### In Files

• matrix.rb
• matrix/eigenvalue_decomposition.rb
• matrix/lup_decomposition.rb

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# Matrix

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.

SELECTORS

### Attributes

column_count[R]

Returns the number of columns.

column_size[R]

Returns the number of columns.

rows[R]

instance creations

### Public Class Methods

I(n) click to toggle source
Alias for: identity
[](*rows) click to toggle source

Creates a matrix where each argument is a row.

```Matrix[ [25, 93], [-1, 66] ]
=>  25 93
-1 66```
```
# File matrix.rb, line 51
def Matrix.[](*rows)
rows(rows, false)
end
```
build(row_count, column_count = row_count) click to toggle source

Creates a matrix of size `row_count` x `column_count`. It fills the values by calling the given block, passing the current row and column. Returns an enumerator if no block is given.

```m = Matrix.build(2, 4) {|row, col| col - row }
=> Matrix[[0, 1, 2, 3], [-1, 0, 1, 2]]
m = Matrix.build(3) { rand }
=> a 3x3 matrix with random elements```
```
# File matrix.rb, line 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
```
column_vector(column) click to toggle source

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 182
def Matrix.column_vector(column)
column = convert_to_array(column)
new [column].transpose, 1
end
```
columns(columns) click to toggle source

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 81
def Matrix.columns(columns)
rows(columns, false).transpose
end
```
combine(*matrices) click to toggle source

Create a matrix by combining matrices entrywise, using the given block

```x = Matrix[[6, 6], [4, 4]]
y = Matrix[[1, 2], [3, 4]]
Matrix.combine(x, y) {|a, b| a - b} # => Matrix[[5, 4], [1, 0]]
```
```
# File matrix.rb, line 259
def Matrix.combine(*matrices)

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
```
diagonal(*values) click to toggle source

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 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
```
empty(row_count = 0, column_count = 0) click to toggle source

Creates a empty matrix of `row_count` x `column_count`. At least one of `row_count` or `column_count` must be 0.

```m = Matrix.empty(2, 0)
m == Matrix[ [], [] ]
=> true
n = Matrix.empty(0, 3)
n == Matrix.columns([ [], [], [] ])
=> true
m * n
=> Matrix[[0, 0, 0], [0, 0, 0]]```
```
# File matrix.rb, line 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
```
hstack(x, *matrices) click to toggle source

Create a matrix by stacking matrices horizontally

```x = Matrix[[1, 2], [3, 4]]
y = Matrix[[5, 6], [7, 8]]
Matrix.hstack(x, y) # => Matrix[[1, 2, 5, 6], [3, 4, 7, 8]]
```
```
# File matrix.rb, line 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
```
identity(n) click to toggle source

Creates an `n` by `n` identity matrix.

```Matrix.identity(2)
=> 1 0
0 1```
```
# File matrix.rb, line 144
def Matrix.identity(n)
scalar(n, 1)
end
```
Also aliased as: unit, I
new(rows, column_count = rows[0].size) click to toggle source

`Matrix.new` is private; use `Matrix.rows`, columns, [], etcâ€¦ to create.

```
# File 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
```
row_vector(row) click to toggle source

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 169
def Matrix.row_vector(row)
row = convert_to_array(row)
new [row]
end
```
rows(rows, copy = true) click to toggle source

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 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
```
scalar(n, value) click to toggle source

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

```Matrix.scalar(2, 5)
=> 5 0
0 5```
```
# File matrix.rb, line 134
def Matrix.scalar(n, value)
diagonal(*Array.new(n, value))
end
```
unit(n) click to toggle source
Alias for: identity
vstack(x, *matrices) click to toggle source

Create a matrix by stacking matrices vertically

```x = Matrix[[1, 2], [3, 4]]
y = Matrix[[5, 6], [7, 8]]
Matrix.vstack(x, y) # => Matrix[[1, 2], [3, 4], [5, 6], [7, 8]]
```
```
# File matrix.rb, line 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
```
zero(row_count, column_count = row_count) click to toggle source

Creates a zero matrix.

```Matrix.zero(2)
=> 0 0
0 0```
```
# File 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
```

### Public Instance Methods

*(m) click to toggle source

`Matrix` multiplication.

```Matrix[[2,4], [6,8]] * Matrix.identity(2)
=> 2 4
6 8```
```
# File matrix.rb, line 1019
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
```
**(other) click to toggle source

`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 1196
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
```
+(m) click to toggle source

`Matrix` addition.

```Matrix.scalar(2,5) + Matrix[[1,0], [-4,7]]
=>  6  0
-4 12```
```
# File matrix.rb, line 1052
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
```
+@() click to toggle source
```
# File matrix.rb, line 1219
def +@
self
end
```
-(m) click to toggle source

`Matrix` subtraction.

```Matrix[[1,5], [4,2]] - Matrix[[9,3], [-4,1]]
=> -8  2
8  1```
```
# File matrix.rb, line 1079
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
```
-@() click to toggle source
```
# File matrix.rb, line 1223
def -@
collect {|e| -e }
end
```
/(other) click to toggle source

`Matrix` division (multiplication by the inverse).

```Matrix[[7,6], [3,9]] / Matrix[[2,9], [3,1]]
=> -7  1
-3 -6```
```
# File matrix.rb, line 1106
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) click to toggle source

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

```
# File matrix.rb, line 982
def ==(other)
return false unless Matrix === other &&
column_count == other.column_count # necessary for empty matrices
rows == other.rows
end
```
[](i, j) click to toggle source

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

```
# File matrix.rb, line 299
def [](i, j)
@rows.fetch(i){return nil}[j]
end
```
Also aliased as: element, component
matrix[range, range] = matrix/element click to toggle source
matrix[range, integer] = vector/column_matrix/element
matrix[integer, range] = vector/row_matrix/element
matrix[integer, integer] = element

Set element or elements of matrix.

```
# File matrix.rb, line 313
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
```
Also aliased as: set_element, set_component
adjugate() click to toggle source

Returns the adjugate of the matrix.

```Matrix[ [7,6],[3,9] ].adjugate
=> 9 -6
-3 7```
```
# File matrix.rb, line 755
Matrix.Raise ErrDimensionMismatch unless square?
Matrix.build(row_count, column_count) do |row, column|
cofactor(column, row)
end
end
```
antisymmetric?() click to toggle source

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

```
# File matrix.rb, line 934
def antisymmetric?
Matrix.Raise ErrDimensionMismatch unless square?
each_with_index(:upper) do |e, row, col|
return false unless e == -rows[col][row]
end
true
end
```
Also aliased as: skew_symmetric?
coerce(other) click to toggle source

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 1532
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) click to toggle source

Returns the (row, column) cofactor which is obtained by multiplying the first minor by (-1)**(row + column).

```Matrix.diagonal(9, 5, -3, 4).cofactor(1, 1)
=> -108```
```
# File matrix.rb, line 740
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
```
cofactor_expansion(row: nil, column: nil) click to toggle source
Alias for: laplace_expansion
collect(which = :all) click to toggle source

Returns a matrix that is the result of iteration of the given block over all elements of the matrix. Elements can be restricted by passing an argument:

• :all (default): yields all elements

• :diagonal: yields only elements on the diagonal

• :off_diagonal: yields all elements except on the diagonal

• :lower: yields only elements on or below the diagonal

• :strict_lower: yields only elements below the diagonal

• :strict_upper: yields only elements above the diagonal

• :upper: yields only elements on or above the diagonal Matrix[ [1,2], [3,4] ].collect { |e| e**2 }

```=> 1  4
9 16```
```
# File matrix.rb, line 470
def collect(which = :all, &block) # :yield: e
dup.collect!(which, &block)
end
```
Also aliased as: map
collect!(which = :all) click to toggle source

Invokes the given block for each element of matrix, replacing the element with the value returned by the block. Elements can be restricted by passing an argument:

• :all (default): yields all elements

• :diagonal: yields only elements on the diagonal

• :off_diagonal: yields all elements except on the diagonal

• :lower: yields only elements on or below the diagonal

• :strict_lower: yields only elements below the diagonal

• :strict_upper: yields only elements above the diagonal

• :upper: yields only elements on or above the diagonal

```
# File matrix.rb, line 488
def collect!(which = :all)
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
```
Also aliased as: map!
column(j) click to toggle source

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 439
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() click to toggle source

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

```
# File matrix.rb, line 1553
def column_vectors
Array.new(column_count) {|i|
column(i)
}
end
```
combine(*matrices, &block) click to toggle source
```
# File matrix.rb, line 277
def combine(*matrices, &block)
Matrix.combine(self, *matrices, &block)
end
```
component(i, j) click to toggle source
Alias for: []
conj() click to toggle source
Alias for: conjugate
conjugate() click to toggle source

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 1478
def conjugate
collect(&:conjugate)
end
```
Also aliased as: conj
det() click to toggle source
Alias for: determinant
det_e() click to toggle source
Alias for: determinant_e
determinant() click to toggle source

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 1241
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
```
Also aliased as: det
determinant_e() click to toggle source

deprecated; use `Matrix#determinant`

```
# File matrix.rb, line 1322
def determinant_e
warn "Matrix#determinant_e is deprecated; use #determinant", uplevel: 1
determinant
end
```
Also aliased as: det_e
diagonal?() click to toggle source

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

```
# File matrix.rb, line 801
def diagonal?
Matrix.Raise ErrDimensionMismatch unless square?
each(:off_diagonal).all?(&:zero?)
end
```
each(which = :all) click to toggle source

Yields all elements of the matrix, starting with those of the first row, or returns an Enumerator if no block given. Elements can be restricted by passing an argument:

• :all (default): yields all elements

• :diagonal: yields only elements on the diagonal

• :off_diagonal: yields all elements except on the diagonal

• :lower: yields only elements on or below the diagonal

• :strict_lower: yields only elements below the diagonal

• :strict_upper: yields only elements above the diagonal

• :upper: yields only elements on or above the diagonal

Matrix[ [1,2], [3,4] ].each { |e| puts e }

```# => prints the numbers 1 to 4
```

Matrix[ [1,2], [3,4] ].each(:strict_lower).to_a # => [3]

```
# File matrix.rb, line 517
def each(which = :all) # :yield: e
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
```
each_with_index(which = :all) click to toggle source

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 578
def each_with_index(which = :all) # :yield: e, row, column
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
```
eigen() click to toggle source
Alias for: eigensystem
eigensystem() click to toggle source

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 1445
def eigensystem
EigenvalueDecomposition.new(self)
end
```
Also aliased as: eigen
element(i, j) click to toggle source
Alias for: []
elements_to_f() click to toggle source

Deprecated.

Use map(&:to_f)

```
# File matrix.rb, line 1576
def elements_to_f
warn "Matrix#elements_to_f is deprecated, use map(&:to_f)", uplevel: 1
map(&:to_f)
end
```
elements_to_i() click to toggle source

Deprecated.

Use map(&:to_i)

```
# File matrix.rb, line 1584
def elements_to_i
warn "Matrix#elements_to_i is deprecated, use map(&:to_i)", uplevel: 1
map(&:to_i)
end
```
elements_to_r() click to toggle source

Deprecated.

Use map(&:to_r)

```
# File matrix.rb, line 1592
def elements_to_r
warn "Matrix#elements_to_r is deprecated, use map(&:to_r)", uplevel: 1
map(&:to_r)
end
```
empty?() click to toggle source

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 810
def empty?
column_count == 0 || row_count == 0
end
```
entrywise_product(m) click to toggle source
eql?(other) click to toggle source
```
# File matrix.rb, line 988
def eql?(other)
return false unless Matrix === other &&
column_count == other.column_count # necessary for empty matrices
rows.eql? other.rows
end
```
find_index(*args) click to toggle source
Alias for: index
first_minor(row, column) click to toggle source

Returns the submatrix obtained by deleting the specified row and column.

```Matrix.diagonal(9, 5, -3, 4).first_minor(1, 2)
=> 9 0 0
0 0 0
0 0 4```
```
# File matrix.rb, line 713
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() click to toggle source
```
# File matrix.rb, line 496
def freeze
@rows.freeze
super
end
```
hadamard_product(m) click to toggle source

```Matrix[[1,2], [3,4]].hadamard_product(Matrix[[1,2], [3,2]])
=> 1  4
9  8```
```
# File matrix.rb, line 1126
combine(m){|a, b| a * b}
end
```
Also aliased as: entrywise_product
hash() click to toggle source

Returns a hash-code for the matrix.

```
# File matrix.rb, line 1005
def hash
@rows.hash
end
```
hermitian?() click to toggle source

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

```
# File matrix.rb, line 818
def hermitian?
Matrix.Raise ErrDimensionMismatch unless square?
each_with_index(:upper).all? do |e, row, col|
e == rows[col][row].conj
end
end
```
hstack(*matrices) click to toggle source

Returns a new matrix resulting by stacking horizontally the receiver with the given matrices

```x = Matrix[[1, 2], [3, 4]]
y = Matrix[[5, 6], [7, 8]]
x.hstack(y) # => Matrix[[1, 2, 5, 6], [3, 4, 7, 8]]
```
```
# File matrix.rb, line 1336
def hstack(*matrices)
self.class.hstack(self, *matrices)
end
```
imag() click to toggle source
Alias for: imaginary
imaginary() click to toggle source

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 1492
def imaginary
collect(&:imaginary)
end
```
Also aliased as: imag
index(value, selector = :all) → [row, column] click to toggle source
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]
```
```
# File matrix.rb, line 641
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
```
Also aliased as: find_index
inspect() click to toggle source

Overrides Object#inspect

```
# File matrix.rb, line 1617
def inspect
if empty?
"#{self.class}.empty(#{row_count}, #{column_count})"
else
"#{self.class}#{@rows.inspect}"
end
end
```
inv() click to toggle source
Alias for: inverse
inverse() click to toggle source

Returns the inverse of the matrix.

```Matrix[[-1, -1], [0, -1]].inverse
=> -1  1
0 -1```
```
# File matrix.rb, line 1137
def inverse
Matrix.Raise ErrDimensionMismatch unless square?
self.class.I(row_count).send(:inverse_from, self)
end
```
Also aliased as: inv
laplace_expansion(row: nil, column: nil) click to toggle source

Returns the Laplace expansion along given row or column.

```Matrix[[7,6], [3,9]].laplace_expansion(column: 1)
=> 45

Matrix[[Vector[1, 0], Vector[0, 1]], [2, 3]].laplace_expansion(row: 0)
=> Vector[3, -2]```
```
# File matrix.rb, line 772
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
```
Also aliased as: cofactor_expansion
lower_triangular?() click to toggle source

Returns `true` if this is a lower triangular matrix.

```
# File matrix.rb, line 828
def lower_triangular?
each(:strict_upper).all?(&:zero?)
end
```
lup() click to toggle source

Returns the LUP decomposition of the matrix; see `LUPDecomposition`.

```a = Matrix[[1, 2], [3, 4]]
l, u, p = a.lup
l.lower_triangular? # => true
u.upper_triangular? # => true
p.permutation?      # => true
l * u == p * a      # => true
a.lup.solve([2, 5]) # => Vector[(1/1), (1/2)]
```
```
# File matrix.rb, line 1460
def lup
LUPDecomposition.new(self)
end
```
Also aliased as: lup_decomposition
lup_decomposition() click to toggle source
Alias for: lup
map(which = :all) click to toggle source
Alias for: collect
map!(which = :all) click to toggle source
Alias for: collect!
minor(*param) click to toggle source

Returns a section of the matrix. The parameters are either:

• start_row, nrows, start_col, ncols; OR

• row_range, col_range

```Matrix.diagonal(9, 5, -3).minor(0..1, 0..2)
=> 9 0 0
0 5 0```

Like Array#[], negative indices count backward from the end of the row or column (-1 is the last element). Returns nil if the starting row or column is greater than `row_count` or `column_count` respectively.

```
# File matrix.rb, line 672
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?() click to toggle source

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

```
# File matrix.rb, line 836
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
```
orthogonal?() click to toggle source

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

```
# File matrix.rb, line 854
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
```
permutation?() click to toggle source

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

```
# File matrix.rb, line 872
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
```
rank() click to toggle source

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 1349
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() click to toggle source

deprecated; use `Matrix#rank`

```
# File matrix.rb, line 1380
def rank_e
warn "Matrix#rank_e is deprecated; use #rank", uplevel: 1
rank
end
```
real() click to toggle source

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 1506
def real
collect(&:real)
end
```
real?() click to toggle source

Returns `true` if all entries of the matrix are real.

```
# File matrix.rb, line 893
def real?
all?(&:real?)
end
```
rect() click to toggle source

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 1516
def rect
[real, imag]
end
```
Also aliased as: rectangular
rectangular() click to toggle source
Alias for: rect
regular?() click to toggle source

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

```
# File matrix.rb, line 900
def regular?
not singular?
end
```
round(ndigits=0) click to toggle source

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

```
# File matrix.rb, line 1388
def round(ndigits=0)
map{|e| e.round(ndigits)}
end
```
row(i) click to toggle source

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 425
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() click to toggle source

Returns the number of rows.

```
# File matrix.rb, line 410
def row_count
@rows.size
end
```
Also aliased as: row_size
row_size() click to toggle source
Alias for: row_count
row_vectors() click to toggle source

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

```
# File matrix.rb, line 1544
def row_vectors
Array.new(row_count) {|i|
row(i)
}
end
```
singular?() click to toggle source

Returns `true` if this is a singular matrix.

```
# File matrix.rb, line 907
def singular?
determinant == 0
end
```
skew_symmetric?() click to toggle source
Alias for: antisymmetric?
square?() click to toggle source

Returns `true` if this is a square matrix.

```
# File matrix.rb, line 914
def square?
column_count == row_count
end
```
symmetric?() click to toggle source

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

```
# File matrix.rb, line 922
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
```
t() click to toggle source
Alias for: transpose
to_a() click to toggle source

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

```
# File matrix.rb, line 1569
def to_a
@rows.collect(&:dup)
end
```
to_matrix() click to toggle source

Explicit conversion to a `Matrix`. Returns self

```
# File matrix.rb, line 1562
def to_matrix
self
end
```
to_s() click to toggle source

Overrides Object#to_s

```
# File matrix.rb, line 1604
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
```
tr() click to toggle source
Alias for: trace
trace() click to toggle source

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

```Matrix[[7,6], [3,9]].trace
=> 16```
```
# File matrix.rb, line 1397
def trace
Matrix.Raise ErrDimensionMismatch unless square?
(0...column_count).inject(0) do |tr, i|
tr + @rows[i][i]
end
end
```
Also aliased as: tr
transpose() click to toggle source

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 1415
def transpose
return self.class.empty(column_count, 0) if row_count.zero?
new_matrix @rows.transpose, row_count
end
```
Also aliased as: t
unitary?() click to toggle source

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

```
# File matrix.rb, line 947
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
```
upper_triangular?() click to toggle source

Returns `true` if this is an upper triangular matrix.

```
# File matrix.rb, line 964
def upper_triangular?
each(:strict_lower).all?(&:zero?)
end
```
vstack(*matrices) click to toggle source

Returns a new matrix resulting by stacking vertically the receiver with the given matrices

```x = Matrix[[1, 2], [3, 4]]
y = Matrix[[5, 6], [7, 8]]
x.vstack(y) # => Matrix[[1, 2], [3, 4], [5, 6], [7, 8]]
```
```
# File matrix.rb, line 1429
def vstack(*matrices)
self.class.vstack(self, *matrices)
end
```
zero?() click to toggle source

Returns `true` if this is a matrix with only zero elements

```
# File matrix.rb, line 971
def zero?
all?(&:zero?)
end
```