for: hadamard_product
eql?(other) Show source
# File lib/matrix.rb, line 1027
def eql?(other)
  return false unless Matrix === other &&
                      column_count == other.column_count # necessary for empty matrices
  rows.eql? other.rows
end
Alias for: index
first_minor(row, column) Show source
# File lib/matrix.rb, line 751
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

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
freeze() Show source
# File lib/matrix.rb, line 534
def freeze
  @rows.each(&:freeze).freeze

  super
end
Calls superclass method Object#freeze
hadamard_product(m) Show source
# File lib/matrix.rb, line 1167
def hadamard_product(m)
  combine(m){|a, b| a * b}
end

Hadamard product

Matrix[[1,2], [3,4]].hadamard_product(Matrix[[1,2], [3,2]])
#  => 1  4
#     9  8
Also aliased as: entrywise_product
hash() Show source
# File lib/matrix.rb, line 1044
def hash
  @rows.hash
end

Returns a hash-code for the matrix.

hermitian?() Show source
# File lib/matrix.rb, line 856
def hermitian?
  raise ErrDimensionMismatch unless square?
  each_with_index(:upper).all? do |e, row, col|
    e == rows[col][row].conj
  end
end

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

hstack(*matrices) Show source
# File lib/matrix.rb, line 1412
def hstack(*matrices)
  self.class.hstack(self, *matrices)
end

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]]
imag()
Alias for: imaginary
imaginary() Show source
# File lib/matrix.rb, line 1579
def imaginary
  collect(&:imaginary)
end

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
Also aliased as: imag
index(value, selector = :all) → [row, column] Show source
index(selector = :all){ block } → [row, column]
index(selector = :all) → an_enumerator
# File lib/matrix.rb, line 679
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

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]
Also aliased as: find_index
inspect() Show source
# File lib/matrix.rb, line 1704
def inspect
  if empty?
    "#{self.class}.empty(#{row_count}, #{column_count})"
  else
    "#{self.class}#{@rows.inspect}"
  end
end

Overrides Object#inspect

inv()
Alias for: inverse
inverse() Show source
# File lib/matrix.rb, line 1178
def inverse
  raise ErrDimensionMismatch unless square?
  self.class.I(row_count).send(:inverse_from, self)
end

Returns the inverse of the matrix.

Matrix[[-1, -1], [0, -1]].inverse
#  => -1  1
#      0 -1
Also aliased as: inv
laplace_expansion(row: nil, column: nil) Show source
# File lib/matrix.rb, line 810
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

  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

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]
Also aliased as: cofactor_expansion
lower_triangular?() Show source
# File lib/matrix.rb, line 866
def lower_triangular?
  each(:strict_upper).all?(&:zero?)
end

Returns true if this is a lower triangular matrix.

lup() Show source
# File lib/matrix.rb, line 1536
def lup
  LUPDecomposition.new(self)
end

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)]
Also aliased as: lup_decomposition
lup_decomposition()
Alias for: lup
map(which = :all)
Alias for: collect
map!(which = :all)
Alias for: collect!
minor(*param) Show source
# File lib/matrix.rb, line 710
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

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.

normal?() Show source
# File lib/matrix.rb, line 874
def normal?
  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

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

orthogonal?() Show source
# File lib/matrix.rb, line 892
def orthogonal?
  raise ErrDimensionMismatch unless square?

  rows.each_with_index do |row_i, i|
    rows.each_with_index do |row_j, j|
      s = 0
      row_count.times do |k|
        s += row_i[k] * row_j[k]
      end
      return false unless s == (i == j ? 1 : 0)
    end
  end
  true
end

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

permutation?() Show source
# File lib/matrix.rb, line 911
def permutation?
  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

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

rank() Show source
# File lib/matrix.rb, line 1425
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

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
rank_e() Show source
# File lib/matrix.rb, line 1456
def rank_e
  warn "Matrix#rank_e is deprecated; use #rank", uplevel: 1
  rank
end

deprecated; use Matrix#rank

real() Show source
# File lib/matrix.rb, line 1593
def real
  collect(&:real)
end

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
real?() Show source
# File lib/matrix.rb, line 932
def real?
  all?(&:real?)
end

Returns true if all entries of the matrix are real.

rect() Show source
# File lib/matrix.rb, line 1603
def rect
  [real, imag]
end

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
Also aliased as: rectangular
rectangular()
Alias for: rect
regular?() Show source
# File lib/matrix.rb, line 939
def regular?
  not singular?
end

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

round(ndigits=0) Show source
# File lib/matrix.rb, line 1464
def round(ndigits=0)
  map{|e| e.round(ndigits)}
end

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

row(i) { |e| ... } Show source
# File lib/matrix.rb, line 463
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

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.

row_count() Show source
# File lib/matrix.rb, line 448
def row_count
  @rows.size
end

Returns the number of rows.

Also aliased as: row_size
row_size()
Alias for: row_count
row_vectors() Show source
# File lib/matrix.rb, line 1631
def row_vectors
  Array.new(row_count) {|i|
    row(i)
  }
end

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

singular?() Show source
# File lib/matrix.rb, line 946
def singular?
  determinant == 0
end

Returns true if this is a singular matrix.

skew_symmetric?()
Alias for: antisymmetric?
square?() Show source
# File lib/matrix.rb, line 953
def square?
  column_count == row_count
end

Returns true if this is a square matrix.

symmetric?() Show source
# File lib/matrix.rb, line 961
def symmetric?
  raise ErrDimensionMismatch unless square?
  each_with_index(:strict_upper) do |e, row, col|
    return false if e != rows[col][row]
  end
  true
end

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

t()
Alias for: transpose
to_a() Show source
# File lib/matrix.rb, line 1656
def to_a
  @rows.collect(&:dup)
end

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

to_matrix() Show source
# File lib/matrix.rb, line 1649
def to_matrix
  self
end

Explicit conversion to a Matrix. Returns self

to_s() Show source
# File lib/matrix.rb, line 1691
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

Overrides Object#to_s

tr()
Alias for: trace
trace() Show source
# File lib/matrix.rb, line 1473
def trace
  raise ErrDimensionMismatch unless square?
  (0...column_count).inject(0) do |tr, i|
    tr + @rows[i][i]
  end
end

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

Matrix[[7,6], [3,9]].trace
#  => 16
Also aliased as: tr
transpose() Show source
# File lib/matrix.rb, line 1491
def transpose
  return self.class.empty(column_count, 0) if row_count.zero?
  new_matrix @rows.transpose, row_count
end

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
Also aliased as: t
unitary?() Show source
# File lib/matrix.rb, line 986
def unitary?
  raise ErrDimensionMismatch unless square?
  rows.each_with_index do |row_i, i|
    rows.each_with_index do |row_j, j|
      s = 0
      row_count.times do |k|
        s += row_i[k].conj * row_j[k]
      end
      return false unless s == (i == j ? 1 : 0)
    end
  end
  true
end

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

upper_triangular?() Show source
# File lib/matrix.rb, line 1003
def upper_triangular?
  each(:strict_lower).all?(&:zero?)
end

Returns true if this is an upper triangular matrix.

vstack(*matrices) Show source
# File lib/matrix.rb, line 1505
def vstack(*matrices)
  self.class.vstack(self, *matrices)
end

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]]
zero?() Show source
# File lib/matrix.rb, line 1010
def zero?
  all?(&:zero?)
end

Returns true if this is a matrix with only zero elements

Protected Instance Methods

power_int(exp) Show source
# File lib/matrix.rb, line 1257
          def power_int(exp)
  # assumes `exp` is an Integer > 0
  #
  # Previous algorithm:
  #   build M**2, M**4 = (M**2)**2, M**8, ... and multiplying those you need
  #   e.g. M**0b1011 = M**11 = M * M**2 * M**8
  #                              ^  ^
  #   (highlighted the 2 out of 5 multiplications involving `M * x`)
  #
  # Current algorithm has same number of multiplications but with lower exponents:
  #    M**11 = M * (M * M**4)**2
  #              ^    ^  ^
  #   (highlighted the 3 out of 5 multiplications involving `M * x`)
  #
  # This should be faster for all (non nil-potent) matrices.
  case
  when exp == 1
    self
  when exp.odd?
    self * power_int(exp - 1)
  else
    sqrt = power_int(exp / 2)
    sqrt * sqrt
  end
end

Private Instance Methods

check_int(val, direction) Show source
# File lib/matrix.rb, line 376
        def check_int(val, direction)
  count = direction == :row ? row_count : column_count
  CoercionHelper.check_int(val, count, direction)
end
check_range(val, direction) Show source
# File lib/matrix.rb, line 370
        def check_range(val, direction)
  return unless val.is_a?(Range)
  count = direction == :row ? row_count : column_count
  CoercionHelper.check_range(val, count, direction)
end

Returns range or nil

determinant_bareiss() Show source
# File lib/matrix.rb, line 1368
        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

Private. Use Matrix#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.

initialize_copy(m) Show source
# File lib/matrix.rb, line 1036
        def initialize_copy(m)
  super
  @rows = @rows.map(&:dup) unless frozen?
end

Called for dup & clone.

Calls superclass method
set_col_range(row, col_range, value) Show source
# File lib/matrix.rb, line 432
        def set_col_range(row, col_range, value)
  value = if value.is_a?(Vector)
    value.to_a
  elsif value.is_a?(Matrix)
    raise ErrDimensionMismatch unless value.row_count == 1
    value.row(0).to_a
  else
    Array.new(col_range.size, value)
  end
  raise ErrDimensionMismatch unless col_range.size == value.size
  @rows[row][col_range] = value
end
set_column_vector(row_range, col, value) Show source
# File lib/matrix.rb, line 425
        def set_column_vector(row_range, col, value)
  value.each_with_index do |e, index|
    r = row_range.begin + index
    @rows[r][col] = e
  end
end
Alias for: []=
Alias for: []=
set_row_and_col_range(row_range, col_range, value) Show source
# File lib/matrix.rb, line 387
        def set_row_and_col_range(row_range, col_range, value)
  if value.is_a?(Matrix)
    if row_range.size != value.row_count || col_range.size != value.column_count
      raise ErrDimensionMismatch, [
        'Expected a Matrix of dimensions',
        "#{row_range.size}x#{col_range.size}",
        'got',
        "#{value.row_count}x#{value.column_count}",
      ].join(' ')
    end
    source = value.instance_variable_get :@rows
    row_range.each_with_index do |row, i|
      @rows[row][col_range] = source[i]
    end
  elsif value.is_a?(Vector)
    raise ErrDimensionMismatch, 'Expected a Matrix or a value, got a Vector'
  else
    value_to_set = Array.new(col_range.size, value)
    row_range.each do |i|
      @rows[i][col_range] = value_to_set
    end
  end
end
set_row_range(row_range, col, value) Show source
# File lib/matrix.rb, line 411
        def set_row_range(row_range, col, value)
  if value.is_a?(Vector)
    raise ErrDimensionMismatch unless row_range.size == value.size
    set_column_vector(row_range, col, value)
  elsif value.is_a?(Matrix)
    raise ErrDimensionMismatch unless value.column_count == 1
    value = value.column(0)
    raise ErrDimensionMismatch unless row_range.size == value.size
    set_column_vector(row_range, col, value)
  else
    @rows[row_range].each{|e| e[col] = value }
  end
end
set_value(row, col, value) Show source
# File lib/matrix.rb, line 381
        def set_value(row, col, value)
  raise ErrDimensionMismatch, "Expected a a value, got a #{value.class}" if value.respond_to?(:to_matrix)

  @rows[row][col] = value
end

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Licensed under the Ruby License.
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Licensed under their own licenses.