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).
To create a matrix:
Matrix
::rows(rows, copy = true)
::build(row_count, #column_count, &block)
::scalar(n, value)
Matrix.I(n)
::empty(row_count, #column_count)
To access Matrix elements/columns/rows/submatrices/properties:
[](i, j)
row_count (row_size)
column_count (column_size)
first_minor(row, column)
cofactor(row, column)
laplace_expansion(row_or_column: num)
cofactor_expansion(row_or_column: num)
Properties of a matrix:
Matrix arithmetic:
Matrix functions:
Matrix decompositions:
Complex arithmetic:
conj
conjugate
imag
imaginary
real
rect
rectangular
Conversion to other data types:
String representations:
Returns the number of columns.
Returns the number of columns.
instance creations
Creates a matrix where each argument is a row.
Matrix[ [25, 93], [-1, 66] ]
=> 25 93
-1 66 # File lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/matrix.rb, line 244 def Matrix.identity(n) scalar(n, 1) end
::new is private; use ::rows, columns, [], etc… to create.
# File lib/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 lib/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 lib/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 lib/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 lib/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 Creates a zero matrix.
Matrix.zero(2)
=> 0 0
0 0 # File lib/matrix.rb, line 258
def Matrix.zero(row_count, column_count = row_count)
rows = Array.new(row_count){Array.new(column_count, 0)}
new rows, column_count
end Matrix multiplication.
Matrix[[2,4], [6,8]] * Matrix.identity(2)
=> 2 4
6 8 # File lib/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 lib/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 lib/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 # File lib/matrix.rb, line 1142 def +@ self end
Matrix subtraction.
Matrix[[1,5], [4,2]] - Matrix[[9,3], [-4,1]]
=> -8 2
8 1 # File lib/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 # File lib/matrix.rb, line 1146
def -@
collect {|e| -e }
end 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 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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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
: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 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 lib/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 lib/matrix.rb, line 1369 def eigensystem EigenvalueDecomposition.new(self) end
# File lib/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 lib/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 lib/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 lib/matrix.rb, line 755 def empty? column_count == 0 || row_count == 0 end
# File lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/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 lib/matrix.rb, line 1430 def real collect(&:real) end
Returns true if all entries of the matrix are real.
# File lib/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 lib/matrix.rb, line 1440 def rect [real, imag] end
Returns true if this is a regular (i.e. non-singular) matrix.
# File lib/matrix.rb, line 845 def regular? not singular? end
Returns a matrix with entries rounded to the given precision (see Float#round)
# File lib/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 lib/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 lib/matrix.rb, line 387 def row_count @rows.size end
Returns an array of the row vectors of the matrix. See Vector.
# File lib/matrix.rb, line 1468
def row_vectors
Array.new(row_count) {|i|
row(i)
}
end Returns true if this is a singular matrix.
# File lib/matrix.rb, line 852 def singular? determinant == 0 end
Returns true if this is a square matrix.
# File lib/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 lib/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 lib/matrix.rb, line 1486 def to_a @rows.collect(&:dup) end
Overrides Object#to_s
# File lib/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 lib/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 lib/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 lib/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 lib/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 lib/matrix.rb, line 1353 def vstack(*matrices) self.class.vstack(self, *matrices) end
Returns true if this is a matrix with only zero elements
# File lib/matrix.rb, line 903 def zero? all?(&:zero?) end
# File lib/matrix.rb, line 377 def []=(i, j, v) @rows[i][j] = v end
Private. Use #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 1215
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
Ruby Core © 1993–2016 Yukihiro Matsumoto
Licensed under the Ruby License.
Ruby Standard Library © contributors
Licensed under their own licenses.