# class 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

### Attributes

column_count[R]

Returns the number of columns.

column_size[R]

Returns the number of columns.

rows[R]

instance creations

### Public Class Methods

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

Creates a matrix where each argument is a row.

    Matrix[ [25, 93], [-1, 66] ]
       =>  25 93
           -1 66

    # File lib/matrix.rb, line 51
    def Matrix.[](*rows)
      rows(rows, false)
    end

build(row_count, column_count = row_count) { |i, j| ... } 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 lib/matrix.rb, line 96
    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 lib/matrix.rb, line 182
    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/matrix.rb, line 81
    def Matrix.columns(columns)
      rows(columns, false).transpose
    end

combine(*matrices) { |map{|m| m}| ... } 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 lib/matrix.rb, line 259
    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|
        Matrix.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 lib/matrix.rb, line 116
    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 lib/matrix.rb, line 200
    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 lib/matrix.rb, line 235
    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 lib/matrix.rb, line 144
    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 `Matrix.rows`, columns, [], etc… to create.

    # File lib/matrix.rb, line 284
    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 lib/matrix.rb, line 169
    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/matrix.rb, line 63
    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 lib/matrix.rb, line 134
    def Matrix.scalar(n, value)
      diagonal(*Array.new(n, value))
    end

unit(n)
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 lib/matrix.rb, line 214
    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 lib/matrix.rb, line 158
    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 lib/matrix.rb, line 1019
    def *(m) # m is matrix or vector or number
      case(m)
      when Numeric
        rows = @rows.collect {|row|
          row.collect {|e| e * m }
        }
        return new_matrix rows, column_count
      when Vector
        m = self.class.column_vector(m)
        r = self * m
        return r.column(0)
      when Matrix
        Matrix.Raise ErrDimensionMismatch if column_count != m.row_count

        rows = Array.new(row_count) {|i|
          Array.new(m.column_count) {|j|
            (0 ... column_count).inject(0) do |vij, k|
              vij + self[i, k] * m[k, j]
            end
          }
        }
        return new_matrix rows, m.column_count
      else
        return apply_through_coercion(m, __method__)
      end
    end

**(other) click to toggle source

`Matrix` exponentiation. Equivalent to multiplying the matrix by itself N times. Non integer exponents will be handled by diagonalizing the matrix.

    Matrix[[7,6], [3,9]] ** 2
      => 67 96
         48 99

    # File lib/matrix.rb, line 1196
    def **(other)
      case other
      when Integer
        x = self
        if other <= 0
          x = self.inverse
          return self.class.identity(self.column_count) if other == 0
          other = -other
        end
        z = nil
        loop do
          z = z ? z * x : x if other[0] == 1
          return z if (other >>= 1).zero?
          x *= x
        end
      when Numeric
        v, d, v_inv = eigensystem
        v * self.class.diagonal(*d.each(:diagonal).map{|e| e ** other}) * v_inv
      else
        Matrix.Raise ErrOperationNotDefined, "**", self.class, other.class
      end
    end

+(m) click to toggle source

`Matrix` addition.

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

    # File lib/matrix.rb, line 1052
    def +(m)
      case m
      when Numeric
        Matrix.Raise ErrOperationNotDefined, "+", self.class, m.class
      when Vector
        m = self.class.column_vector(m)
      when Matrix
      else
        return apply_through_coercion(m, __method__)
      end

      Matrix.Raise ErrDimensionMismatch unless row_count == m.row_count && column_count == m.column_count

      rows = Array.new(row_count) {|i|
        Array.new(column_count) {|j|
          self[i, j] + m[i, j]
        }
      }
      new_matrix rows, column_count
    end

+@() click to toggle source

    # File lib/matrix.rb, line 1219
    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 lib/matrix.rb, line 1079
    def -(m)
      case m
      when Numeric
        Matrix.Raise ErrOperationNotDefined, "-", self.class, m.class
      when Vector
        m = self.class.column_vector(m)
      when Matrix
      else
        return apply_through_coercion(m, __method__)
      end

      Matrix.Raise ErrDimensionMismatch unless row_count == m.row_count && column_count == m.column_count

      rows = Array.new(row_count) {|i|
        Array.new(column_count) {|j|
          self[i, j] - m[i, j]
        }
      }
      new_matrix rows, column_count
    end

-@() click to toggle source

    # File lib/matrix.rb, line 1223
    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 lib/matrix.rb, line 1106
    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 lib/matrix.rb, line 982
    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 lib/matrix.rb, line 299
    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 lib/matrix.rb, line 313
    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
adjugate() click to toggle source

Returns the adjugate of the matrix.

    Matrix[ [7,6],[3,9] ].adjugate
      => 9 -6
         -3 7

    # File lib/matrix.rb, line 755
    def adjugate
      Matrix.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 lib/matrix.rb, line 934
    def antisymmetric?
      Matrix.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 lib/matrix.rb, line 1532
    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 lib/matrix.rb, line 740
    def cofactor(row, column)
      raise RuntimeError, "cofactor of empty matrix is not defined" if empty?
      Matrix.Raise ErrDimensionMismatch unless square?

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

cofactor_expansion(row: nil, column: nil)
Alias for: laplace_expansion
collect(which = :all) { |e| ... } 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 lib/matrix.rb, line 470
    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) { |e| ... } 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 lib/matrix.rb, line 488
    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) { |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/matrix.rb, line 439
    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 lib/matrix.rb, line 1553
    def column_vectors
      Array.new(column_count) {|i|
        column(i)
      }
    end

combine(*matrices, &block) click to toggle source

    # File lib/matrix.rb, line 277
    def combine(*matrices, &block)
      Matrix.combine(self, *matrices, &block)
    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/matrix.rb, line 1478
    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/matrix.rb, line 1241
    def determinant
      Matrix.Raise ErrDimensionMismatch unless square?
      m = @rows
      case row_count
        # Up to 4x4, give result using Laplacian expansion by minors.
        # This will typically be faster, as well as giving good results
        # in case of Floats
      when 0
        +1
      when 1
        + m[0][0]
      when 2
        + m[0][0] * m[1][1] - m[0][1] * m[1][0]
      when 3
        m0, m1, m2 = m
        + m0[0] * m1[1] * m2[2] - m0[0] * m1[2] * m2[1] \
        - m0[1] * m1[0] * m2[2] + m0[1] * m1[2] * m2[0] \
        + m0[2] * m1[0] * m2[1] - m0[2] * m1[1] * m2[0]
      when 4
        m0, m1, m2, m3 = m
        + m0[0] * m1[1] * m2[2] * m3[3] - m0[0] * m1[1] * m2[3] * m3[2] \
        - m0[0] * m1[2] * m2[1] * m3[3] + m0[0] * m1[2] * m2[3] * m3[1] \
        + m0[0] * m1[3] * m2[1] * m3[2] - m0[0] * m1[3] * m2[2] * m3[1] \
        - m0[1] * m1[0] * m2[2] * m3[3] + m0[1] * m1[0] * m2[3] * m3[2] \
        + m0[1] * m1[2] * m2[0] * m3[3] - m0[1] * m1[2] * m2[3] * m3[0] \
        - m0[1] * m1[3] * m2[0] * m3[2] + m0[1] * m1[3] * m2[2] * m3[0] \
        + m0[2] * m1[0] * m2[1] * m3[3] - m0[2] * m1[0] * m2[3] * m3[1] \
        - m0[2] * m1[1] * m2[0] * m3[3] + m0[2] * m1[1] * m2[3] * m3[0] \
        + m0[2] * m1[3] * m2[0] * m3[1] - m0[2] * m1[3] * m2[1] * m3[0] \
        - m0[3] * m1[0] * m2[1] * m3[2] + m0[3] * m1[0] * m2[2] * m3[1] \
        + m0[3] * m1[1] * m2[0] * m3[2] - m0[3] * m1[1] * m2[2] * m3[0] \
        - m0[3] * m1[2] * m2[0] * m3[1] + m0[3] * m1[2] * m2[1] * m3[0]
      else
        # For bigger matrices, use an efficient and general algorithm.
        # Currently, we use the Gauss-Bareiss algorithm
        determinant_bareiss
      end
    end

Also aliased as: det
determinant_e() click to toggle source

deprecated; use `Matrix#determinant`

    # File lib/matrix.rb, line 1322
    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 lib/matrix.rb, line 801
    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` 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 lib/matrix.rb, line 517
    def each(which = :all) # :yield: e
      return to_enum :each, which unless block_given?
      last = column_count - 1
      case which
      when :all
        block = Proc.new
        @rows.each do |row|
          row.each(&block)
        end
      when :diagonal
        @rows.each_with_index do |row, row_index|
          yield row.fetch(row_index){return self}
        end
      when :off_diagonal
        @rows.each_with_index do |row, row_index|
          column_count.times do |col_index|
            yield row[col_index] unless row_index == col_index
          end
        end
      when :lower
        @rows.each_with_index do |row, row_index|
          0.upto([row_index, last].min) do |col_index|
            yield row[col_index]
          end
        end
      when :strict_lower
        @rows.each_with_index do |row, row_index|
          [row_index, column_count].min.times do |col_index|
            yield row[col_index]
          end
        end
      when :strict_upper
        @rows.each_with_index do |row, row_index|
          (row_index+1).upto(last) do |col_index|
            yield row[col_index]
          end
        end
      when :upper
        @rows.each_with_index do |row, row_index|
          row_index.upto(last) do |col_index|
            yield row[col_index]
          end
        end
      else
        raise ArgumentError, "expected #{which.inspect} to be one of :all, :diagonal, :off_diagonal, :lower, :strict_lower, :strict_upper or :upper"
      end
      self
    end

each_with_index(which = :all) { |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/matrix.rb, line 578
    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()
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/matrix.rb, line 1445
    def eigensystem
      EigenvalueDecomposition.new(self)
    end

Also aliased as: eigen
element(i, j)
Alias for: []
elements_to_f() click to toggle source

Deprecated.

Use map(&:to_f)

    # File lib/matrix.rb, line 1576
    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 lib/matrix.rb, line 1584
    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 lib/matrix.rb, line 1592
    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 lib/matrix.rb, line 810
    def empty?
      column_count == 0 || row_count == 0
    end

entrywise_product(m)
Alias for: hadamard_product
eql?(other) click to toggle source

    # File lib/matrix.rb, line 988
    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)
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 lib/matrix.rb, line 713
    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
Calls superclass method `Object#freeze`

    # File lib/matrix.rb, line 496
    def freeze
      @rows.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 lib/matrix.rb, line 1126
    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 lib/matrix.rb, line 1005
    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 lib/matrix.rb, line 818
    def hermitian?
      Matrix.Raise ErrDimensionMismatch unless square?
      each_with_index(:upper).all? do |e, row, col|
        e == rows[col][row].conj
      end
    end

hstack(*matrices) click to toggle source

Returns a new matrix resulting by stacking horizontally the receiver with the given matrices

    x = Matrix[[1, 2], [3, 4]]
    y = Matrix[[5, 6], [7, 8]]
    x.hstack(y) # => Matrix[[1, 2, 5, 6], [3, 4, 7, 8]]

    # File lib/matrix.rb, line 1336
    def hstack(*matrices)
      self.class.hstack(self, *matrices)
    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/matrix.rb, line 1492
    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/matrix.rb, line 641
    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/matrix.rb, line 1617
    def inspect
      if empty?
        "#{self.class}.empty(#{row_count}, #{column_count})"
      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/matrix.rb, line 1137
    def inverse
      Matrix.Raise ErrDimensionMismatch unless square?
      self.class.I(row_count).send(:inverse_from, self)
    end

Also aliased as: inv
laplace_expansion(row: nil, column: nil) click to toggle source

Returns the Laplace expansion along given row or column.

    Matrix[[7,6], [3,9]].laplace_expansion(column: 1)
     => 45

    Matrix[[Vector[1, 0], Vector[0, 1]], [2, 3]].laplace_expansion(row: 0)
     => Vector[3, -2]

    # File lib/matrix.rb, line 772
    def laplace_expansion(row: nil, column: nil)
      num = row || column

      if !num || (row && column)
        raise ArgumentError, "exactly one the row or column arguments must be specified"
      end

      Matrix.Raise ErrDimensionMismatch unless square?
      raise RuntimeError, "laplace_expansion of empty matrix is not defined" if empty?

      unless 0 <= num && num < row_count
        raise ArgumentError, "invalid num (#{num.inspect} for 0..#{row_count - 1})"
      end

      send(row ? :row : :column, num).map.with_index { |e, k|
        e * cofactor(*(row ? [num, k] : [k,num]))
      }.inject(:+)
    end

Also aliased as: cofactor_expansion
lower_triangular?() click to toggle source

Returns `true` if this is a lower triangular matrix.

    # File lib/matrix.rb, line 828
    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 lib/matrix.rb, line 1460
    def lup
      LUPDecomposition.new(self)
    end

Also aliased as: lup_decomposition
lup_decomposition()
Alias for: lup
map(which = :all)
Alias for: collect
map!(which = :all)
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 lib/matrix.rb, line 672
    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 lib/matrix.rb, line 836
    def normal?
      Matrix.Raise ErrDimensionMismatch unless square?
      rows.each_with_index do |row_i, i|
        rows.each_with_index do |row_j, j|
          s = 0
          rows.each_with_index do |row_k, k|
            s += row_i[k] * row_j[k].conj - row_k[i].conj * row_k[j]
          end
          return false unless s == 0
        end
      end
      true
    end

orthogonal?() click to toggle source

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

    # File lib/matrix.rb, line 854
    def orthogonal?
      Matrix.Raise ErrDimensionMismatch unless square?
      rows.each_with_index do |row, i|
        column_count.times do |j|
          s = 0
          row_count.times do |k|
            s += row[k] * rows[k][j]
          end
          return false unless s == (i == j ? 1 : 0)
        end
      end
      true
    end

permutation?() click to toggle source

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

    # File lib/matrix.rb, line 872
    def permutation?
      Matrix.Raise ErrDimensionMismatch unless square?
      cols = Array.new(column_count)
      rows.each_with_index do |row, i|
        found = false
        row.each_with_index do |e, j|
          if e == 1
            return false if found || cols[j]
            found = cols[j] = true
          elsif e != 0
            return false
          end
        end
        return false unless found
      end
      true
    end

rank() click to toggle source

Returns the rank of the matrix. Beware that using `Float` values can yield erroneous results because of their lack of precision. Consider using exact types like `Rational` or `BigDecimal` instead.

    Matrix[[7,6], [3,9]].rank
      => 2

    # File lib/matrix.rb, line 1349
    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 lib/matrix.rb, line 1380
    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 lib/matrix.rb, line 1506
    def real
      collect(&:real)
    end

real?() click to toggle source

Returns `true` if all entries of the matrix are real.

    # File lib/matrix.rb, line 893
    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/matrix.rb, line 1516
    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/matrix.rb, line 900
    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/matrix.rb, line 1388
    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/matrix.rb, line 425
    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 lib/matrix.rb, line 410
    def row_count
      @rows.size
    end

Also aliased as: row_size
row_size()
Alias for: row_count
row_vectors() click to toggle source

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

    # File lib/matrix.rb, line 1544
    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 lib/matrix.rb, line 907
    def singular?
      determinant == 0
    end

skew_symmetric?()
Alias for: antisymmetric?
square?() click to toggle source

Returns `true` if this is a square matrix.

    # File lib/matrix.rb, line 914
    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 lib/matrix.rb, line 922
    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/matrix.rb, line 1569
    def to_a
      @rows.collect(&:dup)
    end

to_matrix() click to toggle source

Explicit conversion to a `Matrix`. Returns self

    # File lib/matrix.rb, line 1562
    def to_matrix
      self
    end

to_s() click to toggle source

Overrides `Object#to_s`

    # File lib/matrix.rb, line 1604
    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()
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/matrix.rb, line 1397
    def trace
      Matrix.Raise ErrDimensionMismatch unless square?
      (0...column_count).inject(0) do |tr, i|
        tr + @rows[i][i]
      end
    end

Also aliased as: tr
transpose() click to toggle source

Returns the transpose of the matrix.

    Matrix[[1,2], [3,4], [5,6]]
      => 1 2
         3 4
         5 6
    Matrix[[1,2], [3,4], [5,6]].transpose
      => 1 3 5
         2 4 6

    # File lib/matrix.rb, line 1415
    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 lib/matrix.rb, line 947
    def unitary?
      Matrix.Raise ErrDimensionMismatch unless square?
      rows.each_with_index do |row, i|
        column_count.times do |j|
          s = 0
          row_count.times do |k|
            s += row[k].conj * rows[k][j]
          end
          return false unless s == (i == j ? 1 : 0)
        end
      end
      true
    end

upper_triangular?() click to toggle source

Returns `true` if this is an upper triangular matrix.

    # File lib/matrix.rb, line 964
    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 lib/matrix.rb, line 1429
    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 lib/matrix.rb, line 971
    def zero?
      all?(&:zero?)
    end

### Private Instance Methods

check_int(val, direction) click to toggle source

    # File lib/matrix.rb, line 338
            def check_int(val, direction)
      count = direction == :row ? row_count : column_count
      CoercionHelper.check_int(val, count, direction)
    end

check_range(val, direction) click to toggle source

Returns range or nil

    # File lib/matrix.rb, line 332
            def check_range(val, direction)
      return unless val.is_a?(Range)
      count = direction == :row ? row_count : column_count
      CoercionHelper.check_range(val, count, direction)
    end

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/matrix.rb, line 1292
            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

initialize_copy(m) click to toggle source

Called for dup & clone.

Calls superclass method

    # File lib/matrix.rb, line 997
            def initialize_copy(m)
      super
      @rows = @rows.map(&:dup) unless frozen?
    end

set_col_range(row, col_range, value) click to toggle source

    # File lib/matrix.rb, line 394
            def set_col_range(row, col_range, value)
      value = if value.is_a?(Vector)
        value.to_a
      elsif value.is_a?(Matrix)
        Matrix.Raise ErrDimensionMismatch unless value.row_count == 1
        value.row(0).to_a
      else
        Array.new(col_range.size, value)
      end
      Matrix.Raise ErrDimensionMismatch unless col_range.size == value.size
      @rows[row][col_range] = value
    end

set_column_vector(row_range, col, value) click to toggle source

    # File lib/matrix.rb, line 387
            def set_column_vector(row_range, col, value)
      value.each_with_index do |e, index|
        r = row_range.begin + index
        @rows[r][col] = e
      end
    end

set_component(i, j, v)
Alias for: []=
set_element(i, j, v)
Alias for: []=
set_row_and_col_range(row_range, col_range, value) click to toggle source

    # File lib/matrix.rb, line 349
            def set_row_and_col_range(row_range, col_range, value)
      if value.is_a?(Matrix)
        if row_range.size != value.row_count || col_range.size != value.column_count
          raise ErrDimensionMismatch, [
            'Expected a Matrix of dimensions',
            "#{row_range.size}x#{col_range.size}",
            'got',
            "#{value.row_count}x#{value.column_count}",
          ].join(' ')
        end
        source = value.instance_variable_get :@rows
        row_range.each_with_index do |row, i|
          @rows[row][col_range] = source[i]
        end
      elsif value.is_a?(Vector)
        raise ErrDimensionMismatch, 'Expected a Matrix or a value, got a Vector'
      else
        value_to_set = Array.new(col_range.size, value)
        row_range.each do |i|
          @rows[i][col_range] = value_to_set
        end
      end
    end

set_row_range(row_range, col, value) click to toggle source

    # File lib/matrix.rb, line 373
            def set_row_range(row_range, col, value)
      if value.is_a?(Vector)
        Matrix.Raise ErrDimensionMismatch unless row_range.size == value.size
        set_column_vector(row_range, col, value)
      elsif value.is_a?(Matrix)
        Matrix.Raise ErrDimensionMismatch unless value.column_count == 1
        value = value.column(0)
        Matrix.Raise ErrDimensionMismatch unless row_range.size == value.size
        set_column_vector(row_range, col, value)
      else
        @rows[row_range].each{|e| e[col] = value }
      end
    end

set_value(row, col, value) click to toggle source

    # File lib/matrix.rb, line 343
            def set_value(row, col, value)
      raise ErrDimensionMismatch, "Expected a a value, got a #{value.class}" if value.respond_to?(:to_matrix)

      @rows[row][col] = value
    end