class Matrix

Constants

SELECTORS

Attributes

column_size[R]

Returns the number of columns.

rows[R]

instance creations

Public Class Methods

I(n)
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 lib/backports/1.9.2/stdlib/matrix.rb, line 141
def Matrix.[](*rows)
  rows(rows, false)
end
build(row_size, column_size = row_size) { |i, j| ... } click to toggle source

Creates a matrix of size row_size x column_size. 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 lib/backports/1.9.2/stdlib/matrix.rb, line 186
def Matrix.build(row_size, column_size = row_size)
  row_size = CoercionHelper.coerce_to_int(row_size)
  column_size = CoercionHelper.coerce_to_int(column_size)
  raise ArgumentError if row_size < 0 || column_size < 0
  return to_enum(:build, row_size, column_size) unless block_given?
  rows = Array.new(row_size) do |i|
    Array.new(column_size) do |j|
      yield i, j
    end
  end
  new rows, column_size
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 lib/backports/1.9.2/stdlib/matrix.rb, line 271
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 lib/backports/1.9.2/stdlib/matrix.rb, line 171
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 lib/backports/1.9.2/stdlib/matrix.rb, line 206
def Matrix.diagonal(*values)
  size = values.size
  rows = Array.new(size) {|j|
    row = Array.new(size, 0)
    row[j] = values[j]
    row
  }
  new rows
end
empty(row_size = 0, column_size = 0) click to toggle source

Creates a empty matrix of row_size x column_size. At least one of row_size or column_size 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 lib/backports/1.9.2/stdlib/matrix.rb, line 289
def Matrix.empty(row_size = 0, column_size = 0)
  Matrix.Raise ArgumentError, "One size must be 0" if column_size != 0 && row_size != 0
  Matrix.Raise ArgumentError, "Negative size" if column_size < 0 || row_size < 0

  new([[]]*row_size, column_size)
end
identity(n) click to toggle source

Creates an n by n identity matrix.

Matrix.identity(2)
  => 1 0
     0 1
# File lib/backports/1.9.2/stdlib/matrix.rb, line 233
def Matrix.identity(n)
  scalar(n, 1)
end
Also aliased as: unit, I
new(rows, column_size = rows[0].size) click to toggle source

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

# File lib/backports/1.9.2/stdlib/matrix.rb, line 299
def initialize(rows, column_size = rows[0].size)
  # No checking is done at this point. rows must be an Array of Arrays.
  # column_size 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_size = column_size
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 lib/backports/1.9.2/stdlib/matrix.rb, line 258
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 lib/backports/1.9.2/stdlib/matrix.rb, line 153
def Matrix.rows(rows, copy = true)
  rows = convert_to_array(rows)
  rows.map! do |row|
    convert_to_array(row, copy)
  end
  size = (rows[0] || []).size
  rows.each do |row|
    Matrix.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 lib/backports/1.9.2/stdlib/matrix.rb, line 223
def Matrix.scalar(n, value)
  diagonal(*Array.new(n, value))
end
unit(n)
Alias for: identity
zero(row_size, column_size = row_size) click to toggle source

Creates a zero matrix.

Matrix.zero(2)
  => 0 0
     0 0
# File lib/backports/1.9.2/stdlib/matrix.rb, line 247
def Matrix.zero(row_size, column_size = row_size)
  rows = Array.new(row_size){Array.new(column_size, 0)}
  new rows, column_size
end

Public Instance Methods

*(m) click to toggle source

Matrix multiplication.

Matrix[[2,4], [6,8]] * Matrix.identity(2)
  => 2 4
     6 8
# File lib/backports/1.9.2/stdlib/matrix.rb, line 807
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_size)
  when Vector
    m = self.class.column_vector(m)
    r = self * m
    return r.column(0)
  when Matrix
    Matrix.Raise ErrDimensionMismatch if column_size != m.row_size

    rows = Array.new(row_size) {|i|
      Array.new(m.column_size) {|j|
        (0 ... column_size).inject(0) do |vij, k|
          vij + self[i, k] * m[k, j]
        end
      }
    }
    return new_matrix(rows, m.column_size)
  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 lib/backports/1.9.2/stdlib/matrix.rb, line 974
def **(other)
  case other
  when Integer
    x = self
    if other <= 0
      x = self.inverse
      return self.class.identity(self.column_size) 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 lib/backports/1.9.2/stdlib/matrix.rb, line 840
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_size == m.row_size and column_size == m.column_size

  rows = Array.new(row_size) {|i|
    Array.new(column_size) {|j|
      self[i, j] + m[i, j]
    }
  }
  new_matrix rows, column_size
end
-(m) click to toggle source

Matrix subtraction.

Matrix[[1,5], [4,2]] - Matrix[[9,3], [-4,1]]
  => -8  2
      8  1
# File lib/backports/1.9.2/stdlib/matrix.rb, line 867
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_size == m.row_size and column_size == m.column_size

  rows = Array.new(row_size) {|i|
    Array.new(column_size) {|j|
      self[i, j] - m[i, j]
    }
  }
  new_matrix rows, column_size
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 lib/backports/1.9.2/stdlib/matrix.rb, line 894
def /(other)
  case other
  when Numeric
    rows = @rows.collect {|row|
      row.collect {|e| e / other }
    }
    return new_matrix(rows, column_size)
  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 lib/backports/1.9.2/stdlib/matrix.rb, line 769
def ==(other)
  return false unless Matrix === other &&
                      column_size == other.column_size # 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 lib/backports/1.9.2/stdlib/matrix.rb, line 315
def [](i, j)
  @rows.fetch(i){return nil}[j]
end
Also aliased as: element, component
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 lib/backports/1.9.2/stdlib/matrix.rb, line 786
def clone
  new_matrix @rows.map(&:dup), column_size
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 lib/backports/1.9.2/stdlib/matrix.rb, line 1280
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
collect() { |e| ... } 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 lib/backports/1.9.2/stdlib/matrix.rb, line 381
def collect(&block) # :yield: e
  return to_enum(:collect) unless block_given?
  rows = @rows.collect{|row| row.collect(&block)}
  new_matrix rows, column_size
end
Also aliased as: map
column(j) { |e| ... } 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 lib/backports/1.9.2/stdlib/matrix.rb, line 358
def column(j) # :yield: e
  if block_given?
    return self if j >= column_size || j < -column_size
    row_size.times do |i|
      yield @rows[i][j]
    end
    self
  else
    return nil if j >= column_size || j < -column_size
    col = Array.new(row_size) {|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 lib/backports/1.9.2/stdlib/matrix.rb, line 1301
def column_vectors
  Array.new(column_size) {|i|
    column(i)
  }
end
component(i, j)
Alias for: []
conj()
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 lib/backports/1.9.2/stdlib/matrix.rb, line 1226
def conjugate
  collect(&:conjugate)
end
Also aliased as: conj
det()
Alias for: determinant
det_e()
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 lib/backports/1.9.2/stdlib/matrix.rb, line 1011
def determinant
  Matrix.Raise ErrDimensionMismatch unless square?
  m = @rows
  case row_size
    # 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 lib/backports/1.9.2/stdlib/matrix.rb, line 1093
def determinant_e
  warn "#{caller(1)[0]}: warning: Matrix#determinant_e is deprecated; use #determinant"
  rank
end
Also aliased as: det_e
diagonal?() click to toggle source

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

# File lib/backports/1.9.2/stdlib/matrix.rb, line 599
def diagonal?
  Matrix.Raise ErrDimensionMismatch unless square?
  each(:off_diagonal).all?(&:zero?)
end
each(which = :all) { |e| ... } click to toggle source

Yields all elements of the matrix, starting with those of the first row, or returns an Enumerator is 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 lib/backports/1.9.2/stdlib/matrix.rb, line 404
def each(which = :all) # :yield: e
  return to_enum(:each, which) unless block_given?
  last = column_size - 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_size.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_size].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
    Matrix.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) { |e, row, column| ... } 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 lib/backports/1.9.2/stdlib/matrix.rb, line 465
def each_with_index(which = :all) # :yield: e, row, column
  return to_enum(:each_with_index, which) unless block_given?
  last = column_size - 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_size.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_size].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
    Matrix.Raise ArgumentError, "expected #{which.inspect} to be one of :all, :diagonal, :off_diagonal, :lower, :strict_lower, :strict_upper or :upper"
  end
  self
end
eigen()
Alias for: eigensystem
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 lib/backports/1.9.2/stdlib/matrix.rb, line 1192
def eigensystem
  EigenvalueDecomposition.new(self)
end
Also aliased as: eigen
element(i, j)
Alias for: []
elements_to_f() click to toggle source
# File lib/backports/1.9.2/stdlib/matrix.rb, line 1314
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 lib/backports/1.9.2/stdlib/matrix.rb, line 1319
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 lib/backports/1.9.2/stdlib/matrix.rb, line 1324
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 lib/backports/1.9.2/stdlib/matrix.rb, line 608
def empty?
  column_size == 0 || row_size == 0
end
eql?(other) click to toggle source
# File lib/backports/1.9.2/stdlib/matrix.rb, line 775
def eql?(other)
  return false unless Matrix === other &&
                      column_size == other.column_size # necessary for empty matrices
  rows.eql? other.rows
end
find_index(*args)
Alias for: index
hash() click to toggle source

Returns a hash-code for the matrix.

# File lib/backports/1.9.2/stdlib/matrix.rb, line 793
def hash
  @rows.hash
end
hermitian?() click to toggle source

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

# File lib/backports/1.9.2/stdlib/matrix.rb, line 618
def hermitian?
  Matrix.Raise ErrDimensionMismatch unless square?
  each_with_index(:strict_upper).all? do |e, row, col|
    e == rows[col][row].conj
  end
end
imag()
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 lib/backports/1.9.2/stdlib/matrix.rb, line 1240
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 lib/backports/1.9.2/stdlib/matrix.rb, line 528
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 lib/backports/1.9.2/stdlib/matrix.rb, line 1349
def inspect
  if empty?
    "#{self.class}.empty(#{row_size}, #{column_size})"
  else
    "#{self.class}#{@rows.inspect}"
  end
end
inv()
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 lib/backports/1.9.2/stdlib/matrix.rb, line 914
def inverse
  Matrix.Raise ErrDimensionMismatch unless square?
  self.class.I(row_size).send(:inverse_from, self)
end
Also aliased as: inv
lower_triangular?() click to toggle source

Returns true is this is a lower triangular matrix.

# File lib/backports/1.9.2/stdlib/matrix.rb, line 628
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 == a * p      # => true
a.lup.solve([2, 5]) # => Vector[(1/1), (1/2)]
# File lib/backports/1.9.2/stdlib/matrix.rb, line 1207
def lup
  LUPDecomposition.new(self)
end
Also aliased as: lup_decomposition
lup_decomposition()
Alias for: lup
map()
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_size or column_size respectively.

# File lib/backports/1.9.2/stdlib/matrix.rb, line 558
def minor(*param)
  case param.size
  when 2
    row_range, col_range = param
    from_row = row_range.first
    from_row += row_size if from_row < 0
    to_row = row_range.end
    to_row += row_size 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_size if from_col < 0
    to_col = col_range.end
    to_col += column_size 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_size if from_row < 0
    from_col += column_size if from_col < 0
  else
    Matrix.Raise ArgumentError, param.inspect
  end

  return nil if from_row > row_size || from_col > column_size || from_row < 0 || from_col < 0
  rows = @rows[from_row, size_row].collect{|row|
    row[from_col, size_col]
  }
  new_matrix rows, [column_size - from_col, size_col].min
end
normal?() click to toggle source

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

# File lib/backports/1.9.2/stdlib/matrix.rb, line 636
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 is this is an orthogonal matrix Raises an error if matrix is not square.

# File lib/backports/1.9.2/stdlib/matrix.rb, line 654
def orthogonal?
  Matrix.Raise ErrDimensionMismatch unless square?
  rows.each_with_index do |row, i|
    column_size.times do |j|
      s = 0
      row_size.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 is this is a permutation matrix Raises an error if matrix is not square.

# File lib/backports/1.9.2/stdlib/matrix.rb, line 672
def permutation?
  Matrix.Raise ErrDimensionMismatch unless square?
  cols = Array.new(column_size)
  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 lib/backports/1.9.2/stdlib/matrix.rb, line 1108
def rank
  # We currently use Bareiss' multistep integer-preserving gaussian elimination
  # (see comments on determinant)
  a = to_a
  last_column = column_size - 1
  last_row = row_size - 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 lib/backports/1.9.2/stdlib/matrix.rb, line 1139
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 lib/backports/1.9.2/stdlib/matrix.rb, line 1254
def real
  collect(&:real)
end
real?() click to toggle source

Returns true if all entries of the matrix are real.

# File lib/backports/1.9.2/stdlib/matrix.rb, line 693
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 lib/backports/1.9.2/stdlib/matrix.rb, line 1264
def rect
  [real, imag]
end
Also aliased as: rectangular
rectangular()
Alias for: rect
regular?() click to toggle source

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

# File lib/backports/1.9.2/stdlib/matrix.rb, line 700
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 lib/backports/1.9.2/stdlib/matrix.rb, line 1147
def round(ndigits=0)
  map{|e| e.round(ndigits)}
end
row(i) { |e| ... } 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 lib/backports/1.9.2/stdlib/matrix.rb, line 344
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_size() click to toggle source

Returns the number of rows.

# File lib/backports/1.9.2/stdlib/matrix.rb, line 331
def row_size
  @rows.size
end
row_vectors() click to toggle source

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

# File lib/backports/1.9.2/stdlib/matrix.rb, line 1292
def row_vectors
  Array.new(row_size) {|i|
    row(i)
  }
end
singular?() click to toggle source

Returns true is this is a singular matrix.

# File lib/backports/1.9.2/stdlib/matrix.rb, line 707
def singular?
  determinant == 0
end
square?() click to toggle source

Returns true is this is a square matrix.

# File lib/backports/1.9.2/stdlib/matrix.rb, line 714
def square?
  column_size == row_size
end
symmetric?() click to toggle source

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

# File lib/backports/1.9.2/stdlib/matrix.rb, line 722
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()
Alias for: transpose
to_a() click to toggle source

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

# File lib/backports/1.9.2/stdlib/matrix.rb, line 1310
def to_a
  @rows.collect(&:dup)
end
to_s() click to toggle source

Overrides Object#to_s

# File lib/backports/1.9.2/stdlib/matrix.rb, line 1336
def to_s
  if empty?
    "#{self.class}.empty(#{row_size}, #{column_size})"
  else
    "#{self.class}[" + @rows.collect{|row|
      "[" + row.collect{|e| e.to_s}.join(", ") + "]"
    }.join(", ")+"]"
  end
end
tr()
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 lib/backports/1.9.2/stdlib/matrix.rb, line 1156
def trace
  Matrix.Raise ErrDimensionMismatch unless square?
  (0...column_size).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 lib/backports/1.9.2/stdlib/matrix.rb, line 1174
def transpose
  return self.class.empty(column_size, 0) if row_size.zero?
  new_matrix @rows.transpose, row_size
end
Also aliased as: t
unitary?() click to toggle source

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

# File lib/backports/1.9.2/stdlib/matrix.rb, line 734
def unitary?
  Matrix.Raise ErrDimensionMismatch unless square?
  rows.each_with_index do |row, i|
    column_size.times do |j|
      s = 0
      row_size.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 is this is an upper triangular matrix.

# File lib/backports/1.9.2/stdlib/matrix.rb, line 751
def upper_triangular?
  each(:strict_lower).all?(&:zero?)
end
zero?() click to toggle source

Returns true is this is a matrix with only zero elements

# File lib/backports/1.9.2/stdlib/matrix.rb, line 758
def zero?
  all?(&:zero?)
end

Private Instance Methods

[]=(i, j, v) click to toggle source
# File lib/backports/1.9.2/stdlib/matrix.rb, line 321
def []=(i, j, v)
  @rows[i][j] = v
end
Also aliased as: set_element, set_component
determinant_bareiss() click to toggle source

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.

# File lib/backports/1.9.2/stdlib/matrix.rb, line 1062
def determinant_bareiss
  size = row_size
  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
set_component(i, j, v)
Alias for: []=
set_element(i, j, v)
Alias for: []=