Extended maintenance of Ruby 1.9.3 ended on February 23, 2015. Read more
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 (trace, rank,
inverse, determinant).
To create a matrix:
Matrix[*rows]
Matrix.[](*rows)
Matrix.rows(rows, copy = true)
Matrix.columns(columns)
Matrix.build(row_size, column_size, &block)
Matrix.diagonal(*values)
Matrix.scalar(n, value)
Matrix.identity(n)
Matrix.unit(n)
Matrix.I(n)
Matrix.zero(n)
Matrix.row_vector(row)
Matrix.column_vector(column)
To access Matrix elements/columns/rows/submatrices/properties:
[](i, j)
#row_size
#column_size
#row(i)
#column(j)
#collect
#map
#each
#each_with_index
#find_index
#minor(*param)
Properties of a matrix:
#diagonal?
#empty?
#hermitian?
#lower_triangular?
#normal?
#orthogonal?
#permutation?
#real?
#regular?
#singular?
#square?
#symmetric?
#unitary?
#upper_triangular?
#zero?
Matrix arithmetic:
*(m)
+(m)
-(m)
#/(m)
#inverse
#inv
**
Matrix functions:
#determinant
#det
#rank
#round
#trace
#tr
#transpose
#t
Matrix decompositions:
#eigen
#eigensystem
#lup
#lup_decomposition
Complex arithmetic:
conj
conjugate
imag
imaginary
real
rect
rectangular
Conversion to other data types:
#coerce(other)
#row_vectors
#column_vectors
#to_a
String representations:
#to_s
#inspect
Creates a matrix where each argument is a row.
Matrix[ [25, 93], [-1, 66] ] => 25 93 -1 66
# File matrix.rb, line 139 def Matrix.[](*rows) Matrix.rows(rows, false) end
Creates a matrix of size row_size
x column_size
.
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 184 def Matrix.build(row_size, column_size = row_size) row_size = CoercionHelper.coerce_to_int(row_size) column_size = CoercionHelper.coerce_to_int(column_size) raise ArgumentError if row_size < 0 || column_size < 0 return to_enum :build, row_size, column_size unless block_given? rows = Array.new(row_size) do |i| Array.new(column_size) do |j| yield i, j end end new rows, column_size 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 269 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 169 def Matrix.columns(columns) Matrix.rows(columns, false).transpose end
Creates a matrix where the diagonal elements are composed of
values
.
Matrix.diagonal(9, 5, -3) => 9 0 0 0 5 0 0 0 -3
# File matrix.rb, line 204 def Matrix.diagonal(*values) size = values.size rows = Array.new(size) {|j| row = Array.new(size, 0) row[j] = values[j] row } new rows end
Creates a empty matrix of row_size
x column_size
.
At least one of row_size
or column_size
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 287 def Matrix.empty(row_size = 0, column_size = 0) Matrix.Raise ArgumentError, "One size must be 0" if column_size != 0 && row_size != 0 Matrix.Raise ArgumentError, "Negative size" if column_size < 0 || row_size < 0 new([[]]*row_size, column_size) end
Creates an n
by n
identity matrix.
Matrix.identity(2) => 1 0 0 1
# File matrix.rb, line 231 def Matrix.identity(n) Matrix.scalar(n, 1) end
::new is private; use ::rows, columns, [], etc... to create.
# File matrix.rb, line 297 def initialize(rows, column_size = rows[0].size) # No checking is done at this point. rows must be an Array of Arrays. # column_size 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_size = column_size 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 256 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 151 def Matrix.rows(rows, copy = true) rows = convert_to_array(rows) rows.map! do |row| convert_to_array(row, copy) end size = (rows[0] || []).size rows.each do |row| Matrix.Raise ErrDimensionMismatch, "row size differs (#{row.size} should be #{size})" unless row.size == size end new rows, size end
Matrix multiplication.
Matrix[[2,4], [6,8]] * Matrix.identity(2) => 2 4 6 8
# File matrix.rb, line 802 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_size when Vector m = Matrix.column_vector(m) r = self * m return r.column(0) when Matrix Matrix.Raise ErrDimensionMismatch if column_size != m.row_size rows = Array.new(row_size) {|i| Array.new(m.column_size) {|j| (0 ... column_size).inject(0) do |vij, k| vij + self[i, k] * m[k, j] end } } return new_matrix rows, m.column_size 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 969 def ** (other) case other when Integer x = self if other <= 0 x = self.inverse return Matrix.identity(self.column_size) 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 * Matrix.diagonal(*d.each(:diagonal).map{|e| e ** other}) * v_inv else Matrix.Raise ErrOperationNotDefined, "**", self.class, other.class end end
Matrix addition.
Matrix.scalar(2,5) + Matrix[[1,0], [-4,7]] => 6 0 -4 12
# File matrix.rb, line 835 def +(m) case m when Numeric Matrix.Raise ErrOperationNotDefined, "+", self.class, m.class when Vector m = Matrix.column_vector(m) when Matrix else return apply_through_coercion(m, __method__) end Matrix.Raise ErrDimensionMismatch unless row_size == m.row_size and column_size == m.column_size rows = Array.new(row_size) {|i| Array.new(column_size) {|j| self[i, j] + m[i, j] } } new_matrix rows, column_size end
Matrix subtraction.
Matrix[[1,5], [4,2]] - Matrix[[9,3], [-4,1]] => -8 2 8 1
# File matrix.rb, line 862 def -(m) case m when Numeric Matrix.Raise ErrOperationNotDefined, "-", self.class, m.class when Vector m = Matrix.column_vector(m) when Matrix else return apply_through_coercion(m, __method__) end Matrix.Raise ErrDimensionMismatch unless row_size == m.row_size and column_size == m.column_size rows = Array.new(row_size) {|i| Array.new(column_size) {|j| self[i, j] - m[i, j] } } new_matrix rows, column_size 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 889 def /(other) case other when Numeric rows = @rows.collect {|row| row.collect {|e| e / other } } return new_matrix rows, column_size 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 764 def ==(other) return false unless Matrix === other && column_size == other.column_size # 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 313 def [](i, j) @rows.fetch(i){return nil}[j] end
Returns a clone of the matrix, so that the contents of each do not reference identical objects. There should be no good reason to do this since Matrices are immutable.
# File matrix.rb, line 781 def clone new_matrix @rows.map(&:dup), column_size 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 1274 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 a matrix that is the result of iteration of the given block over all elements of the matrix.
Matrix[ [1,2], [3,4] ].collect { |e| e**2 } => 1 4 9 16
# File matrix.rb, line 379 def collect(&block) # :yield: e return to_enum(:collect) unless block_given? rows = @rows.collect{|row| row.collect(&block)} new_matrix rows, column_size 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 356 def column(j) # :yield: e if block_given? return self if j >= column_size || j < -column_size row_size.times do |i| yield @rows[i][j] end self else return nil if j >= column_size || j < -column_size col = Array.new(row_size) {|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 1295 def column_vectors Array.new(column_size) {|i| column(i) } end
Returns the conjugate of the matrix.
Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]] => 1+2i i 0 1 2 3 Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]].conjugate => 1-2i -i 0 1 2 3
# File matrix.rb, line 1220 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 1006 def determinant Matrix.Raise ErrDimensionMismatch unless square? m = @rows case row_size # Up to 4x4, give result using Laplacian expansion by minors. # This will typically be faster, as well as giving good results # in case of Floats when 0 +1 when 1 + m[0][0] when 2 + m[0][0] * m[1][1] - m[0][1] * m[1][0] when 3 m0, m1, m2 = m + m0[0] * m1[1] * m2[2] - m0[0] * m1[2] * m2[1] - m0[1] * m1[0] * m2[2] + m0[1] * m1[2] * m2[0] + m0[2] * m1[0] * m2[1] - m0[2] * m1[1] * m2[0] when 4 m0, m1, m2, m3 = m + m0[0] * m1[1] * m2[2] * m3[3] - m0[0] * m1[1] * m2[3] * m3[2] - m0[0] * m1[2] * m2[1] * m3[3] + m0[0] * m1[2] * m2[3] * m3[1] + m0[0] * m1[3] * m2[1] * m3[2] - m0[0] * m1[3] * m2[2] * m3[1] - m0[1] * m1[0] * m2[2] * m3[3] + m0[1] * m1[0] * m2[3] * m3[2] + m0[1] * m1[2] * m2[0] * m3[3] - m0[1] * m1[2] * m2[3] * m3[0] - m0[1] * m1[3] * m2[0] * m3[2] + m0[1] * m1[3] * m2[2] * m3[0] + m0[2] * m1[0] * m2[1] * m3[3] - m0[2] * m1[0] * m2[3] * m3[1] - m0[2] * m1[1] * m2[0] * m3[3] + m0[2] * m1[1] * m2[3] * m3[0] + m0[2] * m1[3] * m2[0] * m3[1] - m0[2] * m1[3] * m2[1] * m3[0] - m0[3] * m1[0] * m2[1] * m3[2] + m0[3] * m1[0] * m2[2] * m3[1] + m0[3] * m1[1] * m2[0] * m3[2] - m0[3] * m1[1] * m2[2] * m3[0] - m0[3] * m1[2] * m2[0] * m3[1] + m0[3] * m1[2] * m2[1] * m3[0] else # For bigger matrices, use an efficient and general algorithm. # Currently, we use the Gauss-Bareiss algorithm determinant_bareiss end end
deprecated; use #determinant
# File matrix.rb, line 1088 def determinant_e warn "#{caller(1)[0]}: warning: Matrix#determinant_e is deprecated; use #determinant" rank end
Returns true
is this is a diagonal matrix. Raises an error if
matrix is not square.
# File matrix.rb, line 597 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 is 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 402 def each(which = :all) # :yield: e return to_enum :each, which unless block_given? last = column_size - 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_size.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_size].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 Matrix.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 463 def each_with_index(which = :all) # :yield: e, row, column return to_enum :each_with_index, which unless block_given? last = column_size - 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_size.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_size].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 Matrix.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 1187 def eigensystem EigenvalueDecomposition.new(self) end
# File matrix.rb, line 1308 def elements_to_f warn "#{caller(1)[0]}: warning: Matrix#elements_to_f is deprecated, use map(&:to_f)" map(&:to_f) end
# File matrix.rb, line 1313 def elements_to_i warn "#{caller(1)[0]}: warning: Matrix#elements_to_i is deprecated, use map(&:to_i)" map(&:to_i) end
# File matrix.rb, line 1318 def elements_to_r warn "#{caller(1)[0]}: warning: Matrix#elements_to_r is deprecated, use map(&:to_r)" map(&:to_r) end
Returns true
if this is an empty matrix, i.e. if the number of
rows or the number of columns is 0.
# File matrix.rb, line 606 def empty? column_size == 0 || row_size == 0 end
# File matrix.rb, line 770 def eql?(other) return false unless Matrix === other && column_size == other.column_size # necessary for empty matrices rows.eql? other.rows end
Returns a hash-code for the matrix.
# File matrix.rb, line 788 def hash @rows.hash end
Returns true
is this is an hermitian matrix. Raises an error
if matrix is not square.
# File matrix.rb, line 614 def hermitian? Matrix.Raise ErrDimensionMismatch unless square? each_with_index(:strict_upper).all? do |e, row, col| e == rows[col][row].conj end 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 1234 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 526 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 1343 def inspect if empty? "Matrix.empty(#{row_size}, #{column_size})" else "Matrix#{@rows.inspect}" end end
Returns the inverse of the matrix.
Matrix[[-1, -1], [0, -1]].inverse => -1 1 0 -1
# File matrix.rb, line 909 def inverse Matrix.Raise ErrDimensionMismatch unless square? Matrix.I(row_size).send(:inverse_from, self) end
Returns true
is this is a lower triangular matrix.
# File matrix.rb, line 624 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 == a * p # => true a.lup.solve([2, 5]) # => Vector[(1/1), (1/2)]
# File matrix.rb, line 1202 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_size or #column_size respectively.
# File matrix.rb, line 556 def minor(*param) case param.size when 2 row_range, col_range = param from_row = row_range.first from_row += row_size if from_row < 0 to_row = row_range.end to_row += row_size 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_size if from_col < 0 to_col = col_range.end to_col += column_size 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_size if from_row < 0 from_col += column_size if from_col < 0 else Matrix.Raise ArgumentError, param.inspect end return nil if from_row > row_size || from_col > column_size || from_row < 0 || from_col < 0 rows = @rows[from_row, size_row].collect{|row| row[from_col, size_col] } new_matrix rows, [column_size - from_col, size_col].min end
Returns true
is this is a normal matrix. Raises an error if
matrix is not square.
# File matrix.rb, line 632 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
is this is an orthogonal matrix Raises an error
if matrix is not square.
# File matrix.rb, line 650 def orthogonal? Matrix.Raise ErrDimensionMismatch unless square? rows.each_with_index do |row, i| column_size.times do |j| s = 0 row_size.times do |k| s += row[k] * rows[k][j] end return false unless s == (i == j ? 1 : 0) end end true end
Returns true
is this is a permutation matrix Raises an error
if matrix is not square.
# File matrix.rb, line 668 def permutation? Matrix.Raise ErrDimensionMismatch unless square? cols = Array.new(column_size) 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 1103 def rank # We currently use Bareiss' multistep integer-preserving gaussian elimination # (see comments on determinant) a = to_a last_column = column_size - 1 last_row = row_size - 1 pivot_row = 0 previous_pivot = 1 0.upto(last_column) do |k| switch_row = (pivot_row .. last_row).find {|row| a[row][k] != 0 } if switch_row a[switch_row], a[pivot_row] = a[pivot_row], a[switch_row] unless pivot_row == switch_row pivot = a[pivot_row][k] (pivot_row+1).upto(last_row) do |i| ai = a[i] (k+1).upto(last_column) do |j| ai[j] = (pivot * ai[j] - ai[k] * a[pivot_row][j]) / previous_pivot end end pivot_row += 1 previous_pivot = pivot end end pivot_row end
deprecated; use #rank
# File matrix.rb, line 1134 def rank_e warn "#{caller(1)[0]}: warning: Matrix#rank_e is deprecated; use #rank" rank end
Returns the real part of the matrix.
Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]] => 1+2i i 0 1 2 3 Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]].real => 1 0 0 1 2 3
# File matrix.rb, line 1248 def real collect(&:real) end
Returns true
if all entries of the matrix are real.
# File matrix.rb, line 689 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 1258 def rect [real, imag] end
Returns true
if this is a regular (i.e. non-singular) matrix.
# File matrix.rb, line 696 def regular? not singular? end
Returns a matrix with entries rounded to the given precision (see Float#round)
# File matrix.rb, line 1142 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 342 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 329 def row_size @rows.size end
Returns an array of the row vectors of the matrix. See Vector.
# File matrix.rb, line 1286 def row_vectors Array.new(row_size) {|i| row(i) } end
Returns true
is this is a singular matrix.
# File matrix.rb, line 703 def singular? determinant == 0 end
Returns true
is this is a square matrix.
# File matrix.rb, line 710 def square? column_size == row_size end
Returns true
is this is a symmetric matrix. Raises an error if
matrix is not square.
# File matrix.rb, line 718 def symmetric? Matrix.Raise ErrDimensionMismatch unless square? each_with_index(:strict_upper).all? do |e, row, col| e == rows[col][row] end end
Returns an array of arrays that describe the rows of the matrix.
# File matrix.rb, line 1304 def to_a @rows.collect(&:dup) end
Overrides Object#to_s
# File matrix.rb, line 1330 def to_s if empty? "Matrix.empty(#{row_size}, #{column_size})" else "Matrix[" + @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 1151 def trace Matrix.Raise ErrDimensionMismatch unless square? (0...column_size).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 1169 def transpose return Matrix.empty(column_size, 0) if row_size.zero? new_matrix @rows.transpose, row_size end
Returns true
is this is a unitary matrix Raises an error if
matrix is not square.
# File matrix.rb, line 729 def unitary? Matrix.Raise ErrDimensionMismatch unless square? rows.each_with_index do |row, i| column_size.times do |j| s = 0 row_size.times do |k| s += row[k].conj * rows[k][j] end return false unless s == (i == j ? 1 : 0) end end true end
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