Class: Matrix

Inherits:
Object
  • Object
show all
Extended by:
ConversionHelper
Includes:
Enumerable, ExceptionForMatrix, CoercionHelper
Defined in:
lib/matrix.rb,
lib/matrix/version.rb,
lib/matrix/lup_decomposition.rb,
lib/matrix/eigenvalue_decomposition.rb

Overview

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.

Defined Under Namespace

Modules: CoercionHelper, ConversionHelper Classes: EigenvalueDecomposition, LUPDecomposition, Scalar

Constant Summary collapse

SELECTORS = 
{all: true, diagonal: true, off_diagonal: true, lower: true, strict_lower: true, strict_upper: true, upper: true}.freeze
VERSION = 
"0.2.0"

Instance Attribute Summary collapse

Class Method Summary collapse

Instance Method Summary collapse

Methods included from CoercionHelper

check_int, check_range, coerce_to, coerce_to_int, coerce_to_matrix

Constructor Details

#initialize(rows, column_count = rows[0].size) ⇒ Matrix

Matrix.new is private; use Matrix.rows, columns, [], etc… to create.



311
312
313
314
315
316
317
 
# File 'lib/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

Instance Attribute Details

#column_count ⇒ Object (readonly) Also known as: column_size

Returns the number of columns.



445
446
447
 
# File 'lib/matrix.rb', line 445

def column_count
  @column_count
end

Class Method Details

.[](*rows) ⇒ Object

Creates a matrix where each argument is a row.

Matrix[ [25, 93], [-1, 66] ]
   =>  25 93
       -1 66



78
79
80
 
# File 'lib/matrix.rb', line 78

def Matrix.[](*rows)
  rows(rows, false)
end

.build(row_count, column_count = row_count) ⇒ Object

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

Raises:

  • (ArgumentError) 


123
124
125
126
127
128
129
130
131
132
133
134
 
# File 'lib/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

.column_vector(column) ⇒ Object

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



209
210
211
212
 
# File 'lib/matrix.rb', line 209

def Matrix.column_vector(column)
  column = convert_to_array(column)
  new [column].transpose, 1
end

.columns(columns) ⇒ Object

Creates a matrix using columns as an array of column vectors.

Matrix.columns([[25, 93], [-1, 66]])
   =>  25 -1
       93 66



108
109
110
 
# File 'lib/matrix.rb', line 108

def Matrix.columns(columns)
  rows(columns, false).transpose
end

.combine(*matrices) ⇒ Object

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]]



286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
 
# File 'lib/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

.diagonal(*values) ⇒ Object

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



143
144
145
146
147
148
149
150
151
152
 
# File 'lib/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

.empty(row_count = 0, column_count = 0) ⇒ Object

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]]

Raises:

  • (ArgumentError) 


227
228
229
230
231
232
 
# File 'lib/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

.hstack(x, *matrices) ⇒ Object

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]]



262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
 
# File 'lib/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

.identity(n) ⇒ Object Also known as: unit, I

Creates an n by n identity matrix.

Matrix.identity(2)
  => 1 0
     0 1



171
172
173
 
# File 'lib/matrix.rb', line 171

def Matrix.identity(n)
  scalar(n, 1)
end

.row_vector(row) ⇒ Object

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



196
197
198
199
 
# File 'lib/matrix.rb', line 196

def Matrix.row_vector(row)
  row = convert_to_array(row)
  new [row]
end

.rows(rows, copy = true) ⇒ Object

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



90
91
92
93
94
95
96
97
98
99
100
 
# File 'lib/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

.scalar(n, value) ⇒ Object

Creates an n by n diagonal matrix where each diagonal element is value.

Matrix.scalar(2, 5)
  => 5 0
     0 5



161
162
163
 
# File 'lib/matrix.rb', line 161

def Matrix.scalar(n, value)
  diagonal(*Array.new(n, value))
end

.vstack(x, *matrices) ⇒ Object

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]]



241
242
243
244
245
246
247
248
249
250
251
252
 
# File 'lib/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

.zero(row_count, column_count = row_count) ⇒ Object

Creates a zero matrix.

Matrix.zero(2)
  => 0 0
     0 0



185
186
187
188
 
# File 'lib/matrix.rb', line 185

def Matrix.zero(row_count, column_count = row_count)
  rows = Array.new(row_count){Array.new(column_count, 0)}
  new rows, column_count
end

Instance Method Details

#*(m) ⇒ Object

Matrix multiplication.

Matrix[[2,4], [6,8]] * Matrix.identity(2)
  => 2 4
     6 8



1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
 
# File 'lib/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

#**(other) ⇒ Object

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



1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
 
# File 'lib/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

#+(m) ⇒ Object

Matrix addition.

Matrix.scalar(2,5) + Matrix[[1,0], [-4,7]]
  =>  6  0
     -4 12

Raises:



1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
 
# File 'lib/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

#+@ ⇒ Object 



1245
1246
1247
 
# File 'lib/matrix.rb', line 1245

def +@
  self
end

#-(m) ⇒ Object

Matrix subtraction.

Matrix[[1,5], [4,2]] - Matrix[[9,3], [-4,1]]
  => -8  2
      8  1

Raises:



1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
 
# File 'lib/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

#-@ ⇒ Object 



1249
1250
1251
 
# File 'lib/matrix.rb', line 1249

def -@
  collect {|e| -e }
end

#/(other) ⇒ Object

Matrix division (multiplication by the inverse).

Matrix[[7,6], [3,9]] / Matrix[[2,9], [3,1]]
  => -7  1
     -3 -6



1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
 
# File 'lib/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

#==(other) ⇒ Object

Returns true if and only if the two matrices contain equal elements.



1008
1009
1010
1011
1012
 
# File 'lib/matrix.rb', line 1008

def ==(other)
  return false unless Matrix === other &&
                      column_count == other.column_count # necessary for empty matrices
  rows == other.rows
end

#[](i, j) ⇒ Object Also known as: element, component

Returns element (i,j) of the matrix. That is: row i, column j.



326
327
328
 
# File 'lib/matrix.rb', line 326

def [](i, j)
  @rows.fetch(i){return nil}[j]
end

#[]=(i, j, v) ⇒ Object Also known as: set_element, set_component

:call-seq:

matrix[range, range] = matrix/element
matrix[range, integer] = vector/column_matrix/element
matrix[integer, range] = vector/row_matrix/element
matrix[integer, integer] = element

Set element or elements of matrix.

Raises:

  • (FrozenError) 


340
341
342
343
344
345
346
347
348
349
350
351
352
353
 
# File 'lib/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

#abs ⇒ Object

Returns the absolute value elementwise



1256
1257
1258
 
# File 'lib/matrix.rb', line 1256

def abs
  collect(&:abs)
end

#adjugate ⇒ Object

Returns the adjugate of the matrix.

Matrix[ [7,6],[3,9] ].adjugate
  => 9 -6
     -3 7

Raises:



781
782
783
784
785
786
 
# File 'lib/matrix.rb', line 781

def adjugate
  raise ErrDimensionMismatch unless square?
  Matrix.build(row_count, column_count) do |row, column|
    cofactor(column, row)
  end
end

#antisymmetric? ⇒ Boolean Also known as: skew_symmetric?

Returns true if this is an antisymmetric matrix. Raises an error if matrix is not square.

Returns:

  • (Boolean)

Raises:



960
961
962
963
964
965
966
 
# File 'lib/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

#coerce(other) ⇒ Object

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.



1565
1566
1567
1568
1569
1570
1571
1572
 
# File 'lib/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

#cofactor(row, column) ⇒ Object

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

Raises:

  • (RuntimeError) 


766
767
768
769
770
771
772
 
# File 'lib/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

#collect(which = :all, &block) ⇒ Object Also known as: map

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



497
498
499
500
 
# File 'lib/matrix.rb', line 497

def collect(which = :all, &block) # :yield: e
  return to_enum(:collect, which) unless block_given?
  dup.collect!(which, &block)
end

#collect!(which = :all) ⇒ Object Also known as: map!

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

Raises:

  • (FrozenError) 


515
516
517
518
519
 
# File 'lib/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

#column(j) ⇒ Object

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.



466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
 
# File 'lib/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

#column_vectors ⇒ Object

Returns an array of the column vectors of the matrix. See Vector.



1586
1587
1588
1589
1590
 
# File 'lib/matrix.rb', line 1586

def column_vectors
  Array.new(column_count) {|i|
    column(i)
  }
end

#combine(*matrices, &block) ⇒ Object 



304
305
306
 
# File 'lib/matrix.rb', line 304

def combine(*matrices, &block)
  Matrix.combine(self, *matrices, &block)
end

#conjugate ⇒ Object Also known as: conj

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



1511
1512
1513
 
# File 'lib/matrix.rb', line 1511

def conjugate
  collect(&:conjugate)
end

#determinant ⇒ Object Also known as: det

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

Raises:



1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
 
# File 'lib/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

#determinant_e ⇒ Object Also known as: det_e

deprecated; use Matrix#determinant



1355
1356
1357
1358
 
# File 'lib/matrix.rb', line 1355

def determinant_e
  warn "Matrix#determinant_e is deprecated; use #determinant", uplevel: 1
  determinant
end

#diagonal? ⇒ Boolean

Returns true if this is a diagonal matrix. Raises an error if matrix is not square.

Returns:

  • (Boolean)

Raises:



827
828
829
830
 
# File 'lib/matrix.rb', line 827

def diagonal?
  raise ErrDimensionMismatch unless square?
  each(:off_diagonal).all?(&:zero?)
end

#each(which = :all, &block) ⇒ Object

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]



544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
 
# File 'lib/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

#each_with_index(which = :all) ⇒ Object

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



604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
 
# File 'lib/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

#eigensystem ⇒ Object Also known as: eigen

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



1478
1479
1480
 
# File 'lib/matrix.rb', line 1478

def eigensystem
  EigenvalueDecomposition.new(self)
end

#elements_to_f ⇒ Object

Deprecated.

Use map(&:to_f)



1609
1610
1611
1612
 
# File 'lib/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

#elements_to_i ⇒ Object

Deprecated.

Use map(&:to_i)



1617
1618
1619
1620
 
# File 'lib/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

#elements_to_r ⇒ Object

Deprecated.

Use map(&:to_r)



1625
1626
1627
1628
 
# File 'lib/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

#empty? ⇒ Boolean

Returns true if this is an empty matrix, i.e. if the number of rows or the number of columns is 0.

Returns:

  • (Boolean) 


836
837
838
 
# File 'lib/matrix.rb', line 836

def empty?
  column_count == 0 || row_count == 0
end

#eql?(other) ⇒ Boolean

Returns:

  • (Boolean) 


1014
1015
1016
1017
1018
 
# File 'lib/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

#first_minor(row, column) ⇒ Object

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

Raises:

  • (RuntimeError) 


739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
 
# File 'lib/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

#freeze ⇒ Object 



523
524
525
526
 
# File 'lib/matrix.rb', line 523

def freeze
  @rows.freeze
  super
end

#hadamard_product(m) ⇒ Object Also known as: entrywise_product

Hadamard product

Matrix[[1,2], [3,4]].hadamard_product(Matrix[[1,2], [3,2]])
  => 1  4
     9  8



1152
1153
1154
 
# File 'lib/matrix.rb', line 1152

def hadamard_product(m)
  combine(m){|a, b| a * b}
end

#hash ⇒ Object

Returns a hash-code for the matrix.



1031
1032
1033
 
# File 'lib/matrix.rb', line 1031

def hash
  @rows.hash
end

#hermitian? ⇒ Boolean

Returns true if this is an hermitian matrix. Raises an error if matrix is not square.

Returns:

  • (Boolean)

Raises:



844
845
846
847
848
849
 
# File 'lib/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

#hstack(*matrices) ⇒ Object

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]]



1369
1370
1371
 
# File 'lib/matrix.rb', line 1369

def hstack(*matrices)
  self.class.hstack(self, *matrices)
end

#imaginary ⇒ Object Also known as: imag

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



1525
1526
1527
 
# File 'lib/matrix.rb', line 1525

def imaginary
  collect(&:imaginary)
end

#index(*args) ⇒ Object Also known as: find_index

:call-seq:

index(value, selector = :all) -> [row, column]
index(selector = :all){ block } -> [row, column]
index(selector = :all) -> an_enumerator

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]

Raises:

  • (ArgumentError) 


667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
 
# File 'lib/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

#inspect ⇒ Object

Overrides Object#inspect



1650
1651
1652
1653
1654
1655
1656
 
# File 'lib/matrix.rb', line 1650

def inspect
  if empty?
    "#{self.class}.empty(#{row_count}, #{column_count})"
  else
    "#{self.class}#{@rows.inspect}"
  end
end

#inverse ⇒ Object Also known as: inv

Returns the inverse of the matrix.

Matrix[[-1, -1], [0, -1]].inverse
  => -1  1
      0 -1

Raises:



1163
1164
1165
1166
 
# File 'lib/matrix.rb', line 1163

def inverse
  raise ErrDimensionMismatch unless square?
  self.class.I(row_count).send(:inverse_from, self)
end

#laplace_expansion(row: nil, column: nil) ⇒ Object Also known as: cofactor_expansion

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]

Raises:



798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
 
# File 'lib/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

#lower_triangular? ⇒ Boolean

Returns true if this is a lower triangular matrix.

Returns:

  • (Boolean) 


854
855
856
 
# File 'lib/matrix.rb', line 854

def lower_triangular?
  each(:strict_upper).all?(&:zero?)
end

#lup ⇒ Object Also known as: lup_decomposition

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)]



1493
1494
1495
 
# File 'lib/matrix.rb', line 1493

def lup
  LUPDecomposition.new(self)
end

#minor(*param) ⇒ Object

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.



698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
 
# File 'lib/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

#normal? ⇒ Boolean

Returns true if this is a normal matrix. Raises an error if matrix is not square.

Returns:

  • (Boolean)

Raises:



862
863
864
865
866
867
868
869
870
871
872
873
874
 
# File 'lib/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

#orthogonal? ⇒ Boolean

Returns true if this is an orthogonal matrix Raises an error if matrix is not square.

Returns:

  • (Boolean)

Raises:



880
881
882
883
884
885
886
887
888
889
890
891
892
 
# File 'lib/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

#permutation? ⇒ Boolean

Returns true if this is a permutation matrix Raises an error if matrix is not square.

Returns:

  • (Boolean)

Raises:



898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
 
# File 'lib/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

#rank ⇒ Object

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



1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
 
# File 'lib/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

#rank_e ⇒ Object

deprecated; use Matrix#rank



1413
1414
1415
1416
 
# File 'lib/matrix.rb', line 1413

def rank_e
  warn "Matrix#rank_e is deprecated; use #rank", uplevel: 1
  rank
end

#real ⇒ Object

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



1539
1540
1541
 
# File 'lib/matrix.rb', line 1539

def real
  collect(&:real)
end

#real? ⇒ Boolean

Returns true if all entries of the matrix are real.

Returns:

  • (Boolean) 


919
920
921
 
# File 'lib/matrix.rb', line 919

def real?
  all?(&:real?)
end

#rect ⇒ Object Also known as: rectangular

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



1549
1550
1551
 
# File 'lib/matrix.rb', line 1549

def rect
  [real, imag]
end

#regular? ⇒ Boolean

Returns true if this is a regular (i.e. non-singular) matrix.

Returns:

  • (Boolean) 


926
927
928
 
# File 'lib/matrix.rb', line 926

def regular?
  not singular?
end

#round(ndigits = 0) ⇒ Object

Returns a matrix with entries rounded to the given precision (see Float#round)



1421
1422
1423
 
# File 'lib/matrix.rb', line 1421

def round(ndigits=0)
  map{|e| e.round(ndigits)}
end

#row(i, &block) ⇒ Object

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.



452
453
454
455
456
457
458
459
 
# File 'lib/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

#row_count ⇒ Object Also known as: row_size

Returns the number of rows.



437
438
439
 
# File 'lib/matrix.rb', line 437

def row_count
  @rows.size
end

#row_vectors ⇒ Object

Returns an array of the row vectors of the matrix. See Vector.



1577
1578
1579
1580
1581
 
# File 'lib/matrix.rb', line 1577

def row_vectors
  Array.new(row_count) {|i|
    row(i)
  }
end

#singular? ⇒ Boolean

Returns true if this is a singular matrix.

Returns:

  • (Boolean) 


933
934
935
 
# File 'lib/matrix.rb', line 933

def singular?
  determinant == 0
end

#square? ⇒ Boolean

Returns true if this is a square matrix.

Returns:

  • (Boolean) 


940
941
942
 
# File 'lib/matrix.rb', line 940

def square?
  column_count == row_count
end

#symmetric? ⇒ Boolean

Returns true if this is a symmetric matrix. Raises an error if matrix is not square.

Returns:

  • (Boolean)

Raises:



948
949
950
951
952
953
954
 
# File 'lib/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

#to_a ⇒ Object

Returns an array of arrays that describe the rows of the matrix.



1602
1603
1604
 
# File 'lib/matrix.rb', line 1602

def to_a
  @rows.collect(&:dup)
end

#to_matrix ⇒ Object

Explicit conversion to a Matrix. Returns self



1595
1596
1597
 
# File 'lib/matrix.rb', line 1595

def to_matrix
  self
end

#to_s ⇒ Object

Overrides Object#to_s



1637
1638
1639
1640
1641
1642
1643
1644
1645
 
# File 'lib/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

#trace ⇒ Object Also known as: tr

Returns the trace (sum of diagonal elements) of the matrix.

Matrix[[7,6], [3,9]].trace
  => 16

Raises:



1430
1431
1432
1433
1434
1435
 
# File 'lib/matrix.rb', line 1430

def trace
  raise ErrDimensionMismatch unless square?
  (0...column_count).inject(0) do |tr, i|
    tr + @rows[i][i]
  end
end

#transpose ⇒ Object Also known as: t

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



1448
1449
1450
1451
 
# File 'lib/matrix.rb', line 1448

def transpose
  return self.class.empty(column_count, 0) if row_count.zero?
  new_matrix @rows.transpose, row_count
end

#unitary? ⇒ Boolean

Returns true if this is a unitary matrix Raises an error if matrix is not square.

Returns:

  • (Boolean)

Raises:



973
974
975
976
977
978
979
980
981
982
983
984
985
 
# File 'lib/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

#upper_triangular? ⇒ Boolean

Returns true if this is an upper triangular matrix.

Returns:

  • (Boolean) 


990
991
992
 
# File 'lib/matrix.rb', line 990

def upper_triangular?
  each(:strict_lower).all?(&:zero?)
end

#vstack(*matrices) ⇒ Object

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]]



1462
1463
1464
 
# File 'lib/matrix.rb', line 1462

def vstack(*matrices)
  self.class.vstack(self, *matrices)
end

#zero? ⇒ Boolean

Returns true if this is a matrix with only zero elements

Returns:

  • (Boolean) 


997
998
999
 
# File 'lib/matrix.rb', line 997

def zero?
  all?(&:zero?)
end