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.
instance creations
Returns the number of columns.
Returns the number of columns.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 51
def Matrix.[](*rows)
rows(rows, false)
end
Creates a matrix where each argument is a row.
Matrix[ [25, 93], [-1, 66] ]
=> 25 93
-1 66
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 96
def Matrix.build(row_count, column_count = row_count)
row_count = CoercionHelper.coerce_to_int(row_count)
column_count = CoercionHelper.coerce_to_int(column_count)
raise ArgumentError if row_count < 0 || column_count < 0
return to_enum :build, row_count, column_count unless block_given?
rows = Array.new(row_count) do |i|
Array.new(column_count) do |j|
yield i, j
end
end
new rows, column_count
end
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 182
def Matrix.column_vector(column)
column = convert_to_array(column)
new [column].transpose, 1
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 81
def Matrix.columns(columns)
rows(columns, false).transpose
end
Creates a matrix using columns as an array of column vectors.
Matrix.columns([[25, 93], [-1, 66]])
=> 25 -1
93 66
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 259
def Matrix.combine(*matrices)
return to_enum(__method__, *matrices) unless block_given?
return Matrix.empty if matrices.empty?
matrices.map!(&CoercionHelper.method(:coerce_to_matrix))
x = matrices.first
matrices.each do |m|
Matrix.Raise ErrDimensionMismatch unless x.row_count == m.row_count && x.column_count == m.column_count
end
rows = Array.new(x.row_count) do |i|
Array.new(x.column_count) do |j|
yield matrices.map{|m| m[i,j]}
end
end
new rows, x.column_count
end
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 116
def Matrix.diagonal(*values)
size = values.size
return Matrix.empty if size == 0
rows = Array.new(size) {|j|
row = Array.new(size, 0)
row[j] = values[j]
row
}
new rows
end
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 200
def Matrix.empty(row_count = 0, column_count = 0)
raise ArgumentError, "One size must be 0" if column_count != 0 && row_count != 0
raise ArgumentError, "Negative size" if column_count < 0 || row_count < 0
new([[]]*row_count, column_count)
end
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 235
def Matrix.hstack(x, *matrices)
x = CoercionHelper.coerce_to_matrix(x)
result = x.send(:rows).map(&:dup)
total_column_count = x.column_count
matrices.each do |m|
m = CoercionHelper.coerce_to_matrix(m)
if m.row_count != x.row_count
raise ErrDimensionMismatch, "The given matrices must have #{x.row_count} rows, but one has #{m.row_count}"
end
result.each_with_index do |row, i|
row.concat m.send(:rows)[i]
end
total_column_count += m.column_count
end
new result, total_column_count
end
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 144
def Matrix.identity(n)
scalar(n, 1)
end
Creates an n by n identity matrix.
Matrix.identity(2)
=> 1 0
0 1
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 284
def initialize(rows, column_count = rows[0].size)
# No checking is done at this point. rows must be an Array of Arrays.
# column_count must be the size of the first row, if there is one,
# otherwise it *must* be specified and can be any integer >= 0
@rows = rows
@column_count = column_count
end
Matrix.new is private; use Matrix.rows, columns, [], etc… to create.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 169
def Matrix.row_vector(row)
row = convert_to_array(row)
new [row]
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 63
def Matrix.rows(rows, copy = true)
rows = convert_to_array(rows, copy)
rows.map! do |row|
convert_to_array(row, copy)
end
size = (rows[0] || []).size
rows.each do |row|
raise ErrDimensionMismatch, "row size differs (#{row.size} should be #{size})" unless row.size == size
end
new rows, size
end
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 134
def Matrix.scalar(n, value)
diagonal(*Array.new(n, value))
end
Creates an n by n diagonal matrix where each diagonal element is value.
Matrix.scalar(2, 5)
=> 5 0
0 5
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 214
def Matrix.vstack(x, *matrices)
x = CoercionHelper.coerce_to_matrix(x)
result = x.send(:rows).map(&:dup)
matrices.each do |m|
m = CoercionHelper.coerce_to_matrix(m)
if m.column_count != x.column_count
raise ErrDimensionMismatch, "The given matrices must have #{x.column_count} columns, but one has #{m.column_count}"
end
result.concat(m.send(:rows))
end
new result, x.column_count
end
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 158
def Matrix.zero(row_count, column_count = row_count)
rows = Array.new(row_count){Array.new(column_count, 0)}
new rows, column_count
end
Creates a zero matrix.
Matrix.zero(2)
=> 0 0
0 0
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 881
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 multiplication.
Matrix[[2,4], [6,8]] * Matrix.identity(2)
=> 2 4
6 8
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1059
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 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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 914
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 addition.
Matrix.scalar(2,5) + Matrix[[1,0], [-4,7]]
=> 6 0
-4 12
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1082
def +@
self
end
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 941
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1086
def -@
collect {|e| -e }
end
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 968
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
Matrix division (multiplication by the inverse).
Matrix[[7,6], [3,9]] / Matrix[[2,9], [3,1]]
=> -7 1
-3 -6
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 843
def ==(other)
return false unless Matrix === other &&
column_count == other.column_count # necessary for empty matrices
rows == other.rows
end
Returns true if and only if the two matrices contain equal elements.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 300
def [](i, j)
@rows.fetch(i){return nil}[j]
end
Returns element (i,j) of the matrix. That is: row i, column j.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 306
def []=(i, j, v)
@rows[i][j] = v
end
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 629
def adjugate
Matrix.Raise ErrDimensionMismatch unless square?
Matrix.build(row_count, column_count) do |row, column|
cofactor(column, row)
end
end
Returns the adjugate of the matrix.
Matrix[ [7,6],[3,9] ].adjugate
=> 9 -6
-3 7
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 860
def clone
new_matrix @rows.map(&:dup), column_count
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1396
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
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 614
def cofactor(row, column)
raise RuntimeError, "cofactor of empty matrix is not defined" if empty?
Matrix.Raise ErrDimensionMismatch unless square?
det_of_minor = first_minor(row, column).determinant
det_of_minor * (-1) ** (row + column)
end
Returns the (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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 368
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 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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 345
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 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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1417
def column_vectors
Array.new(column_count) {|i|
column(i)
}
end
Returns an array of the column vectors of the matrix. See Vector.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 277
def combine(*matrices, &block)
Matrix.combine(self, *matrices, &block)
end
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1342
def conjugate
collect(&:conjugate)
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1104
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
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1155
def determinant_bareiss
size = row_count
last = size - 1
a = to_a
no_pivot = Proc.new{ return 0 }
sign = +1
pivot = 1
size.times do |k|
previous_pivot = pivot
if (pivot = a[k][k]) == 0
switch = (k+1 ... size).find(no_pivot) {|row|
a[row][k] != 0
}
a[switch], a[k] = a[k], a[switch]
pivot = a[k][k]
sign = -sign
end
(k+1).upto(last) do |i|
ai = a[i]
(k+1).upto(last) do |j|
ai[j] = (pivot * ai[j] - ai[k] * a[k][j]) / previous_pivot
end
end
end
sign * pivot
end
Private. Use Matrix#determinant
Returns the determinant of the matrix, using Bareiss’ multistep integer-preserving gaussian elimination. It has the same computational cost order O(n^3) as standard Gaussian elimination. Intermediate results are fraction free and of lower complexity. A matrix of Integers will have thus intermediate results that are also Integers, with smaller bignums (if any), while a matrix of Float will usually have intermediate results with better precision.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1186
def determinant_e
warn "Matrix#determinant_e is deprecated; use #determinant", uplevel: 1
determinant
end
deprecated; use Matrix#determinant
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 675
def diagonal?
Matrix.Raise ErrDimensionMismatch unless square?
each(:off_diagonal).all?(&:zero?)
end
Returns true if this is a diagonal matrix. Raises an error if matrix is not square.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 391
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
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 4Matrix[ [1,2], [3,4] ].each(:strict_lower).to_a # => [3]
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 452
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
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1309
def eigensystem
EigenvalueDecomposition.new(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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1437
def elements_to_f
warn "Matrix#elements_to_f is deprecated, use map(&:to_f)", uplevel: 1
map(&:to_f)
end
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1442
def elements_to_i
warn "Matrix#elements_to_i is deprecated, use map(&:to_i)", uplevel: 1
map(&:to_i)
end
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1447
def elements_to_r
warn "Matrix#elements_to_r is deprecated, use map(&:to_r)", uplevel: 1
map(&:to_r)
end
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 684
def empty?
column_count == 0 || row_count == 0
end
Returns true if this is an empty matrix, i.e. if the number of rows or the number of columns is 0.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 849
def eql?(other)
return false unless Matrix === other &&
column_count == other.column_count # necessary for empty matrices
rows.eql? other.rows
end
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 587
def first_minor(row, column)
raise RuntimeError, "first_minor of empty matrix is not defined" if empty?
unless 0 <= row && row < row_count
raise ArgumentError, "invalid row (#{row.inspect} for 0..#{row_count - 1})"
end
unless 0 <= column && column < column_count
raise ArgumentError, "invalid column (#{column.inspect} for 0..#{column_count - 1})"
end
arrays = to_a
arrays.delete_at(row)
arrays.each do |array|
array.delete_at(column)
end
new_matrix arrays, column_count - 1
end
Returns the 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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 988
def hadamard_product(m)
combine(m){|a, b| a * b}
end
Hadamard product
Matrix[[1,2], [3,4]].hadamard_product(Matrix[[1,2], [3,2]])
=> 1 4
9 8
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 867
def hash
@rows.hash
end
Returns a hash-code for the matrix.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 692
def hermitian?
Matrix.Raise ErrDimensionMismatch unless square?
each_with_index(:upper).all? do |e, row, col|
e == rows[col][row].conj
end
end
Returns true if this is an hermitian matrix. Raises an error if matrix is not square.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1200
def hstack(*matrices)
self.class.hstack(self, *matrices)
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1356
def imaginary
collect(&:imaginary)
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 515
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
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1472
def inspect
if empty?
"#{self.class}.empty(#{row_count}, #{column_count})"
else
"#{self.class}#{@rows.inspect}"
end
end
Overrides Object#inspect
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 999
def inverse
Matrix.Raise ErrDimensionMismatch unless square?
self.class.I(row_count).send(:inverse_from, self)
end
Returns the inverse of the matrix.
Matrix[[-1, -1], [0, -1]].inverse
=> -1 1
0 -1
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 646
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 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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 702
def lower_triangular?
each(:strict_upper).all?(&:zero?)
end
Returns true if this is a lower triangular matrix.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1324
def lup
LUPDecomposition.new(self)
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 546
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 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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 710
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 a normal matrix. Raises an error if matrix is not square.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 728
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 an orthogonal matrix Raises an error if matrix is not square.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 746
def permutation?
Matrix.Raise ErrDimensionMismatch unless square?
cols = Array.new(column_count)
rows.each_with_index do |row, i|
found = false
row.each_with_index do |e, j|
if e == 1
return false if found || cols[j]
found = cols[j] = true
elsif e != 0
return false
end
end
return false unless found
end
true
end
Returns true if this is a permutation matrix Raises an error if matrix is not square.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1213
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
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1244
def rank_e
warn "Matrix#rank_e is deprecated; use #rank", uplevel: 1
rank
end
deprecated; use Matrix#rank
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1370
def real
collect(&:real)
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 767
def real?
all?(&:real?)
end
Returns true if all entries of the matrix are real.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1380
def rect
[real, imag]
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 774
def regular?
not singular?
end
Returns true if this is a regular (i.e. non-singular) matrix.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1252
def round(ndigits=0)
map{|e| e.round(ndigits)}
end
Returns a matrix with entries rounded to the given precision (see Float#round)
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 331
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 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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 316
def row_count
@rows.size
end
Returns the number of rows.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1408
def row_vectors
Array.new(row_count) {|i|
row(i)
}
end
Returns an array of the row vectors of the matrix. See Vector.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 781
def singular?
determinant == 0
end
Returns true if this is a singular matrix.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 788
def square?
column_count == row_count
end
Returns true if this is a square matrix.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 796
def symmetric?
Matrix.Raise ErrDimensionMismatch unless square?
each_with_index(:strict_upper) do |e, row, col|
return false if e != rows[col][row]
end
true
end
Returns true if this is a symmetric matrix. Raises an error if matrix is not square.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1433
def to_a
@rows.collect(&:dup)
end
Returns an array of arrays that describe the rows of the matrix.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1426
def to_matrix
self
end
Explicit conversion to a Matrix. Returns self
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1459
def to_s
if empty?
"#{self.class}.empty(#{row_count}, #{column_count})"
else
"#{self.class}[" + @rows.collect{|row|
"[" + row.collect{|e| e.to_s}.join(", ") + "]"
}.join(", ")+"]"
end
end
Overrides Object#to_s
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1261
def trace
Matrix.Raise ErrDimensionMismatch unless square?
(0...column_count).inject(0) do |tr, i|
tr + @rows[i][i]
end
end
Returns the trace (sum of diagonal elements) of the matrix.
Matrix[[7,6], [3,9]].trace => 16
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1279
def transpose
return self.class.empty(column_count, 0) if row_count.zero?
new_matrix @rows.transpose, row_count
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 808
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 a unitary matrix Raises an error if matrix is not square.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 825
def upper_triangular?
each(:strict_lower).all?(&:zero?)
end
Returns true if this is an upper triangular matrix.
# File tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 1293
def vstack(*matrices)
self.class.vstack(self, *matrices)
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 tmp/rubies/ruby-2.5.9/lib/matrix.rb, line 832
def zero?
all?(&:zero?)
end
Returns true if this is a matrix with only zero elements