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 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) 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 78
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 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

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 209
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 108
def Matrix.columns(columns)
  rows(columns, false).transpose
end

combine(*matrices) { |*elements| ... } click to toggle source

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]]

 
               # File 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

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 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

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 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

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 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

identity(n) click to toggle source

Creates an n by n identity matrix.

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

 
               # File matrix.rb, line 171
def Matrix.identity(n)
  scalar(n, 1)
end

Also aliased as: unit, I

new(rows, column_count = rows[0].size) click to toggle source

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

 
               # File 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

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 196
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 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

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 161
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 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

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 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

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 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

**(exp) 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 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

+(m) click to toggle source

Matrix addition.

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

 
               # File 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

+@() click to toggle source

 
               # File matrix.rb, line 1283
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 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

-@() click to toggle source

Unary matrix negation.

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

 
               # File matrix.rb, line 1292
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 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

==(other) click to toggle source

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

 
               # File matrix.rb, line 1021
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 337
def [](i, j)
  @rows.fetch(i){return nil}[j]
end

Also aliased as: element, component

matrix[range, range] = matrix/element click to toggle source

matrix[range, integer] = vector/column_matrix/element

matrix[integer, range] = vector/row_matrix/element

matrix[integer, integer] = element

Set element or elements of matrix.

 
               # File 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

Also aliased as: set_element, set_component

abs() click to toggle source

Returns the absolute value elementwise

 
               # File matrix.rb, line 1299
def abs
  collect(&:abs)
end

adjoint() click to toggle source

Returns the adjoint of the matrix.

Matrix[ [i,1],[2,-i] ].adjoint
#  => -i 2
#      1 i

 
               # File matrix.rb, line 1566
def adjoint
  conjugate.transpose
end

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 793
def adjugate
  raise ErrDimensionMismatch unless square?
  Matrix.build(row_count, column_count) do |row, column|
    cofactor(column, row)
  end
end

antisymmetric?() click to toggle source

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

 
               # File 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

Also aliased as: skew_symmetric?

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 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

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 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

cofactor_expansion(row: nil, column: nil) click to toggle source

Alias for: laplace_expansion

collect(which = :all) click to toggle source

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

 
               # File matrix.rb, line 508
def collect(which = :all, &block) # :yield: e
  return to_enum(:collect, which) unless block_given?
  dup.collect!(which, &block)
end

Also aliased as: map

collect!(which = :all) click to toggle source

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

 
               # File 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

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 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

column_vectors() click to toggle source

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

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

combine(*other_matrices) { |*elements| ... } click to toggle source

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]]

 
               # File matrix.rb, line 315
def combine(*matrices, &block)
  Matrix.combine(self, *matrices, &block)
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 1554
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 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

Also aliased as: det

determinant_e() click to toggle source

deprecated; use Matrix#determinant

 
               # File matrix.rb, line 1398
def determinant_e
  warn "Matrix#determinant_e is deprecated; use #determinant", uplevel: 1
  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 839
def diagonal?
  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 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

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 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

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 1521
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

Deprecated.

Use map(&:to_f)

 
               # File 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

elements_to_i() click to toggle source

Deprecated.

Use map(&:to_i)

 
               # File 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

elements_to_r() click to toggle source

Deprecated.

Use map(&:to_r)

 
               # File 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

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 848
def empty?
  column_count == 0 || row_count == 0
end

entrywise_product(m) click to toggle source

Alias for: hadamard_product

eql?(other) click to toggle source

 
               # File 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

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 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

freeze() click to toggle source

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

  super
end

hadamard_product(m) click to toggle source

Hadamard product

Matrix[[1,2], [3,4]].hadamard_product(Matrix[[1,2], [3,2]])
#  => 1  4
#     9  8

 
               # File matrix.rb, line 1167
def hadamard_product(m)
  combine(m){|a, b| a * b}
end

Also aliased as: entrywise_product

hash() click to toggle source

Returns a hash-code for the matrix.

 
               # File matrix.rb, line 1044
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 856
def hermitian?
  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 1412
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 1579
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 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

Also aliased as: find_index

inspect() click to toggle source

Overrides Object#inspect

 
               # File matrix.rb, line 1704
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 1178
def inverse
  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 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

Also aliased as: cofactor_expansion

lower_triangular?() click to toggle source

Returns true if this is a lower triangular matrix.

 
               # File matrix.rb, line 866
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 1536
def lup
  LUPDecomposition.new(self)
end

Also aliased as: lup_decomposition

lup_decomposition() click to toggle source

Alias for: lup

map(which = :all) click to toggle source

Alias for: collect

map!(which = :all) 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 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

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 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

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 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

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 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

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 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

rank_e() click to toggle source

deprecated; use Matrix#rank

 
               # File matrix.rb, line 1456
def rank_e
  warn "Matrix#rank_e is deprecated; use #rank", uplevel: 1
  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 1593
def real
  collect(&:real)
end

real?() click to toggle source

Returns true if all entries of the matrix are real.

 
               # File matrix.rb, line 932
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 1603
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 939
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 1464
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 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

row_count() click to toggle source

Returns the number of rows.

 
               # File matrix.rb, line 448
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 1631
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 946
def singular?
  determinant == 0
end

skew_symmetric?() click to toggle source

Alias for: antisymmetric?

square?() click to toggle source

Returns true if this is a square matrix.

 
               # File matrix.rb, line 953
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 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

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 1656
def to_a
  @rows.collect(&:dup)
end

to_matrix() click to toggle source

Explicit conversion to a Matrix. Returns self

 
               # File matrix.rb, line 1649
def to_matrix
  self
end

to_s() click to toggle source

Overrides Object#to_s

 
               # File 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

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 1473
def trace
  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 1491
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 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

upper_triangular?() click to toggle source

Returns true if this is an upper triangular matrix.

 
               # File matrix.rb, line 1003
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 1505
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 1010
def zero?
  all?(&:zero?)
end

Protected Instance Methods

power_int(exp) click to toggle source

 
               # File 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

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