Object
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.
Creates a matrix where each argument is a row.
Matrix[ [25, 93], [-1, 66] ] => 25 93 -1 66
# File matrix.rb, line 78 def Matrix.[](*rows) rows(rows, false) end
Creates a matrix of size row_count
x column_count
. It fills the values by calling the given block, passing the current row and column. Returns an enumerator if no block is given.
m = Matrix.build(2, 4) {|row, col| col - row } => Matrix[[0, 1, 2, 3], [-1, 0, 1, 2]] m = Matrix.build(3) { rand } => a 3x3 matrix with random elements
# File matrix.rb, line 123 def Matrix.build(row_count, column_count = row_count) row_count = CoercionHelper.coerce_to_int(row_count) column_count = CoercionHelper.coerce_to_int(column_count) raise ArgumentError if row_count < 0 || column_count < 0 return to_enum :build, row_count, column_count unless block_given? rows = Array.new(row_count) do |i| Array.new(column_count) do |j| yield i, j end end new rows, column_count end
Creates a single-column matrix where the values of that column are as given in column
.
Matrix.column_vector([4,5,6]) => 4 5 6
# File matrix.rb, line 209 def Matrix.column_vector(column) column = convert_to_array(column) new [column].transpose, 1 end
Creates a matrix using columns
as an array of column vectors.
Matrix.columns([[25, 93], [-1, 66]]) => 25 -1 93 66
# File matrix.rb, line 108 def Matrix.columns(columns) rows(columns, false).transpose end
Create a matrix by combining matrices entrywise, using the given block
x = Matrix[[6, 6], [4, 4]] y = Matrix[[1, 2], [3, 4]] Matrix.combine(x, y) {|a, b| a - b} # => Matrix[[5, 4], [1, 0]]
# File matrix.rb, line 286 def Matrix.combine(*matrices) return to_enum(__method__, *matrices) unless block_given? return Matrix.empty if matrices.empty? matrices.map!(&CoercionHelper.method(:coerce_to_matrix)) x = matrices.first matrices.each do |m| raise ErrDimensionMismatch unless x.row_count == m.row_count && x.column_count == m.column_count end rows = Array.new(x.row_count) do |i| Array.new(x.column_count) do |j| yield matrices.map{|m| m[i,j]} end end new rows, x.column_count end
Creates a matrix where the diagonal elements are composed of values
.
Matrix.diagonal(9, 5, -3) => 9 0 0 0 5 0 0 0 -3
# File matrix.rb, line 143 def Matrix.diagonal(*values) size = values.size return Matrix.empty if size == 0 rows = Array.new(size) {|j| row = Array.new(size, 0) row[j] = values[j] row } new rows end
Creates a empty matrix of row_count
x column_count
. At least one of row_count
or column_count
must be 0.
m = Matrix.empty(2, 0) m == Matrix[ [], [] ] => true n = Matrix.empty(0, 3) n == Matrix.columns([ [], [], [] ]) => true m * n => Matrix[[0, 0, 0], [0, 0, 0]]
# File matrix.rb, line 227 def Matrix.empty(row_count = 0, column_count = 0) raise ArgumentError, "One size must be 0" if column_count != 0 && row_count != 0 raise ArgumentError, "Negative size" if column_count < 0 || row_count < 0 new([[]]*row_count, column_count) end
Create a matrix by stacking matrices horizontally
x = Matrix[[1, 2], [3, 4]] y = Matrix[[5, 6], [7, 8]] Matrix.hstack(x, y) # => Matrix[[1, 2, 5, 6], [3, 4, 7, 8]]
# File matrix.rb, line 262 def Matrix.hstack(x, *matrices) x = CoercionHelper.coerce_to_matrix(x) result = x.send(:rows).map(&:dup) total_column_count = x.column_count matrices.each do |m| m = CoercionHelper.coerce_to_matrix(m) if m.row_count != x.row_count raise ErrDimensionMismatch, "The given matrices must have #{x.row_count} rows, but one has #{m.row_count}" end result.each_with_index do |row, i| row.concat m.send(:rows)[i] end total_column_count += m.column_count end new result, total_column_count end
Creates an n
by n
identity matrix.
Matrix.identity(2) => 1 0 0 1
# File matrix.rb, line 171 def Matrix.identity(n) scalar(n, 1) end
Matrix.new
is private; use Matrix.rows
, columns, [], etc… to create.
# File matrix.rb, line 311 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 196 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 90 def Matrix.rows(rows, copy = true) rows = convert_to_array(rows, copy) rows.map! do |row| convert_to_array(row, copy) end size = (rows[0] || []).size rows.each do |row| raise ErrDimensionMismatch, "row size differs (#{row.size} should be #{size})" unless row.size == size end new rows, size end
Creates an n
by n
diagonal matrix where each diagonal element is value
.
Matrix.scalar(2, 5) => 5 0 0 5
# File matrix.rb, line 161 def Matrix.scalar(n, value) diagonal(*Array.new(n, value)) end
Create a matrix by stacking matrices vertically
x = Matrix[[1, 2], [3, 4]] y = Matrix[[5, 6], [7, 8]] Matrix.vstack(x, y) # => Matrix[[1, 2], [3, 4], [5, 6], [7, 8]]
# File matrix.rb, line 241 def Matrix.vstack(x, *matrices) x = CoercionHelper.coerce_to_matrix(x) result = x.send(:rows).map(&:dup) matrices.each do |m| m = CoercionHelper.coerce_to_matrix(m) if m.column_count != x.column_count raise ErrDimensionMismatch, "The given matrices must have #{x.column_count} columns, but one has #{m.column_count}" end result.concat(m.send(:rows)) end new result, x.column_count end
Matrix
multiplication.
Matrix[[2,4], [6,8]] * Matrix.identity(2) => 2 4 6 8
# File matrix.rb, line 1045 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 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 1222 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 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 1078 def +(m) case m when Numeric raise ErrOperationNotDefined, ["+", self.class, m.class] when Vector m = self.class.column_vector(m) when Matrix else return apply_through_coercion(m, __method__) end raise ErrDimensionMismatch unless row_count == m.row_count && column_count == m.column_count rows = Array.new(row_count) {|i| Array.new(column_count) {|j| self[i, j] + m[i, j] } } new_matrix rows, column_count end
Matrix
subtraction.
Matrix[[1,5], [4,2]] - Matrix[[9,3], [-4,1]] => -8 2 8 1
# File matrix.rb, line 1105 def -(m) case m when Numeric raise ErrOperationNotDefined, ["-", self.class, m.class] when Vector m = self.class.column_vector(m) when Matrix else return apply_through_coercion(m, __method__) end raise ErrDimensionMismatch unless row_count == m.row_count && column_count == m.column_count rows = Array.new(row_count) {|i| Array.new(column_count) {|j| self[i, j] - m[i, j] } } new_matrix rows, column_count end
Matrix
division (multiplication by the inverse).
Matrix[[7,6], [3,9]] / Matrix[[2,9], [3,1]] => -7 1 -3 -6
# File matrix.rb, line 1132 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 1008 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 326 def [](i, j) @rows.fetch(i){return nil}[j] end
Set element or elements of matrix.
# File matrix.rb, line 340 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
Returns the absolute value elementwise
# File matrix.rb, line 1256 def abs collect(&:abs) end
Returns the adjugate of the matrix.
Matrix[ [7,6],[3,9] ].adjugate => 9 -6 -3 7
# File matrix.rb, line 781 def adjugate raise ErrDimensionMismatch unless square? Matrix.build(row_count, column_count) do |row, column| cofactor(column, row) end end
Returns true
if this is an antisymmetric matrix. Raises an error if matrix is not square.
# File matrix.rb, line 960 def antisymmetric? raise ErrDimensionMismatch unless square? each_with_index(:upper) do |e, row, col| return false unless e == -rows[col][row] end true 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 1565 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 766 def cofactor(row, column) raise RuntimeError, "cofactor of empty matrix is not defined" if empty? raise ErrDimensionMismatch unless square? det_of_minor = first_minor(row, column).determinant det_of_minor * (-1) ** (row + column) end
Returns a matrix that is the result of iteration of the given block over all elements of the matrix. Elements can be restricted by passing an argument:
:all (default): yields all elements
:diagonal: yields only elements on the diagonal
:off_diagonal: yields all elements except on the diagonal
:lower: yields only elements on or below the diagonal
:strict_lower: yields only elements below the diagonal
:strict_upper: yields only elements above the diagonal
:upper: yields only elements on or above the diagonal Matrix[ [1,2], [3,4] ].collect { |e| e**2 }
=> 1 4 9 16
# File matrix.rb, line 497 def collect(which = :all, &block) # :yield: e return to_enum(:collect, which) unless block_given? dup.collect!(which, &block) end
Invokes the given block for each element of matrix, replacing the element with the value returned by the block. Elements can be restricted by passing an argument:
:all (default): yields all elements
:diagonal: yields only elements on the diagonal
:off_diagonal: yields all elements except on the diagonal
:lower: yields only elements on or below the diagonal
:strict_lower: yields only elements below the diagonal
:strict_upper: yields only elements above the diagonal
:upper: yields only elements on or above the diagonal
# File matrix.rb, line 515 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
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 466 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 1586 def column_vectors Array.new(column_count) {|i| column(i) } end
# File matrix.rb, line 304 def combine(*matrices, &block) Matrix.combine(self, *matrices, &block) 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 1511 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 1274 def determinant raise ErrDimensionMismatch unless square? m = @rows case row_count # Up to 4x4, give result using Laplacian expansion by minors. # This will typically be faster, as well as giving good results # in case of Floats when 0 +1 when 1 + m[0][0] when 2 + m[0][0] * m[1][1] - m[0][1] * m[1][0] when 3 m0, m1, m2 = m + m0[0] * m1[1] * m2[2] - m0[0] * m1[2] * m2[1] \ - m0[1] * m1[0] * m2[2] + m0[1] * m1[2] * m2[0] \ + m0[2] * m1[0] * m2[1] - m0[2] * m1[1] * m2[0] when 4 m0, m1, m2, m3 = m + m0[0] * m1[1] * m2[2] * m3[3] - m0[0] * m1[1] * m2[3] * m3[2] \ - m0[0] * m1[2] * m2[1] * m3[3] + m0[0] * m1[2] * m2[3] * m3[1] \ + m0[0] * m1[3] * m2[1] * m3[2] - m0[0] * m1[3] * m2[2] * m3[1] \ - m0[1] * m1[0] * m2[2] * m3[3] + m0[1] * m1[0] * m2[3] * m3[2] \ + m0[1] * m1[2] * m2[0] * m3[3] - m0[1] * m1[2] * m2[3] * m3[0] \ - m0[1] * m1[3] * m2[0] * m3[2] + m0[1] * m1[3] * m2[2] * m3[0] \ + m0[2] * m1[0] * m2[1] * m3[3] - m0[2] * m1[0] * m2[3] * m3[1] \ - m0[2] * m1[1] * m2[0] * m3[3] + m0[2] * m1[1] * m2[3] * m3[0] \ + m0[2] * m1[3] * m2[0] * m3[1] - m0[2] * m1[3] * m2[1] * m3[0] \ - m0[3] * m1[0] * m2[1] * m3[2] + m0[3] * m1[0] * m2[2] * m3[1] \ + m0[3] * m1[1] * m2[0] * m3[2] - m0[3] * m1[1] * m2[2] * m3[0] \ - m0[3] * m1[2] * m2[0] * m3[1] + m0[3] * m1[2] * m2[1] * m3[0] else # For bigger matrices, use an efficient and general algorithm. # Currently, we use the Gauss-Bareiss algorithm determinant_bareiss end end
deprecated; use Matrix#determinant
# File matrix.rb, line 1355 def determinant_e warn "Matrix#determinant_e is deprecated; use #determinant", uplevel: 1 determinant end
Returns true
if this is a diagonal matrix. Raises an error if matrix is not square.
# File matrix.rb, line 827 def diagonal? 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
:off_diagonal: yields all elements except on the diagonal
:lower: yields only elements on or below the diagonal
:strict_lower: yields only elements below the diagonal
:strict_upper: yields only elements above the diagonal
:upper: yields only elements on or above the diagonal
Matrix[ [1,2], [3,4] ].each { |e| puts e }
# => prints the numbers 1 to 4
Matrix[ [1,2], [3,4] ].each(:strict_lower).to_a # => [3]
# File matrix.rb, line 544 def each(which = :all, &block) # :yield: e return to_enum :each, which unless block_given? last = column_count - 1 case which when :all @rows.each do |row| row.each(&block) end when :diagonal @rows.each_with_index do |row, row_index| yield row.fetch(row_index){return self} end when :off_diagonal @rows.each_with_index do |row, row_index| column_count.times do |col_index| yield row[col_index] unless row_index == col_index end end when :lower @rows.each_with_index do |row, row_index| 0.upto([row_index, last].min) do |col_index| yield row[col_index] end end when :strict_lower @rows.each_with_index do |row, row_index| [row_index, column_count].min.times do |col_index| yield row[col_index] end end when :strict_upper @rows.each_with_index do |row, row_index| (row_index+1).upto(last) do |col_index| yield row[col_index] end end when :upper @rows.each_with_index do |row, row_index| row_index.upto(last) do |col_index| yield row[col_index] end end else raise ArgumentError, "expected #{which.inspect} to be one of :all, :diagonal, :off_diagonal, :lower, :strict_lower, :strict_upper or :upper" end self end
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 604 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 1478 def eigensystem EigenvalueDecomposition.new(self) end
Deprecated.
Use map(&:to_f)
# File matrix.rb, line 1609 def elements_to_f warn "Matrix#elements_to_f is deprecated, use map(&:to_f)", uplevel: 1 map(&:to_f) end
Deprecated.
Use map(&:to_i)
# File matrix.rb, line 1617 def elements_to_i warn "Matrix#elements_to_i is deprecated, use map(&:to_i)", uplevel: 1 map(&:to_i) end
Deprecated.
Use map(&:to_r)
# File matrix.rb, line 1625 def elements_to_r warn "Matrix#elements_to_r is deprecated, use map(&:to_r)", uplevel: 1 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 836 def empty? column_count == 0 || row_count == 0 end
# File matrix.rb, line 1014 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 739 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
Hadamard product
Matrix[[1,2], [3,4]].hadamard_product(Matrix[[1,2], [3,2]]) => 1 4 9 8
# File matrix.rb, line 1152 def hadamard_product(m) combine(m){|a, b| a * b} end
Returns a hash-code for the matrix.
# File matrix.rb, line 1031 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 844 def hermitian? 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 1369 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 1525 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 667 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 1650 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 1163 def inverse 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 798 def laplace_expansion(row: nil, column: nil) num = row || column if !num || (row && column) raise ArgumentError, "exactly one the row or column arguments must be specified" end raise ErrDimensionMismatch unless square? raise RuntimeError, "laplace_expansion of empty matrix is not defined" if empty? unless 0 <= num && num < row_count raise ArgumentError, "invalid num (#{num.inspect} for 0..#{row_count - 1})" end send(row ? :row : :column, num).map.with_index { |e, k| e * cofactor(*(row ? [num, k] : [k,num])) }.inject(:+) end
Returns true
if this is a lower triangular matrix.
# File matrix.rb, line 854 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 1493 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 698 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 862 def normal? raise ErrDimensionMismatch unless square? rows.each_with_index do |row_i, i| rows.each_with_index do |row_j, j| s = 0 rows.each_with_index do |row_k, k| s += row_i[k] * row_j[k].conj - row_k[i].conj * row_k[j] end return false unless s == 0 end end true end
Returns true
if this is an orthogonal matrix Raises an error if matrix is not square.
# File matrix.rb, line 880 def orthogonal? 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 898 def permutation? raise ErrDimensionMismatch unless square? cols = Array.new(column_count) rows.each_with_index do |row, i| found = false row.each_with_index do |e, j| if e == 1 return false if found || cols[j] found = cols[j] = true elsif e != 0 return false end end return false unless found end true end
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 1382 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
# File matrix.rb, line 1413 def rank_e warn "Matrix#rank_e is deprecated; use #rank", uplevel: 1 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 1539 def real collect(&:real) end
Returns true
if all entries of the matrix are real.
# File matrix.rb, line 919 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 1549 def rect [real, imag] end
Returns true
if this is a regular (i.e. non-singular) matrix.
# File matrix.rb, line 926 def regular? not singular? end
Returns a matrix with entries rounded to the given precision (see Float#round)
# File matrix.rb, line 1421 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 452 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 437 def row_count @rows.size end
Returns an array of the row vectors of the matrix. See Vector
.
# File matrix.rb, line 1577 def row_vectors Array.new(row_count) {|i| row(i) } end
Returns true
if this is a singular matrix.
# File matrix.rb, line 933 def singular? determinant == 0 end
Returns true
if this is a square matrix.
# File matrix.rb, line 940 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 948 def symmetric? raise ErrDimensionMismatch unless square? each_with_index(:strict_upper) do |e, row, col| return false if e != rows[col][row] end true end
Returns an array of arrays that describe the rows of the matrix.
# File matrix.rb, line 1602 def to_a @rows.collect(&:dup) end
Explicit conversion to a Matrix
. Returns self
# File matrix.rb, line 1595 def to_matrix self end
Overrides Object#to_s
# File matrix.rb, line 1637 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 1430 def trace 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 1448 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 973 def unitary? 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 990 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 1462 def vstack(*matrices) self.class.vstack(self, *matrices) end