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).

## Method Catalogue

To create a matrix:

To access Matrix elements/columns/rows/submatrices/properties:

Properties of a matrix:

Matrix arithmetic:

Matrix functions:

Matrix decompositions:

Complex arithmetic:

• conj

• conjugate

• imag

• imaginary

• real

• rect

• rectangular

Conversion to other data types:

String representations:

Namespace
Methods
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Included Modules
Constants
 SELECTORS = {all: true, diagonal: true, off_diagonal: true, lower: true, strict_lower: true, strict_upper: true, upper: true}.freeze
Attributes
 [R] column_count Returns the number of columns. [R] column_size Returns the number of columns. [R] rows instance creations
Class Public methods
I(n)
Alias for: identity
[](*rows)

Creates a matrix where each argument is a row.

``````Matrix[ [25, 93], [-1, 66] ]
=>  25 93
-1 66``````
```# File lib/matrix.rb, line 139
def Matrix.[](*rows)
rows(rows, false)
end```
build(row_count, column_count = row_count)

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

``````m = Matrix.build(2, 4) {|row, col| col - row }
=> Matrix[[0, 1, 2, 3], [-1, 0, 1, 2]]
m = Matrix.build(3) { rand }
=> a 3x3 matrix with random elements``````
```# File lib/matrix.rb, line 184
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
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)

Creates a single-column matrix where the values of that column are as given in `column`.

``````Matrix.column_vector([4,5,6])
=> 4
5
6``````
```# File lib/matrix.rb, line 269
def Matrix.column_vector(column)
column = convert_to_array(column)
new [column].transpose, 1
end```
columns(columns)

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

``````Matrix.columns([[25, 93], [-1, 66]])
=>  25 -1
93 66``````
```# File lib/matrix.rb, line 169
def Matrix.columns(columns)
rows(columns, false).transpose
end```
diagonal(*values)

Creates a matrix where the diagonal elements are composed of `values`.

``````Matrix.diagonal(9, 5, -3)
=>  9  0  0
0  5  0
0  0 -3``````
```# File lib/matrix.rb, line 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```
empty(row_count = 0, column_count = 0)

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

``````m = Matrix.empty(2, 0)
m == Matrix[ [], [] ]
=> true
n = Matrix.empty(0, 3)
n == Matrix.columns([ [], [], [] ])
=> true
m * n
=> Matrix[[0, 0, 0], [0, 0, 0]]``````
```# File lib/matrix.rb, line 287
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```
identity(n)

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

``````Matrix.identity(2)
=> 1 0
0 1``````
Also aliased as: unit, I
```# File lib/matrix.rb, line 231
def Matrix.identity(n)
scalar(n, 1)
end```
new(rows, column_count = rows[0].size)

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

```# File lib/matrix.rb, line 297
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)

Creates a single-row matrix where the values of that row are as given in `row`.

``````Matrix.row_vector([4,5,6])
=> 4 5 6``````
```# File lib/matrix.rb, line 256
def Matrix.row_vector(row)
row = convert_to_array(row)
new [row]
end```
rows(rows, copy = true)

Creates a matrix where `rows` is an array of arrays, each of which is a row of the matrix. If the optional argument `copy` is false, use the given arrays as the internal structure of the matrix without copying.

``````Matrix.rows([[25, 93], [-1, 66]])
=>  25 93
-1 66``````
```# File lib/matrix.rb, line 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|
raise ErrDimensionMismatch, "row size differs (#{row.size} should be #{size})" unless row.size == size
end
new rows, size
end```
scalar(n, value)

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

``````Matrix.scalar(2, 5)
=> 5 0
0 5``````
```# File lib/matrix.rb, line 221
def Matrix.scalar(n, value)
diagonal(*Array.new(n, value))
end```
unit(n)
Alias for: identity
zero(row_count, column_count = row_count)

Creates a zero matrix.

``````Matrix.zero(2)
=> 0 0
0 0``````
```# File lib/matrix.rb, line 245
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 Public methods
*(m)

Matrix multiplication.

``````Matrix[[2,4], [6,8]] * Matrix.identity(2)
=> 2 4
6 8``````
```# File lib/matrix.rb, line 805
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)

Matrix exponentiation. Equivalent to multiplying the matrix by itself N times. Non integer exponents will be handled by diagonalizing the matrix.

``````Matrix[[7,6], [3,9]] ** 2
=> 67 96
48 99``````
```# File lib/matrix.rb, line 972
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)

``````Matrix.scalar(2,5) + Matrix[[1,0], [-4,7]]
=>  6  0
-4 12``````
```# File lib/matrix.rb, line 838
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 and 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```
-(m)

Matrix subtraction.

``````Matrix[[1,5], [4,2]] - Matrix[[9,3], [-4,1]]
=> -8  2
8  1``````
```# File lib/matrix.rb, line 865
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 and 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```
/(other)

Matrix division (multiplication by the inverse).

``````Matrix[[7,6], [3,9]] / Matrix[[2,9], [3,1]]
=> -7  1
-3 -6``````
```# File lib/matrix.rb, line 892
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)

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

```# File lib/matrix.rb, line 767
def ==(other)
return false unless Matrix === other &&
column_count == other.column_count # necessary for empty matrices
rows == other.rows
end```
[](i, j)

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

Also aliased as: element, component
```# File lib/matrix.rb, line 313
def [](i, j)
@rows.fetch(i){return nil}[j]
end```
clone()

Returns a clone of the matrix, so that the contents of each do not reference identical objects. There should be no good reason to do this since Matrices are immutable.

```# File lib/matrix.rb, line 784
def clone
new_matrix @rows.map(&:dup), column_count
end```
coerce(other)

The coerce method provides support for Ruby type coercion. This coercion mechanism is used by Ruby to handle mixed-type numeric operations: it is intended to find a compatible common type between the two operands of the operator. See also Numeric#coerce.

```# File lib/matrix.rb, line 1277
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```
collect()

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``````
Also aliased as: map
```# File lib/matrix.rb, line 381
def collect(&block) # :yield: e
rows = @rows.collect{|row| row.collect(&block)}
new_matrix rows, column_count
end```
column(j)

Returns column vector number `j` of the matrix as a Vector (starting at 0 like an array). When a block is given, the elements of that vector are iterated.

```# File lib/matrix.rb, line 358
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()

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

```# File lib/matrix.rb, line 1298
def column_vectors
Array.new(column_count) {|i|
column(i)
}
end```
component(i, j)
Alias for: []
conj()
Alias for: conjugate
conjugate()

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``````
Also aliased as: conj
```# File lib/matrix.rb, line 1223
def conjugate
collect(&:conjugate)
end```
det()
Alias for: determinant
det_e()
Alias for: determinant_e
determinant()

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``````
Also aliased as: det
```# File lib/matrix.rb, line 1009
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```
determinant_e()

deprecated; use #determinant

Also aliased as: det_e
```# File lib/matrix.rb, line 1091
def determinant_e
warn "#{caller(1)[0]}: warning: Matrix#determinant_e is deprecated; use #determinant"
determinant
end```
diagonal?()

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

```# File lib/matrix.rb, line 599
def diagonal?
Matrix.Raise ErrDimensionMismatch unless square?
each(:off_diagonal).all?(&:zero?)
end```
each(which = :all)

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 lib/matrix.rb, line 404
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)

Same as each, but the row index and column index in addition to the element

``````Matrix[ [1,2], [3,4] ].each_with_index do |e, row, col|
puts "#{e} at #{row}, #{col}"
end
# => Prints:
#    1 at 0, 0
#    2 at 0, 1
#    3 at 1, 0
#    4 at 1, 1
``````
```# File lib/matrix.rb, line 465
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()
Alias for: eigensystem
eigensystem()

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
``````
Also aliased as: eigen
```# File lib/matrix.rb, line 1190
def eigensystem
EigenvalueDecomposition.new(self)
end```
element(i, j)
Alias for: []
elements_to_f()
```# File lib/matrix.rb, line 1311
def elements_to_f
warn "#{caller(1)[0]}: warning: Matrix#elements_to_f is deprecated, use map(&:to_f)"
map(&:to_f)
end```
elements_to_i()
```# File lib/matrix.rb, line 1316
def elements_to_i
warn "#{caller(1)[0]}: warning: Matrix#elements_to_i is deprecated, use map(&:to_i)"
map(&:to_i)
end```
elements_to_r()
```# File lib/matrix.rb, line 1321
def elements_to_r
warn "#{caller(1)[0]}: warning: Matrix#elements_to_r is deprecated, use map(&:to_r)"
map(&:to_r)
end```
empty?()

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

```# File lib/matrix.rb, line 608
def empty?
column_count == 0 || row_count == 0
end```
eql?(other)
```# File lib/matrix.rb, line 773
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)
Alias for: index
hash()

Returns a hash-code for the matrix.

```# File lib/matrix.rb, line 791
def hash
@rows.hash
end```
hermitian?()

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

```# File lib/matrix.rb, line 616
def hermitian?
Matrix.Raise ErrDimensionMismatch unless square?
each_with_index(:upper).all? do |e, row, col|
e == rows[col][row].conj
end
end```
imag()
Alias for: imaginary
imaginary()

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``````
Also aliased as: imag
```# File lib/matrix.rb, line 1237
def imaginary
collect(&:imaginary)
end```
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]
``````
Also aliased as: find_index
```# File lib/matrix.rb, line 528
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
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()

Overrides Object#inspect

```# File lib/matrix.rb, line 1346
def inspect
if empty?
"#{self.class}.empty(#{row_count}, #{column_count})"
else
"#{self.class}#{@rows.inspect}"
end
end```
inv()
Alias for: inverse
inverse()

Returns the inverse of the matrix.

``````Matrix[[-1, -1], [0, -1]].inverse
=> -1  1
0 -1``````
Also aliased as: inv
```# File lib/matrix.rb, line 912
def inverse
Matrix.Raise ErrDimensionMismatch unless square?
self.class.I(row_count).send(:inverse_from, self)
end```
lower_triangular?()

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

```# File lib/matrix.rb, line 626
def lower_triangular?
each(:strict_upper).all?(&:zero?)
end```
lup()

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)]
``````
Also aliased as: lup_decomposition
```# File lib/matrix.rb, line 1205
def lup
LUPDecomposition.new(self)
end```
lup_decomposition()
Alias for: lup
map()
Alias for: collect
minor(*param)

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

• start_row, nrows, start_col, ncols; OR

• row_range, col_range

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

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

```# File lib/matrix.rb, line 558
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?()

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

```# File lib/matrix.rb, line 634
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?()

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

```# File lib/matrix.rb, line 652
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?()

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

```# File lib/matrix.rb, line 670
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()

Returns the rank of the matrix. Beware that using Float values can yield erroneous results because of their lack of precision. Consider using exact types like Rational or BigDecimal instead.

``````Matrix[[7,6], [3,9]].rank
=> 2``````
```# File lib/matrix.rb, line 1106
def rank
# We currently use Bareiss' multistep integer-preserving gaussian elimination
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()

deprecated; use #rank

```# File lib/matrix.rb, line 1137
def rank_e
warn "#{caller(1)[0]}: warning: Matrix#rank_e is deprecated; use #rank"
rank
end```
real()

Returns the real part of the matrix.

``````Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]]
=> 1+2i  i  0
1  2  3
Matrix[[Complex(1,2), Complex(0,1), 0], [1, 2, 3]].real
=>    1  0  0
1  2  3``````
```# File lib/matrix.rb, line 1251
def real
collect(&:real)
end```
real?()

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

```# File lib/matrix.rb, line 691
def real?
all?(&:real?)
end```
rect()

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

Also aliased as: rectangular
```# File lib/matrix.rb, line 1261
def rect
[real, imag]
end```
rectangular()
Alias for: rect
regular?()

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

```# File lib/matrix.rb, line 698
def regular?
not singular?
end```
round(ndigits=0)

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

```# File lib/matrix.rb, line 1145
def round(ndigits=0)
map{|e| e.round(ndigits)}
end```
row(i)

Returns row vector number `i` of the matrix as a Vector (starting at 0 like an array). When a block is given, the elements of that vector are iterated.

```# File lib/matrix.rb, line 344
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()

Returns the number of rows.

Also aliased as: row_size
```# File lib/matrix.rb, line 329
def row_count
@rows.size
end```
row_size()
Alias for: row_count
row_vectors()

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

```# File lib/matrix.rb, line 1289
def row_vectors
Array.new(row_count) {|i|
row(i)
}
end```
singular?()

Returns `true` is this is a singular matrix.

```# File lib/matrix.rb, line 705
def singular?
determinant == 0
end```
square?()

Returns `true` is this is a square matrix.

```# File lib/matrix.rb, line 712
def square?
column_count == row_count
end```
symmetric?()

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

```# File lib/matrix.rb, line 720
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()
Alias for: transpose
to_a()

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

```# File lib/matrix.rb, line 1307
def to_a
@rows.collect(&:dup)
end```
to_s()

Overrides Object#to_s

```# File lib/matrix.rb, line 1333
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()
Alias for: trace
trace()

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

``````Matrix[[7,6], [3,9]].trace
=> 16``````
Also aliased as: tr
```# File lib/matrix.rb, line 1154
def trace
Matrix.Raise ErrDimensionMismatch unless square?
(0...column_count).inject(0) do |tr, i|
tr + @rows[i][i]
end
end```
transpose()

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``````
Also aliased as: t
```# File lib/matrix.rb, line 1172
def transpose
return self.class.empty(column_count, 0) if row_count.zero?
new_matrix @rows.transpose, row_count
end```
unitary?()

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

```# File lib/matrix.rb, line 732
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?()

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

```# File lib/matrix.rb, line 749
def upper_triangular?
each(:strict_lower).all?(&:zero?)
end```
zero?()

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

```# File lib/matrix.rb, line 756
def zero?
all?(&:zero?)
end```