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+# encoding: utf-8
+#
+# = matrix.rb
+#
+# An implementation of Matrix and Vector classes.
+#
+# See classes Matrix and Vector for documentation.
+#
+# Current Maintainer:: Marc-André Lafortune
+# Original Author:: Keiju ISHITSUKA
+# Original Documentation:: Gavin Sinclair (sourced from <i>Ruby in a Nutshell</i> (Matsumoto, O'Reilly))
+##
+
+require "e2mmap.rb"
+
+module ExceptionForMatrix # :nodoc:
+  extend Exception2MessageMapper
+  def_e2message(TypeError, "wrong argument type %s (expected %s)")
+  def_e2message(ArgumentError, "Wrong # of arguments(%d for %d)")
+
+  def_exception("ErrDimensionMismatch", "\#{self.name} dimension mismatch")
+  def_exception("ErrNotRegular", "Not Regular Matrix")
+  def_exception("ErrOperationNotDefined", "Operation(%s) can\\'t be defined: %s op %s")
+  def_exception("ErrOperationNotImplemented", "Sorry, Operation(%s) not implemented: %s op %s")
+end
+
+#
+# The +Matrix+ class represents a mathematical matrix. It provides methods for creating
+# matrices, operating on them arithmetically and algebraically,
+# and determining their mathematical properties (trace, rank, inverse, determinant).
+#
+# == Method Catalogue
+#
+# To create a matrix:
+# * Matrix[*rows]
+# * Matrix.[](*rows)
+# * Matrix.rows(rows, copy = true)
+# * Matrix.columns(columns)
+# * Matrix.build(row_count, column_count, &block)
+# * Matrix.diagonal(*values)
+# * Matrix.scalar(n, value)
+# * Matrix.identity(n)
+# * Matrix.unit(n)
+# * Matrix.I(n)
+# * Matrix.zero(n)
+# * Matrix.row_vector(row)
+# * Matrix.column_vector(column)
+# * Matrix.hstack(*matrices)
+# * Matrix.vstack(*matrices)
+#
+# To access Matrix elements/columns/rows/submatrices/properties:
+# * #[](i, j)
+# * #row_count (row_size)
+# * #column_count (column_size)
+# * #row(i)
+# * #column(j)
+# * #collect
+# * #map
+# * #each
+# * #each_with_index
+# * #find_index
+# * #minor(*param)
+# * #first_minor(row, column)
+# * #cofactor(row, column)
+# * #adjugate
+# * #laplace_expansion(row_or_column: num)
+# * #cofactor_expansion(row_or_column: num)
+#
+# Properties of a matrix:
+# * #diagonal?
+# * #empty?
+# * #hermitian?
+# * #lower_triangular?
+# * #normal?
+# * #orthogonal?
+# * #permutation?
+# * #real?
+# * #regular?
+# * #singular?
+# * #square?
+# * #symmetric?
+# * #unitary?
+# * #upper_triangular?
+# * #zero?
+#
+# Matrix arithmetic:
+# * #*(m)
+# * #+(m)
+# * #-(m)
+# * #/(m)
+# * #inverse
+# * #inv
+# * #**
+# * #+@
+# * #-@
+#
+# Matrix functions:
+# * #determinant
+# * #det
+# * #hstack(*matrices)
+# * #rank
+# * #round
+# * #trace
+# * #tr
+# * #transpose
+# * #t
+# * #vstack(*matrices)
+#
+# Matrix decompositions:
+# * #eigen
+# * #eigensystem
+# * #lup
+# * #lup_decomposition
+#
+# Complex arithmetic:
+# * conj
+# * conjugate
+# * imag
+# * imaginary
+# * real
+# * rect
+# * rectangular
+#
+# Conversion to other data types:
+# * #coerce(other)
+# * #row_vectors
+# * #column_vectors
+# * #to_a
+#
+# String representations:
+# * #to_s
+# * #inspect
+#
+class Matrix
+  include Enumerable
+  include ExceptionForMatrix
+  autoload :EigenvalueDecomposition, "matrix/eigenvalue_decomposition"
+  autoload :LUPDecomposition, "matrix/lup_decomposition"
+
+  # instance creations
+  private_class_method :new
+  attr_reader :rows
+  protected :rows
+
+  #
+  # Creates a matrix where each argument is a row.
+  #   Matrix[ [25, 93], [-1, 66] ]
+  #      =>  25 93
+  #          -1 66
+  #
+  def Matrix.[](*rows)
+    rows(rows, false)
+  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
+  #
+  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 using +columns+ as an array of column vectors.
+  #   Matrix.columns([[25, 93], [-1, 66]])
+  #      =>  25 -1
+  #          93 66
+  #
+  def Matrix.columns(columns)
+    rows(columns, false).transpose
+  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
+  #
+  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 where the diagonal elements are composed of +values+.
+  #   Matrix.diagonal(9, 5, -3)
+  #     =>  9  0  0
+  #         0  5  0
+  #         0  0 -3
+  #
+  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 an +n+ by +n+ diagonal matrix where each diagonal element is
+  # +value+.
+  #   Matrix.scalar(2, 5)
+  #     => 5 0
+  #        0 5
+  #
+  def Matrix.scalar(n, value)
+    diagonal(*Array.new(n, value))
+  end
+
+  #
+  # Creates an +n+ by +n+ identity matrix.
+  #   Matrix.identity(2)
+  #     => 1 0
+  #        0 1
+  #
+  def Matrix.identity(n)
+    scalar(n, 1)
+  end
+  class << Matrix
+    alias unit identity
+    alias I identity
+  end
+
+  #
+  # Creates a zero matrix.
+  #   Matrix.zero(2)
+  #     => 0 0
+  #        0 0
+  #
+  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 single-row matrix where the values of that row are as given in
+  # +row+.
+  #   Matrix.row_vector([4,5,6])
+  #     => 4 5 6
+  #
+  def Matrix.row_vector(row)
+    row = convert_to_array(row)
+    new [row]
+  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
+  #
+  def Matrix.column_vector(column)
+    column = convert_to_array(column)
+    new [column].transpose, 1
+  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]]
+  #
+  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
+
+  #
+  # 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]]
+  #
+  def Matrix.vstack(x, *matrices)
+    raise TypeError, "Expected a Matrix, got a #{x.class}" unless x.is_a?(Matrix)
+    result = x.send(:rows).map(&:dup)
+    matrices.each do |m|
+      raise TypeError, "Expected a Matrix, got a #{m.class}" unless m.is_a?(Matrix)
+      if m.column_count != x.column_count
+        raise ErrDimensionMismatch, "The given matrices must have #{x.column_count} columns, but one has #{m.column_count}"
+      end
+      result.concat(m.send(:rows))
+    end
+    new result, x.column_count
+  end
+
+
+  #
+  # 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]]
+  #
+  def Matrix.hstack(x, *matrices)
+    raise TypeError, "Expected a Matrix, got a #{x.class}" unless x.is_a?(Matrix)
+    result = x.send(:rows).map(&:dup)
+    total_column_count = x.column_count
+    matrices.each do |m|
+      raise TypeError, "Expected a Matrix, got a #{m.class}" unless m.is_a?(Matrix)
+      if m.row_count != x.row_count
+        raise ErrDimensionMismatch, "The given matrices must have #{x.row_count} rows, but one has #{m.row_count}"
+      end
+      result.each_with_index do |row, i|
+        row.concat m.send(:rows)[i]
+      end
+      total_column_count += m.column_count
+    end
+    new result, total_column_count
+  end
+
+  #
+  # Matrix.new is private; use Matrix.rows, columns, [], etc... to create.
+  #
+  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
+
+  def new_matrix(rows, column_count = rows[0].size) # :nodoc:
+    self.class.send(:new, rows, column_count) # bypass privacy of Matrix.new
+  end
+  private :new_matrix
+
+  #
+  # Returns element (+i+,+j+) of the matrix.  That is: row +i+, column +j+.
+  #
+  def [](i, j)
+    @rows.fetch(i){return nil}[j]
+  end
+  alias element []
+  alias component []
+
+  def []=(i, j, v)
+    @rows[i][j] = v
+  end
+  alias set_element []=
+  alias set_component []=
+  private :[]=, :set_element, :set_component
+
+  #
+  # Returns the number of rows.
+  #
+  def row_count
+    @rows.size
+  end
+
+  alias_method :row_size, :row_count
+  #
+  # Returns the number of columns.
+  #
+  attr_reader :column_count
+  alias_method :column_size, :column_count
+
+  #
+  # 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.
+  #
+  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 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.
+  #
+  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 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
+  #
+  def collect(&block) # :yield: e
+    return to_enum(:collect) unless block_given?
+    rows = @rows.collect{|row| row.collect(&block)}
+    new_matrix rows, column_count
+  end
+  alias map collect
+
+  #
+  # 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]
+  #
+  def each(which = :all) # :yield: e
+    return to_enum :each, which unless block_given?
+    last = column_count - 1
+    case which
+    when :all
+      block = Proc.new
+      @rows.each do |row|
+        row.each(&block)
+      end
+    when :diagonal
+      @rows.each_with_index do |row, row_index|
+        yield row.fetch(row_index){return self}
+      end
+    when :off_diagonal
+      @rows.each_with_index do |row, row_index|
+        column_count.times do |col_index|
+          yield row[col_index] unless row_index == col_index
+        end
+      end
+    when :lower
+      @rows.each_with_index do |row, row_index|
+        0.upto([row_index, last].min) do |col_index|
+          yield row[col_index]
+        end
+      end
+    when :strict_lower
+      @rows.each_with_index do |row, row_index|
+        [row_index, column_count].min.times do |col_index|
+          yield row[col_index]
+        end
+      end
+    when :strict_upper
+      @rows.each_with_index do |row, row_index|
+        (row_index+1).upto(last) do |col_index|
+          yield row[col_index]
+        end
+      end
+    when :upper
+      @rows.each_with_index do |row, row_index|
+        row_index.upto(last) do |col_index|
+          yield row[col_index]
+        end
+      end
+    else
+      raise ArgumentError, "expected #{which.inspect} to be one of :all, :diagonal, :off_diagonal, :lower, :strict_lower, :strict_upper or :upper"
+    end
+    self
+  end
+
+  #
+  # 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
+  #
+  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
+
+  SELECTORS = {all: true, diagonal: true, off_diagonal: true, lower: true, strict_lower: true, strict_upper: true, upper: true}.freeze
+  #
+  # :call-seq:
+  #   index(value, selector = :all) -> [row, column]
+  #   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]
+  #
+  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
+  alias_method :find_index, :index
+
+  #
+  # 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.
+  #
+  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 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
+  #
+  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 (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
+  #
+  def cofactor(row, column)
+    raise RuntimeError, "cofactor of empty matrix is not defined" if empty?
+    Matrix.Raise ErrDimensionMismatch unless square?
+
+    det_of_minor = first_minor(row, column).determinant
+    det_of_minor * (-1) ** (row + column)
+  end
+
+  #
+  # Returns the adjugate of the matrix.
+  #
+  #   Matrix[ [7,6],[3,9] ].adjugate
+  #     => 9 -6
+  #        -3 7
+  #
+  def adjugate
+    Matrix.Raise ErrDimensionMismatch unless square?
+    Matrix.build(row_count, column_count) do |row, column|
+      cofactor(column, row)
+    end
+  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]
+  #
+  #
+  def laplace_expansion(row: nil, column: nil)
+    num = row || column
+
+    if !num || (row && column)
+      raise ArgumentError, "exactly one the row or column arguments must be specified"
+    end
+
+    Matrix.Raise ErrDimensionMismatch unless square?
+    raise RuntimeError, "laplace_expansion of empty matrix is not defined" if empty?
+
+    unless 0 <= num && num < row_count
+      raise ArgumentError, "invalid num (#{num.inspect} for 0..#{row_count - 1})"
+    end
+
+    send(row ? :row : :column, num).map.with_index { |e, k|
+      e * cofactor(*(row ? [num, k] : [k,num]))
+    }.inject(:+)
+  end
+  alias_method :cofactor_expansion, :laplace_expansion
+
+
+  #--
+  # TESTING -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
+  #++
+
+  #
+  # Returns +true+ if this is a diagonal matrix.
+  # Raises an error if matrix is not square.
+  #
+  def diagonal?
+    Matrix.Raise ErrDimensionMismatch unless square?
+    each(:off_diagonal).all?(&:zero?)
+  end
+
+  #
+  # Returns +true+ if this is an empty matrix, i.e. if the number of rows
+  # or the number of columns is 0.
+  #
+  def empty?
+    column_count == 0 || row_count == 0
+  end
+
+  #
+  # Returns +true+ if this is an hermitian matrix.
+  # Raises an error if matrix is not square.
+  #
+  def hermitian?
+    Matrix.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 a lower triangular matrix.
+  #
+  def lower_triangular?
+    each(:strict_upper).all?(&:zero?)
+  end
+
+  #
+  # Returns +true+ if this is a normal matrix.
+  # Raises an error if matrix is not square.
+  #
+  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
+
+  #
+  # Returns +true+ if this is an orthogonal matrix
+  # Raises an error if matrix is not square.
+  #
+  def orthogonal?
+    Matrix.Raise ErrDimensionMismatch unless square?
+    rows.each_with_index do |row, i|
+      column_count.times do |j|
+        s = 0
+        row_count.times do |k|
+          s += row[k] * rows[k][j]
+        end
+        return false unless s == (i == j ? 1 : 0)
+      end
+    end
+    true
+  end
+
+  #
+  # Returns +true+ if this is a permutation matrix
+  # Raises an error if matrix is not square.
+  #
+  def permutation?
+    Matrix.Raise ErrDimensionMismatch unless square?
+    cols = Array.new(column_count)
+    rows.each_with_index do |row, i|
+      found = false
+      row.each_with_index do |e, j|
+        if e == 1
+          return false if found || cols[j]
+          found = cols[j] = true
+        elsif e != 0
+          return false
+        end
+      end
+      return false unless found
+    end
+    true
+  end
+
+  #
+  # Returns +true+ if all entries of the matrix are real.
+  #
+  def real?
+    all?(&:real?)
+  end
+
+  #
+  # Returns +true+ if this is a regular (i.e. non-singular) matrix.
+  #
+  def regular?
+    not singular?
+  end
+
+  #
+  # Returns +true+ if this is a singular matrix.
+  #
+  def singular?
+    determinant == 0
+  end
+
+  #
+  # Returns +true+ if this is a square matrix.
+  #
+  def square?
+    column_count == row_count
+  end
+
+  #
+  # Returns +true+ if this is a symmetric matrix.
+  # Raises an error if matrix is not square.
+  #
+  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
+
+  #
+  # Returns +true+ if this is a unitary matrix
+  # Raises an error if matrix is not square.
+  #
+  def unitary?
+    Matrix.Raise ErrDimensionMismatch unless square?
+    rows.each_with_index do |row, i|
+      column_count.times do |j|
+        s = 0
+        row_count.times do |k|
+          s += row[k].conj * rows[k][j]
+        end
+        return false unless s == (i == j ? 1 : 0)
+      end
+    end
+    true
+  end
+
+  #
+  # Returns +true+ if this is an upper triangular matrix.
+  #
+  def upper_triangular?
+    each(:strict_lower).all?(&:zero?)
+  end
+
+  #
+  # Returns +true+ if this is a matrix with only zero elements
+  #
+  def zero?
+    all?(&:zero?)
+  end
+
+  #--
+  # OBJECT METHODS -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
+  #++
+
+  #
+  # Returns +true+ if and only if the two matrices contain equal elements.
+  #
+  def ==(other)
+    return false unless Matrix === other &&
+                        column_count == other.column_count # necessary for empty matrices
+    rows == other.rows
+  end
+
+  def eql?(other)
+    return false unless Matrix === other &&
+                        column_count == other.column_count # necessary for empty matrices
+    rows.eql? other.rows
+  end
+
+  #
+  # 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.
+  #
+  def clone
+    new_matrix @rows.map(&:dup), column_count
+  end
+
+  #
+  # Returns a hash-code for the matrix.
+  #
+  def hash
+    @rows.hash
+  end
+
+  #--
+  # ARITHMETIC -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
+  #++
+
+  #
+  # Matrix multiplication.
+  #   Matrix[[2,4], [6,8]] * Matrix.identity(2)
+  #     => 2 4
+  #        6 8
+  #
+  def *(m) # m is matrix or vector or number
+    case(m)
+    when Numeric
+      rows = @rows.collect {|row|
+        row.collect {|e| e * m }
+      }
+      return new_matrix rows, column_count
+    when Vector
+      m = self.class.column_vector(m)
+      r = self * m
+      return r.column(0)
+    when Matrix
+      Matrix.Raise ErrDimensionMismatch if column_count != m.row_count
+
+      rows = Array.new(row_count) {|i|
+        Array.new(m.column_count) {|j|
+          (0 ... column_count).inject(0) do |vij, k|
+            vij + self[i, k] * m[k, j]
+          end
+        }
+      }
+      return new_matrix rows, m.column_count
+    else
+      return apply_through_coercion(m, __method__)
+    end
+  end
+
+  #
+  # Matrix addition.
+  #   Matrix.scalar(2,5) + Matrix[[1,0], [-4,7]]
+  #     =>  6  0
+  #        -4 12
+  #
+  def +(m)
+    case m
+    when Numeric
+      Matrix.Raise ErrOperationNotDefined, "+", self.class, m.class
+    when Vector
+      m = self.class.column_vector(m)
+    when Matrix
+    else
+      return apply_through_coercion(m, __method__)
+    end
+
+    Matrix.Raise ErrDimensionMismatch unless row_count == m.row_count && column_count == m.column_count
+
+    rows = Array.new(row_count) {|i|
+      Array.new(column_count) {|j|
+        self[i, j] + m[i, j]
+      }
+    }
+    new_matrix rows, column_count
+  end
+
+  #
+  # Matrix subtraction.
+  #   Matrix[[1,5], [4,2]] - Matrix[[9,3], [-4,1]]
+  #     => -8  2
+  #         8  1
+  #
+  def -(m)
+    case m
+    when Numeric
+      Matrix.Raise ErrOperationNotDefined, "-", self.class, m.class
+    when Vector
+      m = self.class.column_vector(m)
+    when Matrix
+    else
+      return apply_through_coercion(m, __method__)
+    end
+
+    Matrix.Raise ErrDimensionMismatch unless row_count == m.row_count && column_count == m.column_count
+
+    rows = Array.new(row_count) {|i|
+      Array.new(column_count) {|j|
+        self[i, j] - m[i, j]
+      }
+    }
+    new_matrix rows, column_count
+  end
+
+  #
+  # Matrix division (multiplication by the inverse).
+  #   Matrix[[7,6], [3,9]] / Matrix[[2,9], [3,1]]
+  #     => -7  1
+  #        -3 -6
+  #
+  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
+
+  #
+  # Returns the inverse of the matrix.
+  #   Matrix[[-1, -1], [0, -1]].inverse
+  #     => -1  1
+  #         0 -1
+  #
+  def inverse
+    Matrix.Raise ErrDimensionMismatch unless square?
+    self.class.I(row_count).send(:inverse_from, self)
+  end
+  alias inv inverse
+
+  def inverse_from(src) # :nodoc:
+    last = row_count - 1
+    a = src.to_a
+
+    0.upto(last) do |k|
+      i = k
+      akk = a[k][k].abs
+      (k+1).upto(last) do |j|
+        v = a[j][k].abs
+        if v > akk
+          i = j
+          akk = v
+        end
+      end
+      Matrix.Raise ErrNotRegular if akk == 0
+      if i != k
+        a[i], a[k] = a[k], a[i]
+        @rows[i], @rows[k] = @rows[k], @rows[i]
+      end
+      akk = a[k][k]
+
+      0.upto(last) do |ii|
+        next if ii == k
+        q = a[ii][k].quo(akk)
+        a[ii][k] = 0
+
+        (k + 1).upto(last) do |j|
+          a[ii][j] -= a[k][j] * q
+        end
+        0.upto(last) do |j|
+          @rows[ii][j] -= @rows[k][j] * q
+        end
+      end
+
+      (k+1).upto(last) do |j|
+        a[k][j] = a[k][j].quo(akk)
+      end
+      0.upto(last) do |j|
+        @rows[k][j] = @rows[k][j].quo(akk)
+      end
+    end
+    self
+  end
+  private :inverse_from
+
+  #
+  # 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
+  #
+  def ** (other)
+    case other
+    when Integer
+      x = self
+      if other <= 0
+        x = self.inverse
+        return self.class.identity(self.column_count) if other == 0
+        other = -other
+      end
+      z = nil
+      loop do
+        z = z ? z * x : x if other[0] == 1
+        return z if (other >>= 1).zero?
+        x *= x
+      end
+    when Numeric
+      v, d, v_inv = eigensystem
+      v * self.class.diagonal(*d.each(:diagonal).map{|e| e ** other}) * v_inv
+    else
+      Matrix.Raise ErrOperationNotDefined, "**", self.class, other.class
+    end
+  end
+
+  def +@
+    self
+  end
+
+  def -@
+    collect {|e| -e }
+  end
+
+  #--
+  # MATRIX FUNCTIONS -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
+  #++
+
+  #
+  # 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
+  #
+  def determinant
+    Matrix.Raise ErrDimensionMismatch unless square?
+    m = @rows
+    case row_count
+      # Up to 4x4, give result using Laplacian expansion by minors.
+      # This will typically be faster, as well as giving good results
+      # in case of Floats
+    when 0
+      +1
+    when 1
+      + m[0][0]
+    when 2
+      + m[0][0] * m[1][1] - m[0][1] * m[1][0]
+    when 3
+      m0, m1, m2 = m
+      + m0[0] * m1[1] * m2[2] - m0[0] * m1[2] * m2[1] \
+      - m0[1] * m1[0] * m2[2] + m0[1] * m1[2] * m2[0] \
+      + m0[2] * m1[0] * m2[1] - m0[2] * m1[1] * m2[0]
+    when 4
+      m0, m1, m2, m3 = m
+      + m0[0] * m1[1] * m2[2] * m3[3] - m0[0] * m1[1] * m2[3] * m3[2] \
+      - m0[0] * m1[2] * m2[1] * m3[3] + m0[0] * m1[2] * m2[3] * m3[1] \
+      + m0[0] * m1[3] * m2[1] * m3[2] - m0[0] * m1[3] * m2[2] * m3[1] \
+      - m0[1] * m1[0] * m2[2] * m3[3] + m0[1] * m1[0] * m2[3] * m3[2] \
+      + m0[1] * m1[2] * m2[0] * m3[3] - m0[1] * m1[2] * m2[3] * m3[0] \
+      - m0[1] * m1[3] * m2[0] * m3[2] + m0[1] * m1[3] * m2[2] * m3[0] \
+      + m0[2] * m1[0] * m2[1] * m3[3] - m0[2] * m1[0] * m2[3] * m3[1] \
+      - m0[2] * m1[1] * m2[0] * m3[3] + m0[2] * m1[1] * m2[3] * m3[0] \
+      + m0[2] * m1[3] * m2[0] * m3[1] - m0[2] * m1[3] * m2[1] * m3[0] \
+      - m0[3] * m1[0] * m2[1] * m3[2] + m0[3] * m1[0] * m2[2] * m3[1] \
+      + m0[3] * m1[1] * m2[0] * m3[2] - m0[3] * m1[1] * m2[2] * m3[0] \
+      - m0[3] * m1[2] * m2[0] * m3[1] + m0[3] * m1[2] * m2[1] * m3[0]
+    else
+      # For bigger matrices, use an efficient and general algorithm.
+      # Currently, we use the Gauss-Bareiss algorithm
+      determinant_bareiss
+    end
+  end
+  alias_method :det, :determinant
+
+  #
+  # 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.
+  #
+  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 :determinant_bareiss
+
+  #
+  # deprecated; use Matrix#determinant
+  #
+  def determinant_e
+    warn "#{caller(1)[0]}: warning: Matrix#determinant_e is deprecated; use #determinant"
+    determinant
+  end
+  alias det_e determinant_e
+
+  #
+  # 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]]
+  #
+  def hstack(*matrices)
+    self.class.hstack(self, *matrices)
+  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
+  #
+  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
+
+  #
+  # deprecated; use Matrix#rank
+  #
+  def rank_e
+    warn "#{caller(1)[0]}: warning: Matrix#rank_e is deprecated; use #rank"
+    rank
+  end
+
+  # Returns a matrix with entries rounded to the given precision
+  # (see Float#round)
+  #
+  def round(ndigits=0)
+    map{|e| e.round(ndigits)}
+  end
+
+  #
+  # Returns the trace (sum of diagonal elements) of the matrix.
+  #   Matrix[[7,6], [3,9]].trace
+  #     => 16
+  #
+  def trace
+    Matrix.Raise ErrDimensionMismatch unless square?
+    (0...column_count).inject(0) do |tr, i|
+      tr + @rows[i][i]
+    end
+  end
+  alias tr trace
+
+  #
+  # 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
+  #
+  def transpose
+    return self.class.empty(column_count, 0) if row_count.zero?
+    new_matrix @rows.transpose, row_count
+  end
+  alias t transpose
+
+  #
+  # 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]]
+  #
+  def vstack(*matrices)
+    self.class.vstack(self, *matrices)
+  end
+
+  #--
+  # DECOMPOSITIONS -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
+  #++
+
+  #
+  # 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
+  #
+  def eigensystem
+    EigenvalueDecomposition.new(self)
+  end
+  alias eigen eigensystem
+
+  #
+  # 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)]
+  #
+  def lup
+    LUPDecomposition.new(self)
+  end
+  alias lup_decomposition lup
+
+  #--
+  # COMPLEX ARITHMETIC -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
+  #++
+
+  #
+  # 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
+  #
+  def conjugate
+    collect(&:conjugate)
+  end
+  alias conj conjugate
+
+  #
+  # 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
+  #
+  def imaginary
+    collect(&:imaginary)
+  end
+  alias imag imaginary
+
+  #
+  # 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
+  #
+  def real
+    collect(&:real)
+  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
+  #
+  def rect
+    [real, imag]
+  end
+  alias rectangular rect
+
+  #--
+  # CONVERTING -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
+  #++
+
+  #
+  # 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.
+  #
+  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
+
+  #
+  # Returns an array of the row vectors of the matrix.  See Vector.
+  #
+  def row_vectors
+    Array.new(row_count) {|i|
+      row(i)
+    }
+  end
+
+  #
+  # Returns an array of the column vectors of the matrix.  See Vector.
+  #
+  def column_vectors
+    Array.new(column_count) {|i|
+      column(i)
+    }
+  end
+
+  #
+  # Returns an array of arrays that describe the rows of the matrix.
+  #
+  def to_a
+    @rows.collect(&:dup)
+  end
+
+  def elements_to_f
+    warn "#{caller(1)[0]}: warning: Matrix#elements_to_f is deprecated, use map(&:to_f)"
+    map(&:to_f)
+  end
+
+  def elements_to_i
+    warn "#{caller(1)[0]}: warning: Matrix#elements_to_i is deprecated, use map(&:to_i)"
+    map(&:to_i)
+  end
+
+  def elements_to_r
+    warn "#{caller(1)[0]}: warning: Matrix#elements_to_r is deprecated, use map(&:to_r)"
+    map(&:to_r)
+  end
+
+  #--
+  # PRINTING -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
+  #++
+
+  #
+  # Overrides Object#to_s
+  #
+  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#inspect
+  #
+  def inspect
+    if empty?
+      "#{self.class}.empty(#{row_count}, #{column_count})"
+    else
+      "#{self.class}#{@rows.inspect}"
+    end
+  end
+
+  # Private helper modules
+
+  module ConversionHelper # :nodoc:
+    #
+    # Converts the obj to an Array. If copy is set to true
+    # a copy of obj will be made if necessary.
+    #
+    def convert_to_array(obj, copy = false) # :nodoc:
+      case obj
+      when Array
+        copy ? obj.dup : obj
+      when Vector
+        obj.to_a
+      else
+        begin
+          converted = obj.to_ary
+        rescue Exception => e
+          raise TypeError, "can't convert #{obj.class} into an Array (#{e.message})"
+        end
+        raise TypeError, "#{obj.class}#to_ary should return an Array" unless converted.is_a? Array
+        converted
+      end
+    end
+    private :convert_to_array
+  end
+
+  extend ConversionHelper
+
+  module CoercionHelper # :nodoc:
+    #
+    # Applies the operator +oper+ with argument +obj+
+    # through coercion of +obj+
+    #
+    def apply_through_coercion(obj, oper)
+      coercion = obj.coerce(self)
+      raise TypeError unless coercion.is_a?(Array) && coercion.length == 2
+      coercion[0].public_send(oper, coercion[1])
+    rescue
+      raise TypeError, "#{obj.inspect} can't be coerced into #{self.class}"
+    end
+    private :apply_through_coercion
+
+    #
+    # Helper method to coerce a value into a specific class.
+    # Raises a TypeError if the coercion fails or the returned value
+    # is not of the right class.
+    # (from Rubinius)
+    #
+    def self.coerce_to(obj, cls, meth) # :nodoc:
+      return obj if obj.kind_of?(cls)
+
+      begin
+        ret = obj.__send__(meth)
+      rescue Exception => e
+        raise TypeError, "Coercion error: #{obj.inspect}.#{meth} => #{cls} failed:\n" \
+                         "(#{e.message})"
+      end
+      raise TypeError, "Coercion error: obj.#{meth} did NOT return a #{cls} (was #{ret.class})" unless ret.kind_of? cls
+      ret
+    end
+
+    def self.coerce_to_int(obj)
+      coerce_to(obj, Integer, :to_int)
+    end
+  end
+
+  include CoercionHelper
+
+  # Private CLASS
+
+  class Scalar < Numeric # :nodoc:
+    include ExceptionForMatrix
+    include CoercionHelper
+
+    def initialize(value)
+      @value = value
+    end
+
+    # ARITHMETIC
+    def +(other)
+      case other
+      when Numeric
+        Scalar.new(@value + other)
+      when Vector, Matrix
+        Scalar.Raise ErrOperationNotDefined, "+", @value.class, other.class
+      else
+        apply_through_coercion(other, __method__)
+      end
+    end
+
+    def -(other)
+      case other
+      when Numeric
+        Scalar.new(@value - other)
+      when Vector, Matrix
+        Scalar.Raise ErrOperationNotDefined, "-", @value.class, other.class
+      else
+        apply_through_coercion(other, __method__)
+      end
+    end
+
+    def *(other)
+      case other
+      when Numeric
+        Scalar.new(@value * other)
+      when Vector, Matrix
+        other.collect{|e| @value * e}
+      else
+        apply_through_coercion(other, __method__)
+      end
+    end
+
+    def / (other)
+      case other
+      when Numeric
+        Scalar.new(@value / other)
+      when Vector
+        Scalar.Raise ErrOperationNotDefined, "/", @value.class, other.class
+      when Matrix
+        self * other.inverse
+      else
+        apply_through_coercion(other, __method__)
+      end
+    end
+
+    def ** (other)
+      case other
+      when Numeric
+        Scalar.new(@value ** other)
+      when Vector
+        Scalar.Raise ErrOperationNotDefined, "**", @value.class, other.class
+      when Matrix
+        #other.powered_by(self)
+        Scalar.Raise ErrOperationNotImplemented, "**", @value.class, other.class
+      else
+        apply_through_coercion(other, __method__)
+      end
+    end
+  end
+
+end
+
+
+#
+# The +Vector+ class represents a mathematical vector, which is useful in its own right, and
+# also constitutes a row or column of a Matrix.
+#
+# == Method Catalogue
+#
+# To create a Vector:
+# * Vector.[](*array)
+# * Vector.elements(array, copy = true)
+# * Vector.basis(size: n, index: k)
+#
+# To access elements:
+# * #[](i)
+#
+# To enumerate the elements:
+# * #each2(v)
+# * #collect2(v)
+#
+# Properties of vectors:
+# * #angle_with(v)
+# * Vector.independent?(*vs)
+# * #independent?(*vs)
+#
+# Vector arithmetic:
+# * #*(x) "is matrix or number"
+# * #+(v)
+# * #-(v)
+# * #+@
+# * #-@
+#
+# Vector functions:
+# * #inner_product(v), dot(v)
+# * #cross_product(v), cross(v)
+# * #collect
+# * #magnitude
+# * #map
+# * #map2(v)
+# * #norm
+# * #normalize
+# * #r
+# * #size
+#
+# Conversion to other data types:
+# * #covector
+# * #to_a
+# * #coerce(other)
+#
+# String representations:
+# * #to_s
+# * #inspect
+#
+class Vector
+  include ExceptionForMatrix
+  include Enumerable
+  include Matrix::CoercionHelper
+  extend Matrix::ConversionHelper
+  #INSTANCE CREATION
+
+  private_class_method :new
+  attr_reader :elements
+  protected :elements
+
+  #
+  # Creates a Vector from a list of elements.
+  #   Vector[7, 4, ...]
+  #
+  def Vector.[](*array)
+    new convert_to_array(array, false)
+  end
+
+  #
+  # Creates a vector from an Array.  The optional second argument specifies
+  # whether the array itself or a copy is used internally.
+  #
+  def Vector.elements(array, copy = true)
+    new convert_to_array(array, copy)
+  end
+
+  #
+  # Returns a standard basis +n+-vector, where k is the index.
+  #
+  #    Vector.basis(size:, index:) # => Vector[0, 1, 0]
+  #
+  def Vector.basis(size:, index:)
+    raise ArgumentError, "invalid size (#{size} for 1..)" if size < 1
+    raise ArgumentError, "invalid index (#{index} for 0...#{size})" unless 0 <= index && index < size
+    array = Array.new(size, 0)
+    array[index] = 1
+    new convert_to_array(array, false)
+  end
+
+  #
+  # Vector.new is private; use Vector[] or Vector.elements to create.
+  #
+  def initialize(array)
+    # No checking is done at this point.
+    @elements = array
+  end
+
+  # ACCESSING
+
+  #
+  # Returns element number +i+ (starting at zero) of the vector.
+  #
+  def [](i)
+    @elements[i]
+  end
+  alias element []
+  alias component []
+
+  def []=(i, v)
+    @elements[i]= v
+  end
+  alias set_element []=
+  alias set_component []=
+  private :[]=, :set_element, :set_component
+
+  #
+  # Returns the number of elements in the vector.
+  #
+  def size
+    @elements.size
+  end
+
+  #--
+  # ENUMERATIONS -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
+  #++
+
+  #
+  # Iterate over the elements of this vector
+  #
+  def each(&block)
+    return to_enum(:each) unless block_given?
+    @elements.each(&block)
+    self
+  end
+
+  #
+  # Iterate over the elements of this vector and +v+ in conjunction.
+  #
+  def each2(v) # :yield: e1, e2
+    raise TypeError, "Integer is not like Vector" if v.kind_of?(Integer)
+    Vector.Raise ErrDimensionMismatch if size != v.size
+    return to_enum(:each2, v) unless block_given?
+    size.times do |i|
+      yield @elements[i], v[i]
+    end
+    self
+  end
+
+  #
+  # Collects (as in Enumerable#collect) over the elements of this vector and +v+
+  # in conjunction.
+  #
+  def collect2(v) # :yield: e1, e2
+    raise TypeError, "Integer is not like Vector" if v.kind_of?(Integer)
+    Vector.Raise ErrDimensionMismatch if size != v.size
+    return to_enum(:collect2, v) unless block_given?
+    Array.new(size) do |i|
+      yield @elements[i], v[i]
+    end
+  end
+
+  #--
+  # PROPERTIES -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
+  #++
+
+  #
+  # Returns +true+ iff all of vectors are linearly independent.
+  #
+  #   Vector.independent?(Vector[1,0], Vector[0,1])
+  #     => true
+  #
+  #   Vector.independent?(Vector[1,2], Vector[2,4])
+  #     => false
+  #
+  def Vector.independent?(*vs)
+    vs.each do |v|
+      raise TypeError, "expected Vector, got #{v.class}" unless v.is_a?(Vector)
+      Vector.Raise ErrDimensionMismatch unless v.size == vs.first.size
+    end
+    return false if vs.count > vs.first.size
+    Matrix[*vs].rank.eql?(vs.count)
+  end
+
+  #
+  # Returns +true+ iff all of vectors are linearly independent.
+  #
+  #   Vector[1,0].independent?(Vector[0,1])
+  #     => true
+  #
+  #   Vector[1,2].independent?(Vector[2,4])
+  #     => false
+  #
+  def independent?(*vs)
+    self.class.independent?(self, *vs)
+  end
+
+  #--
+  # COMPARING -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
+  #++
+
+  #
+  # Returns +true+ iff the two vectors have the same elements in the same order.
+  #
+  def ==(other)
+    return false unless Vector === other
+    @elements == other.elements
+  end
+
+  def eql?(other)
+    return false unless Vector === other
+    @elements.eql? other.elements
+  end
+
+  #
+  # Returns a copy of the vector.
+  #
+  def clone
+    self.class.elements(@elements)
+  end
+
+  #
+  # Returns a hash-code for the vector.
+  #
+  def hash
+    @elements.hash
+  end
+
+  #--
+  # ARITHMETIC -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
+  #++
+
+  #
+  # Multiplies the vector by +x+, where +x+ is a number or another vector.
+  #
+  def *(x)
+    case x
+    when Numeric
+      els = @elements.collect{|e| e * x}
+      self.class.elements(els, false)
+    when Matrix
+      Matrix.column_vector(self) * x
+    when Vector
+      Vector.Raise ErrOperationNotDefined, "*", self.class, x.class
+    else
+      apply_through_coercion(x, __method__)
+    end
+  end
+
+  #
+  # Vector addition.
+  #
+  def +(v)
+    case v
+    when Vector
+      Vector.Raise ErrDimensionMismatch if size != v.size
+      els = collect2(v) {|v1, v2|
+        v1 + v2
+      }
+      self.class.elements(els, false)
+    when Matrix
+      Matrix.column_vector(self) + v
+    else
+      apply_through_coercion(v, __method__)
+    end
+  end
+
+  #
+  # Vector subtraction.
+  #
+  def -(v)
+    case v
+    when Vector
+      Vector.Raise ErrDimensionMismatch if size != v.size
+      els = collect2(v) {|v1, v2|
+        v1 - v2
+      }
+      self.class.elements(els, false)
+    when Matrix
+      Matrix.column_vector(self) - v
+    else
+      apply_through_coercion(v, __method__)
+    end
+  end
+
+  #
+  # Vector division.
+  #
+  def /(x)
+    case x
+    when Numeric
+      els = @elements.collect{|e| e / x}
+      self.class.elements(els, false)
+    when Matrix, Vector
+      Vector.Raise ErrOperationNotDefined, "/", self.class, x.class
+    else
+      apply_through_coercion(x, __method__)
+    end
+  end
+
+  def +@
+    self
+  end
+
+  def -@
+    collect {|e| -e }
+  end
+
+  #--
+  # VECTOR FUNCTIONS -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
+  #++
+
+  #
+  # Returns the inner product of this vector with the other.
+  #   Vector[4,7].inner_product Vector[10,1]  => 47
+  #
+  def inner_product(v)
+    Vector.Raise ErrDimensionMismatch if size != v.size
+
+    p = 0
+    each2(v) {|v1, v2|
+      p += v1 * v2.conj
+    }
+    p
+  end
+  alias_method :dot, :inner_product
+
+  #
+  # Returns the cross product of this vector with the others.
+  #   Vector[1, 0, 0].cross_product Vector[0, 1, 0]   => Vector[0, 0, 1]
+  #
+  # It is generalized to other dimensions to return a vector perpendicular
+  # to the arguments.
+  #   Vector[1, 2].cross_product # => Vector[-2, 1]
+  #   Vector[1, 0, 0, 0].cross_product(
+  #      Vector[0, 1, 0, 0],
+  #      Vector[0, 0, 1, 0]
+  #   )  #=> Vector[0, 0, 0, 1]
+  #
+  def cross_product(*vs)
+    raise ErrOperationNotDefined, "cross product is not defined on vectors of dimension #{size}" unless size >= 2
+    raise ArgumentError, "wrong number of arguments (#{vs.size} for #{size - 2})" unless vs.size == size - 2
+    vs.each do |v|
+      raise TypeError, "expected Vector, got #{v.class}" unless v.is_a? Vector
+      Vector.Raise ErrDimensionMismatch unless v.size == size
+    end
+    case size
+    when 2
+      Vector[-@elements[1], @elements[0]]
+    when 3
+      v = vs[0]
+      Vector[ v[2]*@elements[1] - v[1]*@elements[2],
+        v[0]*@elements[2] - v[2]*@elements[0],
+        v[1]*@elements[0] - v[0]*@elements[1] ]
+    else
+      rows = self, *vs, Array.new(size) {|i| Vector.basis(size: size, index: i) }
+      Matrix.rows(rows).laplace_expansion(row: size - 1)
+    end
+  end
+  alias_method :cross, :cross_product
+
+  #
+  # Like Array#collect.
+  #
+  def collect(&block) # :yield: e
+    return to_enum(:collect) unless block_given?
+    els = @elements.collect(&block)
+    self.class.elements(els, false)
+  end
+  alias map collect
+
+  #
+  # Returns the modulus (Pythagorean distance) of the vector.
+  #   Vector[5,8,2].r => 9.643650761
+  #
+  def magnitude
+    Math.sqrt(@elements.inject(0) {|v, e| v + e.abs2})
+  end
+  alias r magnitude
+  alias norm magnitude
+
+  #
+  # Like Vector#collect2, but returns a Vector instead of an Array.
+  #
+  def map2(v, &block) # :yield: e1, e2
+    return to_enum(:map2, v) unless block_given?
+    els = collect2(v, &block)
+    self.class.elements(els, false)
+  end
+
+  class ZeroVectorError < StandardError
+  end
+  #
+  # Returns a new vector with the same direction but with norm 1.
+  #   v = Vector[5,8,2].normalize
+  #   # => Vector[0.5184758473652127, 0.8295613557843402, 0.20739033894608505]
+  #   v.norm => 1.0
+  #
+  def normalize
+    n = magnitude
+    raise ZeroVectorError, "Zero vectors can not be normalized" if n == 0
+    self / n
+  end
+
+  #
+  # Returns an angle with another vector. Result is within the [0...Math::PI].
+  #   Vector[1,0].angle_with(Vector[0,1])
+  #   # => Math::PI / 2
+  #
+  def angle_with(v)
+    raise TypeError, "Expected a Vector, got a #{v.class}" unless v.is_a?(Vector)
+    Vector.Raise ErrDimensionMismatch if size != v.size
+    prod = magnitude * v.magnitude
+    raise ZeroVectorError, "Can't get angle of zero vector" if prod == 0
+
+    Math.acos( inner_product(v) / prod )
+  end
+
+  #--
+  # CONVERTING
+  #++
+
+  #
+  # Creates a single-row matrix from this vector.
+  #
+  def covector
+    Matrix.row_vector(self)
+  end
+
+  #
+  # Returns the elements of the vector in an array.
+  #
+  def to_a
+    @elements.dup
+  end
+
+  def elements_to_f
+    warn "#{caller(1)[0]}: warning: Vector#elements_to_f is deprecated"
+    map(&:to_f)
+  end
+
+  def elements_to_i
+    warn "#{caller(1)[0]}: warning: Vector#elements_to_i is deprecated"
+    map(&:to_i)
+  end
+
+  def elements_to_r
+    warn "#{caller(1)[0]}: warning: Vector#elements_to_r is deprecated"
+    map(&:to_r)
+  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.
+  #
+  def coerce(other)
+    case other
+    when Numeric
+      return Matrix::Scalar.new(other), self
+    else
+      raise TypeError, "#{self.class} can't be coerced into #{other.class}"
+    end
+  end
+
+  #--
+  # PRINTING -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
+  #++
+
+  #
+  # Overrides Object#to_s
+  #
+  def to_s
+    "Vector[" + @elements.join(", ") + "]"
+  end
+
+  #
+  # Overrides Object#inspect
+  #
+  def inspect
+    "Vector" + @elements.inspect
+  end
+end