In Files
- matrix.rb
- matrix/eigenvalue_decomposition.rb
- matrix/lup_decomposition.rb
Parent
Object
Namespace
Methods
- ::I
- ::[]
- ::build
- ::column_vector
- ::columns
- ::diagonal
- ::empty
- ::hstack
- ::identity
- ::new
- ::row_vector
- ::rows
- ::scalar
- ::unit
- ::vstack
- ::zero
- #*
- #**
- #+
- #+@
- #-
- #-@
- #/
- #==
- #[]
- #adjugate
- #clone
- #coerce
- #cofactor
- #cofactor_expansion
- #collect
- #column
- #column_vectors
- #component
- #conj
- #conjugate
- #det
- #det_e
- #determinant
- #determinant_e
- #diagonal?
- #each
- #each_with_index
- #eigen
- #eigensystem
- #element
- #elements_to_f
- #elements_to_i
- #elements_to_r
- #empty?
- #eql?
- #find_index
- #first_minor
- #hash
- #hermitian?
- #hstack
- #imag
- #imaginary
- #index
- #inspect
- #inv
- #inverse
- #laplace_expansion
- #lower_triangular?
- #lup
- #lup_decomposition
- #map
- #minor
- #normal?
- #orthogonal?
- #permutation?
- #rank
- #rank_e
- #real
- #real?
- #rect
- #rectangular
- #regular?
- #round
- #row
- #row_count
- #row_size
- #row_vectors
- #singular?
- #square?
- #symmetric?
- #t
- #to_a
- #to_s
- #tr
- #trace
- #transpose
- #unitary?
- #upper_triangular?
- #vstack
- #zero?
Included Modules
- Enumerable
Class/Module Index ![[+]](http://ruby-doc.org/./images/find.png "show/hide quicksearch")
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 (trace, rank, inverse, determinant).
Method Catalogue¶ ↑
To create a matrix:
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::rows(rows, copy = true)
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::build(row_count, #column_count, &block)
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::scalar(n, value)
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Matrix.I(n)
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::empty(row_count, #column_count)
To access Matrix elements/columns/rows/submatrices/properties:
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[](i, j)
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row_count (row_size)
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column_count (column_size)
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first_minor(row, column)
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cofactor(row, column)
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laplace_expansion(row_or_column: num)
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cofactor_expansion(row_or_column: num)
Properties of a matrix:
Matrix arithmetic:
Matrix functions:
Matrix decompositions:
Complex arithmetic:
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conj
-
conjugate
-
imag
-
imaginary
-
real
-
rect
-
rectangular
Conversion to other data types:
String representations:
Constants
Attributes
Public Class Methods
Creates a matrix where each argument is a row.
Matrix[ [25, 93], [-1, 66] ] => 25 93 -1 66
# File matrix.rb, line 151 def Matrix.[](*rows) rows(rows, false) 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 # File matrix.rb, line 196 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 single-column matrix where the values of that column are as given in column.
Matrix.column_vector([4,5,6]) => 4 5 6
# File matrix.rb, line 282 def Matrix.column_vector(column) column = convert_to_array(column) new [column].transpose, 1 end
Creates a matrix using columns as an array of column vectors.
Matrix.columns([[25, 93], [-1, 66]]) => 25 -1 93 66
# File matrix.rb, line 181 def Matrix.columns(columns) rows(columns, false).transpose 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
# File matrix.rb, line 216 def Matrix.diagonal(*values) size = values.size return Matrix.empty if size == 0 rows = Array.new(size) {|j| row = Array.new(size, 0) row[j] = values[j] row } new rows end
Creates a empty matrix of row_count x column_count. At least one of row_count or column_count must be 0.
m = Matrix.empty(2, 0) m == Matrix[ [], [] ] => true n = Matrix.empty(0, 3) n == Matrix.columns([ [], [], [] ]) => true m * n => Matrix[[0, 0, 0], [0, 0, 0]]
# File matrix.rb, line 300 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 horizontally
x = Matrix[[1, 2], [3, 4]] y = Matrix[[5, 6], [7, 8]] Matrix.hstack(x, y) # => Matrix[[1, 2, 5, 6], [3, 4, 7, 8]]
# File matrix.rb, line 335 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
Creates an n by n identity matrix.
Matrix.identity(2) => 1 0 0 1
# File matrix.rb, line 244 def Matrix.identity(n) scalar(n, 1) end
::new is private; use ::rows, columns, [], etc… to create.
# File matrix.rb, line 355 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
Creates a single-row matrix where the values of that row are as given in row.
Matrix.row_vector([4,5,6]) => 4 5 6
# File matrix.rb, line 269 def Matrix.row_vector(row) row = convert_to_array(row) new [row] 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
# File matrix.rb, line 163 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 an n by n diagonal matrix where each diagonal element is value.
Matrix.scalar(2, 5) => 5 0 0 5
# File matrix.rb, line 234 def Matrix.scalar(n, value) diagonal(*Array.new(n, value)) 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]]
# File matrix.rb, line 314 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
Public Instance Methods
Matrix multiplication.
Matrix[[2,4], [6,8]] * Matrix.identity(2) => 2 4 6 8
# File matrix.rb, line 952 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 exponentiation. Equivalent to multiplying the matrix by itself N times. Non integer exponents will be handled by diagonalizing the matrix.
Matrix[[7,6], [3,9]] ** 2 => 67 96 48 99
# File matrix.rb, line 1119 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
Matrix addition.
Matrix.scalar(2,5) + Matrix[[1,0], [-4,7]] => 6 0 -4 12
# File matrix.rb, line 985 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
# File matrix.rb, line 1012 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
# File matrix.rb, line 1039 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 true if and only if the two matrices contain equal elements.
# File matrix.rb, line 914 def ==(other) return false unless Matrix === other && column_count == other.column_count # necessary for empty matrices rows == other.rows end
Returns element (i,j) of the matrix. That is: row i, column j.
# File matrix.rb, line 371 def [](i, j) @rows.fetch(i){return nil}[j] end
Returns the adjugate of the matrix.
Matrix[ [7,6],[3,9] ].adjugate => 9 -6 -3 7
# File matrix.rb, line 700 def adjugate Matrix.Raise ErrDimensionMismatch unless square? Matrix.build(row_count, column_count) do |row, column| cofactor(column, row) end 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.
# File matrix.rb, line 931 def clone new_matrix @rows.map(&:dup), column_count 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.
# File matrix.rb, line 1456 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 the (row, column) cofactor which is obtained by multiplying the first minor by (-1)**(row + column).
Matrix.diagonal(9, 5, -3, 4).cofactor(1, 1) => -108
# File matrix.rb, line 685 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 a matrix that is the result of iteration of the given block over all elements of the matrix.
Matrix[ [1,2], [3,4] ].collect { |e| e**2 } => 1 4 9 16 # File matrix.rb, line 439 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
Returns column vector number j of the matrix as a Vector (starting at 0 like an array). When a block is given, the elements of that vector are iterated.
# File matrix.rb, line 416 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 an array of the column vectors of the matrix. See Vector.
# File matrix.rb, line 1477 def column_vectors Array.new(column_count) {|i| column(i) } end
Returns the conjugate of the matrix.
Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]] => 1+2i i 0 1 2 3 Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]].conjugate => 1-2i -i 0 1 2 3
# File matrix.rb, line 1402 def conjugate collect(&:conjugate) end
Returns the determinant of the matrix.
Beware that using Float values can yield erroneous results because of their lack of precision. Consider using exact types like Rational or BigDecimal instead.
Matrix[[7,6], [3,9]].determinant => 45
# File matrix.rb, line 1164 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
deprecated; use #determinant
# File matrix.rb, line 1246 def determinant_e warn "#{caller(1)[0]}: warning: Matrix#determinant_e is deprecated; use #determinant" determinant end
Returns true if this is a diagonal matrix. Raises an error if matrix is not square.
# File matrix.rb, line 746 def diagonal? Matrix.Raise ErrDimensionMismatch unless square? each(:off_diagonal).all?(&:zero?) end
Yields all elements of the matrix, starting with those of the first row, or returns an Enumerator if no block given. Elements can be restricted by passing an argument:
-
:all (default): yields all elements
-
:diagonal: yields only elements on the diagonal
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:off_diagonal: yields all elements except on the diagonal
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:lower: yields only elements on or below the diagonal
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:strict_lower: yields only elements below the diagonal
-
:strict_upper: yields only elements above the diagonal
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:upper: yields only elements on or above the diagonal
Matrix[ [1,2], [3,4] ].each { |e| puts e }
# => prints the numbers 1 to 4Matrix[ [1,2], [3,4] ].each(:strict_lower).to_a # => [3]
# File matrix.rb, line 462 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
# File matrix.rb, line 523 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
Returns the Eigensystem of the matrix; see EigenvalueDecomposition.
m = Matrix[[1, 2], [3, 4]] v, d, v_inv = m.eigensystem d.diagonal? # => true v.inv == v_inv # => true (v * d * v_inv).round(5) == m # => true
# File matrix.rb, line 1369 def eigensystem EigenvalueDecomposition.new(self) end
# File matrix.rb, line 1490 def elements_to_f warn "#{caller(1)[0]}: warning: Matrix#elements_to_f is deprecated, use map(&:to_f)" map(&:to_f) end
# File matrix.rb, line 1495 def elements_to_i warn "#{caller(1)[0]}: warning: Matrix#elements_to_i is deprecated, use map(&:to_i)" map(&:to_i) end
# File matrix.rb, line 1500 def elements_to_r warn "#{caller(1)[0]}: warning: Matrix#elements_to_r is deprecated, use map(&:to_r)" map(&:to_r) end
Returns true if this is an empty matrix, i.e. if the number of rows or the number of columns is 0.
# File matrix.rb, line 755 def empty? column_count == 0 || row_count == 0 end
# File matrix.rb, line 920 def eql?(other) return false unless Matrix === other && column_count == other.column_count # necessary for empty matrices rows.eql? other.rows 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
# File matrix.rb, line 658 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 a hash-code for the matrix.
# File matrix.rb, line 938 def hash @rows.hash end
Returns true if this is an hermitian matrix. Raises an error if matrix is not square.
# File matrix.rb, line 763 def hermitian? Matrix.Raise ErrDimensionMismatch unless square? each_with_index(:upper).all? do |e, row, col| e == rows[col][row].conj end end
Returns a new matrix resulting by stacking horizontally the receiver with the given matrices
x = Matrix[[1, 2], [3, 4]] y = Matrix[[5, 6], [7, 8]] x.hstack(y) # => Matrix[[1, 2, 5, 6], [3, 4, 7, 8]]
# File matrix.rb, line 1260 def hstack(*matrices) self.class.hstack(self, *matrices) end
Returns the imaginary part of the matrix.
Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]] => 1+2i i 0 1 2 3 Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]].imaginary => 2i i 0 0 0 0
# File matrix.rb, line 1416 def imaginary collect(&:imaginary) end
The index method is specialized to return the index as [row, column] It also accepts an optional selector argument, see each for details.
Matrix[ [1,2], [3,4] ].index(&:even?) # => [0, 1] Matrix[ [1,1], [1,1] ].index(1, :strict_lower) # => [1, 0]
# File matrix.rb, line 586 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
Overrides Object#inspect
# File matrix.rb, line 1525 def inspect if empty? "#{self.class}.empty(#{row_count}, #{column_count})" else "#{self.class}#{@rows.inspect}" end end
Returns the inverse of the matrix.
Matrix[[-1, -1], [0, -1]].inverse => -1 1 0 -1
# File matrix.rb, line 1059 def inverse Matrix.Raise ErrDimensionMismatch unless square? self.class.I(row_count).send(:inverse_from, self) 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]
# File matrix.rb, line 717 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
Returns true if this is a lower triangular matrix.
# File matrix.rb, line 773 def lower_triangular? each(:strict_upper).all?(&:zero?) end
Returns the LUP decomposition of the matrix; see LUPDecomposition.
a = Matrix[[1, 2], [3, 4]] l, u, p = a.lup l.lower_triangular? # => true u.upper_triangular? # => true p.permutation? # => true l * u == p * a # => true a.lup.solve([2, 5]) # => Vector[(1/1), (1/2)]
# File matrix.rb, line 1384 def lup LUPDecomposition.new(self) end
Returns a section of the matrix. The parameters are either:
-
start_row, nrows, start_col, ncols; OR
-
row_range, col_range
Matrix.diagonal(9, 5, -3).minor(0..1, 0..2) => 9 0 0 0 5 0
Like Array#[], negative indices count backward from the end of the row or column (-1 is the last element). Returns nil if the starting row or column is greater than #row_count or #column_count respectively.
# File matrix.rb, line 617 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 true if this is a normal matrix. Raises an error if matrix is not square.
# File matrix.rb, line 781 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.
# File matrix.rb, line 799 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.
# File matrix.rb, line 817 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 the rank of the matrix. Beware that using Float values can yield erroneous results because of their lack of precision. Consider using exact types like Rational or BigDecimal instead.
Matrix[[7,6], [3,9]].rank => 2
# File matrix.rb, line 1273 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 #rank
# File matrix.rb, line 1304 def rank_e warn "#{caller(1)[0]}: warning: Matrix#rank_e is deprecated; use #rank" rank end
Returns the real part of the matrix.
Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]] => 1+2i i 0 1 2 3 Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]].real => 1 0 0 1 2 3
# File matrix.rb, line 1430 def real collect(&:real) end
Returns true if all entries of the matrix are real.
# File matrix.rb, line 838 def real? all?(&: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
# File matrix.rb, line 1440 def rect [real, imag] end
Returns true if this is a regular (i.e. non-singular) matrix.
# File matrix.rb, line 845 def regular? not singular? end
Returns a matrix with entries rounded to the given precision (see Float#round)
# File matrix.rb, line 1312 def round(ndigits=0) map{|e| e.round(ndigits)} end
Returns row vector number i of the matrix as a Vector (starting at 0 like an array). When a block is given, the elements of that vector are iterated.
# File matrix.rb, line 402 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 the number of rows.
# File matrix.rb, line 387 def row_count @rows.size end
Returns an array of the row vectors of the matrix. See Vector.
# File matrix.rb, line 1468 def row_vectors Array.new(row_count) {|i| row(i) } end
Returns true if this is a singular matrix.
# File matrix.rb, line 852 def singular? determinant == 0 end
Returns true if this is a square matrix.
# File matrix.rb, line 859 def square? column_count == row_count end
Returns true if this is a symmetric matrix. Raises an error if matrix is not square.
# File matrix.rb, line 867 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 an array of arrays that describe the rows of the matrix.
# File matrix.rb, line 1486 def to_a @rows.collect(&:dup) end
Overrides Object#to_s
# File matrix.rb, line 1512 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
Returns the trace (sum of diagonal elements) of the matrix.
Matrix[[7,6], [3,9]].trace => 16
# File matrix.rb, line 1321 def trace Matrix.Raise ErrDimensionMismatch unless square? (0...column_count).inject(0) do |tr, i| tr + @rows[i][i] end end
Returns the transpose of the matrix.
Matrix[[1,2], [3,4], [5,6]] => 1 2 3 4 5 6 Matrix[[1,2], [3,4], [5,6]].transpose => 1 3 5 2 4 6
# File matrix.rb, line 1339 def transpose return self.class.empty(column_count, 0) if row_count.zero? new_matrix @rows.transpose, row_count end
Returns true if this is a unitary matrix Raises an error if matrix is not square.
# File matrix.rb, line 879 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.
# File matrix.rb, line 896 def upper_triangular? each(:strict_lower).all?(&:zero?) end
Returns a new matrix resulting by stacking vertically the receiver with the given matrices
x = Matrix[[1, 2], [3, 4]] y = Matrix[[5, 6], [7, 8]] x.vstack(y) # => Matrix[[1, 2], [3, 4], [5, 6], [7, 8]]
# File matrix.rb, line 1353 def vstack(*matrices) self.class.vstack(self, *matrices) end