# NAME

Math::Matrix - create, compare, compute and measure 2D matrices

# VERSION

0.3.9

# !MOVING!

This repo will be moved to a different place soon.

# SYNOPSIS

Matrices are tables with rows and columns (index counting from 0) of numbers (Numeric type - Bool or Int or Num or Rat or FatRat or Complex):

```
transpose, invert, negate, add, multiply, dot product, tensor product, 22 ops, determinant, rank, norm
14 numerical properties, 26 boolean properties, 3 decompositions, submatrix, splice, map, reduce and more
```

Table of Content:

# DESCRIPTION

Because the list based, functional toolbox of Raku is not enough to calculate matrices comfortably, there is a need for a dedicated data type. The aim is to provide a full featured set of structural and mathematical operations that integrate fully with the Raku conventions. This module is pure perl and we plan to use native shaped arrays one day.

Matrices are readonly - methods and operators do create new matrix objects. All methods return readonly data or deep clones - also the constructor does a deep clone of provided data. In that sense the library is thread safe.

All computation heavy properties will be calculated lazily and cached. Mathematically or otherwise undefined operations will cause an exception.

# METHODS

**constructors**: new [], new (), new "", new-zero, new-identity, new-diagonal, new-vector-product**accessors**: element, AT-POS, row, column, diagonal, skew-diagonal, submatrix**converter**: Bool, Str, Numeric, Range, Array, list, list-rows, list-columns, Hash, gist, perl**boolean properties**: zero, identity, square, triangular, tri-, anti-, diagonal, -dominant, -constant, catalecticant, anti-, symmetric, unitary, self-adjoint, invertible, orthogonal, positive-definite, positive-semidefinite**numeric properties**: size, density, bandwith, trace, rank, nullity, determinant, minor, norm, condition, element-type**derived matrices**: transposed, negated, conjugated, adjugated, inverted, reduced-row-echelon-form**decompositions**: LU, LDU, Cholesky, LDL**list like ops**: elems, elem, cont, map, map-with-index, reduce, reduce-rows, reduce-columns**structural ops**: move-row, move-column, swap-rows, swap-columns, splice-rows, splice-columns**operators**: MM, ?, ~, |, @, %, +, -, *, **, dot, ⋅, ÷, X*, ⊗, ==, ~~, ❘ ❘, ‖ ‖, [ ]

## Constructors

Methods that create a new Math::Matrix object. The default is of course .new, which can take array of array of values (fastest) or one string. Additional constructers: new-zero, new-identity, new-diagonal and new-vector-product are there for convenience and to optimize property calculation.

### new( [[...],...,[...]] )

The default constructor, takes arrays of arrays of numbers as the only required parameter. Each second level array represents a row in the matrix. That is why their length has to be the same. Empty rows or columns we not be accepted.

```
say Math::Matrix.new( [[1,2],[3,4]] ) :
1 2
3 4
Math::Matrix.new([<1 2>,<3 4>]); # does the same, WARNING: doesn't work with complex numbers
Math::Matrix.new( [[1]] ); # one element 1*1 matrix (exception where you don't have to mind comma)
Math::Matrix.new( [[1,2,3],] ); # one row 1*3 matrix, mind the trailing comma
Math::Matrix.new( [$[1,2,3]] ); # does the same, if you don't like trailing comma
Math::Matrix.new( [[1],[2]] ); # one column 2*1 matrix
use Math::Matrix :MM; # tag :ALL works too
MM [[1,2],[3,4]]; # shortcut
```

### new( ((...),...,(...)) )

Instead of square brackets you can use round ones too and use a list of lists as argument too.

```
say Math::Matrix.new( ((1,2),(3,4)) );
say MM ((1,2),(3,4)) :
1 2
3 4
```

### new( "..." )

Alternatively you can define the matrix from a string, which makes most sense while using heredocs.

```
Math::Matrix.new("1 2 \n 3 4"); # our demo matrix as string
Math::Matrix.new: q:to/END/; # indent as you wish
1 2
3 4
END
use Math::Matrix :ALL; #
MM '1'; # 1 * 1 matrix, this case begs for a shortcut
```

### new-zero

This method is a constructor, that returns a *zero* matrix, which is sometimes called empty. That is a matrix, which holds zeros in all its elements (as checked by is-zero).

*new-zero* needs one ore two integer arguments. These are the numbers of rows and columns - the size size of the new matrix. If both numbers are the same, you can omit the ladder and get a quadratic matrix filld with zeros.

```
say Math::Matrix.new-zero( 3, 4 ) :
0 0 0 0
0 0 0 0
0 0 0 0
say Math::Matrix.new-zero( 2 ) :
0 0
0 0
```

### new-identity

This method creates a new *identity matrix* (as checked by is-identity). It contains only zeros, except in the main diagonal, which consist of ones. Since identity matrices have to be quadratic, *new-identity* requires only one integer arguments, which sets the number of rows and columns of the new matrix.

```
say Math::Matrix.new-identity( 3 ) :
1 0 0
0 1 0
0 0 1
```

### new-diagonal

creates a *diagonal* matrix (as checked by is-diagonal), which contains zeros, except in the main diagonal, that can hold arbitrary values. The only required argument is a list these arbitrary numbers, which will become the content of the main diagonal.

```
say Math::Matrix.new-diagonal( 2, 4, 5 ) :
2 0 0
0 4 0
0 0 5
```

### new-vector-product

creates a matrix by calculating the tensor-product of two vectors, which are in a list or Array form the two required arguments of the method. Both lists don't have to be of the same size. The length of the first list will determine the amount of rows and the length of the second the amount of columns.

```
say Math::Matrix.new-vector-product([1,2,3],[2,3,4]) :
* 2 3 4
1 1*2 1*3 1*4 2 3 4
2 2*2 2*3 2*4 = 4 6 8
3 3*2 3*3 3*4 6 9 12
```

## Accessors

Methods that return the content of selected elements.

element, AT-POS, row, column, diagonal, skew-diagonal, **submatrix**: (leaving out one, leaving out more, reordering)

### element

Gets value of one element in row (first parameter) and column (second parameter - counting always from 0). Sometimes its called matrix cell, to distinct from other type of elements. See: elems, elem, element-type

```
my $matrix = Math::Matrix.new([[1,2],[3,4]]);
say $matrix.element(0,1) : 2
say $matrix[0][1] # array syntax alias
```

### AT-POS

Gets row as array to enable direct postcircumfix syntax as shown in last example.

```
say $matrix.AT-POS(0) : [1,2]
say $matrix[0] # operator alias
say $matrix.Array[0] # long alias with converter method Array
```

### row

returns the values of specified row (first required parameter) as a list.

```
say Math::Matrix.new([[1,2],[3,4]]).row(0) : (1, 2)
```

### column

returns values of specified column (first required parameter) as a list.

```
say Math::Matrix.new([[1,2],[3,4]]).column(0) : (1, 3)
```

### diagonal

Called without an argument, it returns the values of main diagonal as a list. The main diagonal starts at the left upper corner (index [0][0]). Every next cell of the diagonal is reached by going one step below and one step to the right (indices [x][x] - x being an integer within the size of the matrix).

Use the optional parameter of the method to get any other parallel diagonal. A positive value for a parallel diagonale above and to the right of the main one - a negative value to access a parallel diagonal to the left and below the main diagonal. 0 is the default value. The matrix does not have to be a quadratic (square).

```
say Math::Matrix.new([[1,2],[3,4]] ).diagonal : (1, 4)
say Math::Matrix.new([[1,2],[3,4]] ).diagonal(1) : (2)
say Math::Matrix.new([[1,2],[3,4],[5,6]]).diagonal(-1): (3, 6)
```

### skew-diagonal

Unlike a *diagonal*, which runs from the left upper corner to the right lower, the (main) skew diagonal is only defined for square matrixes and runs from left lower corner to the right upper (from $matrix[m][0] to $matrix[0][m]). Use the optional argument to get any other parallel skew diagonal. Positive value for the ones below - negative above.

```
say $matrix.skew-diagonal : (2, 3)
say $matrix.skew-diagonal(0) : (2, 3)
say $matrix.skew-diagonal(-1) : (1)
say $matrix.skew-diagonal(1): (4)
```

### submatrix

Returns a matrix that might miss certain rows and columns of the original. This method accepts arguments in three different formats. The first follows the strict mathematical definition of a submatrix, the second supports a rather visual understanding of the term and the third is a way to get almost any combination rows and columns you might wish for. To properly present these functions, we base the next examples upon this matrix:

```
say $m: 1 2 3 4
2 3 4 5
3 4 5 6
4 5 6 7
```

#### leaving out one

In mathematics, a submatrix is built by leaving out one row and one column. In the two positional argument format you name these by their index ($row, $column).

```
say $m.submatrix(1,2) : 1 2 4
3 4 6
4 5 7
```

#### leaving out more

If you provide two ranges (row-min .. row-max, col-min .. col-max - both optional) to the appropriately named arguments, you get the excerpt of the matrix, that contains only the requested rows and columns - in the original order.

say $m.submatrix( rows => 1..1, columns => 1..2) : 3 4 say $m.submatrix( rows => 1..1, columns => 2..*) : 4 5 say $m.submatrix( rows => 1..2 ) : 2 3 4 5 3 4 5 6

#### reordering

Alternatively each (as previously) named argument can also take a list (or array) of values, as created my the sequence operator (...). The result will be a matrix with that selection of rows and columns. Please note, you may pick rows/columns in any order and as many times you prefer.

```
$m.submatrix(rows => (1,2), columns => (3,2)): 5 4
6 5
$m.submatrix(rows => (1...2), columns => (3,2)) # same thing
```

Arguments with ranges and lists can be mixed and are in both cases optional. If you provide none of them, the result will be the original matrix.

```
say $m.submatrix( rows => (1,) ) : 3 4 5
$m.submatrix(rows => (1..*), columns => (3,2)): 5 4
6 5
```

Even more powerful or explicit in syntax are the structural ops.

## Converter

Methods that convert a matrix into other types: Bool, Str, Numeric, Range, Array, Hash, list, list-rows, list-columns or allow different views on the overall content (output formats): gist, perl.

### Bool

Conversion into Bool context. Returns False if matrix is zero (all elements equal zero as in is-zero), otherwise True.

```
$matrix.Bool
? $matrix # alias op
if $matrix # matrix in Bool context too
```

### Str

Returns values of all elements, separated by one whitespace, rows by new line. This is the same format as expected by new(""). Str is called implicitly by put and print. A shortened version is provided by gist

```
say Math::Matrix.new([[1,2],[3,4]]).Str:
1 2 # meaning: "1 2\n3 4"
3 4
~$matrix # alias op
```

### Numeric

Conversion into Numeric context. Returns Euclidean norm. Please note, only a prefix operator + (as in: + $matrix) will call this Method. An infix (as in $matrix + $number) calls $matrix.add($number).

```
$matrix.Numeric
+ $matrix # alias op
```

### Range

Returns an range object that reflects the content of all elements. Please note that complex number can not be endpoints of ranges.

```
say $matrix.Range: 1..4
```

To get single endpoints you could write:

```
say $matrix.Range.min: 1
say $matrix.list.max: 4
```

### Array

Content of all elements as an array of arrays (same format that was put into new([...])).

```
say Math::Matrix.new([[1,2],[3,4]]).Array : [[1 2] [3 4]]
say @ $matrix # alias op, space between @ and $ needed
```

### list

Returns a flat list with all elements (same as .list-rows.flat.list).

```
say $matrix.list : (1 2 3 4)
say |$matrix # alias op
```

### list-rows

Returns a list of lists, reflecting the row-wise content of the matrix. Same format as new () takes in.

```
say Math::Matrix.new( [[1,2],[3,4]] ).list-rows : ((1 2) (3 4))
say Math::Matrix.new( [[1,2],[3,4]] ).list-rows.flat : (1 2 3 4)
```

### list-columns

Returns a list of lists, reflecting the row-wise content of the matrix.

```
say Math::Matrix.new( [[1,2],[3,4]] ).list-columns : ((1 3) (2 4))
say Math::Matrix.new( [[1,2],[3,4]] ).list-columns.flat : (1 3 2 4)
```

### Hash

Gets you a nested key - value hash.

```
say $matrix.Hash : { 0 => { 0 => 1, 1 => 2}, 1 => {0 => 3, 1 => 4} }
say % $matrix # alias op, space between % and $ still needed
```

### gist

Limited tabular view, optimized for shell output. Just cuts off excessive columns that do not fit into standard terminal and also stops after 20 rows. If you call it explicitly, you can add width and height (char count) as optional arguments. Might even not show all decimals. Several dots will hint that something is missing. It is implicitly called by say. For a full view use Str.

```
say $matrix; # output of a matrix with more than 100 elements
1 2 3 4 5 ..
3 4 5 6 7 ..
...
say $matrix.gist(max-chars => 100, max-rows => 5 );
```

max-chars is the maximum amount of characters in any row of output (default is 80). max-rows is the maximum amount of rows gist will put out (default is 20). After gist ist called once (with or without arguments) the output will be cached. So next time you call:

```
say $matrix # 100 x 5 is still the max
```

You change the cache by calling gist with arguments again.

### perl

Conversion into String that can reevaluated into the same object later using default constructor.

```
my $clone = eval $matrix.perl; # same as: $matrix.clone
```

## Boolean Properties

These are mathematical properties, a given matrix has or not. Thus, the return value is a always of boolean type (Bool). Arguments, like in case of triangular and is-diagonally-dominant are only necessary, when a method can tell you about a group of closely related properties.

zero, identity, square, **triangular**: (upper, lower, strict, unit, atomic), diagonal, anti-diagonal, tridiagonal, diagonal-constant, catalecticant, symmetric, anti-symmetric, self-adjoint, invertible, orthogonal, unitary, **diagonally-dominant**, positive-definite, positive-semidefinite

### is-zero

True if every element has value of 0 (as created by new-zero). Density of zero matrices is 0. This matrix behaves like a zero element regarding to the dot-product. Every matrix multiplied with a zero matrix is a zero matrix (A * 0 = 0 * A = 0).

```
Example: 0 0 0
0 0 0
```

### is-identity

True if every element on the main diagonal (where row index equals column index) is 1 and any other element is 0. Such a matrix will be created by new-identity. It is the neutral element in redgards to the dot-product. Every matrix multiplied with a fitting identiy matrix results in the same matrix again (A * I = I * A = A). Every identity matrix has to be a square.

```
Example: 1 0 0
0 1 0
0 0 1
```

### is-square

True if number of rows and colums are the same (see size).

### is-triangular

square matrix where all elements above the main diagonal or all elements below the main diagonal are zero. The method accepts five optional, boolean arguments: :upper, :lower, :strict, :unit and :atomic. Each argument can be used in a positive form (*:upper*), as a negative (*:!upper*), or omitted. Positively they demand a certain property, negatively the absence or opposite. When omitted, both states are acceptable.

Please note that a triangular matrix can never be *:unit* and *:strict* or *:atomic* and *:strict* at the same time, nor *:!upper* and *:!lower*. A triangular matrix that is *:upper* and *:lower* is-diagonal. Identity matrices are *:upper*, *:lower* and *:unit*.

#### upper triangular

a.k.a *right triangular matrix*: all elements left-below the main *diagonal* are zero. In other words: the lower-bandwith has to be zero.

```
$tri-matrix.is-triangular(:upper); # matrix in the example below would pass this test
$tri-matrix.is-triangular(); # True, there is a lower or upper triangle
Example: 1 2 5
0 3 8
0 0 7
```

#### lower triangular

a.k.a *left triangular* matrix: every element right and above the diagonal (where row index is greater than column index) are zero. In other words: the upper-bandwith has to be zero.

```
$tri-matrix.is-triangular(:lower); # matrix in the example below would pass this test
$tri-matrix.is-triangular(); # True too
$tri-matrix.is-triangular(:!upper); # True, because the is a triangle, but not an upper
$tri-matrix.is-triangular(:lower,:!upper);# True still, even a bit redundant
Example: 1 0 0
2 3 0
5 8 7
```

#### strict triangular

are *triangular* matrices, that have a diagonal consisting only of values equal zero.

```
$tri-matrix.is-triangular(:strict); # matrix in the example below would pass this test
$tri-matrix.is-triangular(:!unit, :lower);# True too
Example: 0 0 0
5 0 0
0 6 0
```

#### unit triangular

a.k.a *unitriangular* matrices have a diagonal consisting only of values equal one. An identity matrix is *:unit*, *:upper* and *:lower*.

```
$tri-matrix.is-triangular(:unit); # matrix in the example below would pass this test
$tri-matrix.is-triangular(:unit, :lower); # False, because unit upper triangular
Example: 1 2 5
0 1 8
0 0 1
```

#### atomic triangular

a.k.a *Frobenius* matrix is a unit triangular matrix with one column of 'none zero values.

```
$tri-matrix.is-triangular(:atomic); # matrix in the example below would pass this test
Example: 1 0 0 0
0 1 0 0
0 2 1 0
0 5 0 1
```

### is-diagonal

square matrix, where only elements on the main diagonal differ from 0 (as created by new-diagonal).

```
$tri-matrix.is-diagonal();
$tri-matrix.is-triangular(:upper, :lower); # same thing
$tri-matrix.lower-bandwith() == $tri-matrix.upper-bandwith() == 0; # True
Example: 1 0 0
0 5 0
0 0 9
```

### is-anti-diagonal

square matrix, where only elements on the main skew-diagonal differ from 0.

```
$tri-matrix.is-anti-diagonal();
Example: 0 0 3
0 5 0
7 0 0
```

### is-tridiagonal

square matrix, where only elements on the main diagonal and their direct parallels differ from 0.

```
$tri-matrix.is-tridiagonal();
$tri-matrix.lower-bandwith() < 2 and $tri-matrix.upper-bandwith() < 2; # True
Example: 1 2 0 0
3 4 5 0
0 6 7 8
0 0 9 1
```

### is-diagonal-constant

a.k.a. or *Töplitz* matrix is a matrix where every diagonal is the a collection of elements that hold the same value.

```
Example: 0 1 2
-1 0 1
-2 -1 0
```

### is-catalecticant

a.k.a *Hankel* matrix is a square matrix, where every skew-diagonal is the a collection of elements that hold the same value. *Catalecticant* matrices are symmetric.

```
Example: 0 1 2
1 2 3
2 3 4
```

### is-symmetric

True if every element with coordinates x y has same value as the element on y x. In other words: $matrix and $matrix.transposed (alias T) are the same.

```
Example: 1 2 3
2 5 4
3 4 7
```

### is-antisymmetric

Means the transposed and negated matrix are the same.

```
Example: 0 2 3
-2 0 4
-3 -4 0
```

### is-self-adjoint

A Hermitian or self-adjoint matrix is equal to its transposed and complex conjugated.

```
Example: 1 2 3+i
2 5 4
3-i 4 7
```

### is-invertible

, also called *nonsingular* or *nondegenerate*, is a square matrix, which has a none zerot determinant. That means all rows or colums have to be independent vectors. Please check this property before calling $matrix.inverted, or you will get an exception.

### is-orthogonal

An orthogonal matrix multiplied (dot-product) with its transposed derivative (T) is an identity matrix or in other words: transposed and inverted matrices are equal.

### is-unitary

An unitery matrix multiplied (dot-product) with its concjugated and transposed derivative (.conj.T) is an identity matrix, or said differently: the concjugate transposed matrix equals the inverted matrix.

### is-diagonally-dominant

True when elements on the diagonal have a bigger (if strict) or at least equal (in none strict) absolute value than the sum of its row (sum of absolute values of the row except diagonal element).

```
if $matrix.is-diagonally-dominant {
$matrix.is-diagonally-dominant(:!strict) # same thing (default)
$matrix.is-diagonally-dominant(:strict) # diagonal elements (DE) are stricly greater (>)
$matrix.is-diagonally-dominant(:!strict, :along<column>) # default
$matrix.is-diagonally-dominant(:strict, :along<row>) # DE > sum of rest row
$matrix.is-diagonally-dominant(:!strict, :along<both>) # DE >= sum of rest row and rest column
```

### is-positive-definite

True if all main minors or all Eigenvalues are strictly greater zero.

### is-positive-semidefinite

True if all main minors or all Eigenvalues are greater equal zero.

## Numeric Properties

Matrix properties that are expressed with a single number, which will be calculated without further input.

size, density, **bandwith**: (lower-, upper-), trace, rank, nullity, determinant, minor, norm, condition, **element-type**: (narrowest-, widest-)

### size

List of two values: number of rows and number of columns.

```
say $matrix.size();
my $dim = min $matrix.size;
```

### density

*Density* is the percentage of elements which are not zero. *sparsity* = 1 - *density*.

```
my $d = $matrix.density;
```

### bandwith

Is roughly the greatest distance of a none zero value from the main diagonal. A matrix that is-diagonal has a bandwith of 0. If there is a none zero value on an diagonal above or below the main diagonal, tha bandwith would be one.

```
my $bw = $matrix.bandwith;
```

#### lower-bandwith

In a matrix with the lower bandwith of k, every element with row index m and column index n, for which holds m - k > n, the content has to be zero. Literally speaking: there are k diagonals below the main diagonal, that contain none zero values.

#### upper-bandwith

Analogously, every element with m + k < n is zero if matrix has an upper bandwith of k.

```
Example: 1 0 0
4 2 0
0 5 3
```

The *bidiagonal* example matrix has an upper bandwith of zero and lower bandwith of one, so the overall bandwith is one.

### trace

The trace of a square matrix is the sum of the elements on the main diagonal. In other words: sum of elements which row and column value is identical.

```
my $tr = $matrix.trace;
```

### rank

Rank is the number of independent row or column vectors or also called independent dimensions (thats why this command is sometimes calles dim)

```
my $r = $matrix.rank;
```

### nullity

Nullity of a matrix is the number of dependent rows or columns (rank + nullity = dim). Or number of dimensions of the kernel (vector space mapped by the matrix into zero).

```
my $n = $matrix.nullity;
```

### determinant

Only a square matrice has a defined determinant, which tells the volume, spanned by the row or column vectors. So if the volume is just in one dimension flat, the determinant is zero, and has a kernel (not a full rank - thus is not invertible).

```
my $det = $matrix.determinant;
my $d = $matrix.det; # same thing
my $d = ❘ $matrix ❘; # unicode operator shortcut
```

### minor

A Minor is the determinant of a submatrix (first variant with same 2 scalar arguments a minor method). The two required positional arguments are row and column indices of an existing element.

```
my $m = $matrix.minor(1,2);
```

### norm

A norm is a single positive number, which is an abstraction to the concept of size. Most common form for matrices is the p-norm, where in step 1 the absolute value of every element is taken to the power of p. The sum of these results is taken to the power of 1/p. The p-q-Norm extents this process. In his step 2 every column-sum is taken to the power of (p/q). In step 3 the sum of these are taken to the power of (1/q).

```
my $norm = $matrix.norm( ); # euclidian norm aka L2 (p = 2, q = 2)
my $norm = + $matrix; # context op shortcut
my $norm = ‖ $matrix ‖; # unicode op shortcut
my $norm = $matrix.norm(1); # p-norm aka L1 = sum of all elements absolute values (p = 1, q = 1)
my $norm = $matrix.norm(p:<4>,q:<3>); # p,q - norm, p = 4, q = 3
my $norm = $matrix.norm(p:<2>,q:<2>); # L2 aka Euclidean aka Frobenius norm
my $norm = $matrix.norm('euclidean'); # same thing, more expressive to some
my $norm = $matrix.norm('frobenius'); # same thing, more expressive to some
my $norm = $matrix.norm('max'); # maximum norm - biggest absolute value of a element
$matrix.norm('row-sum'); # row sum norm - biggest abs. value-sum of a row
$matrix.norm('column-sum'); # column sum norm - same column wise
```

### condition

Condition number of a matrix is L2 norm * L2 of inverted matrix.

```
my $c = $matrix.condition( );
```

### element-type

#### narrowest-element-type

#### widest-element-type

Matrix elements can be (from most narrow to widest), of type (Bool), (Int), (Num), (Rat), (FatRat) or (Complex). The widest type of any element will returned as type object.

In the next example the smartmatch returns true, because no element of our default example matrix has wider type than (Int). After such a test all elements can be safely treated as Int or Bool.

```
if $matrix.widest-element-type ~~ Int { ...
```

You can also check if all elements have the same type:

```
if $matrix.widest-element-type eqv $matrix.narrowest-element-type
```

## Derived Matrices

Single matrices, that can be computed with only our original matrix as input.

transposed, negated, conjugated, adjugated, inverted, reduced-row-echelon-form

### transposed

Returns a new, transposed Matrix, where rows became colums and vice versa.

```
Math::Matrix.new([[1,2,3],[3,4,6]]).transposed :
[[1 2 3] = [[1 4]
[4 5 6]].T [2 5]
[3 6]]
Math::Matrix.new([[1,2,3],[3,4,6]]).T # same but shorter
```

### negated

Creates a matrix where every element has the negated value of the original (invertion of sign).

```
my $new = $matrix.negated(); # invert sign of all elements
my $neg = - $matrix; # operator alias
say $neg: -1 -2
-3 -4
```

### conjugated

Creates a matrix where every element is the complex conjugated of the original.

```
my $c = $matrix.conjugated(); # change every value to its complex conjugated
my $c = $matrix.conj(); # short alias (official Perl 6 shortcut for conjugation of numbers)
say Math::Matrix.new([[1+i,2],[3,4-i]]).conj :
1-1i 2
3 4+1i
my $ct = $matrix.H; # conjugated and transposed matrix (Hermitian transpose)
```

### adjugated

Creates a matrix out of the properly signed minors of the original. It is called adjugate, classical adjoint or sometimes adjunct.

```
$matrix.adjugated.say :
4 -3
-2 1
```

### inverted

Matrices that have a square form and a full rank can be inverted (see is-invertible). Inverse matrix regarding to matrix multiplication (see dot-product). The dot product of a matrix with it's inverted results in a identity matrix (neutral element in this group).

```
my $i = $matrix.inverted(); # invert matrix
my $i = $matrix ** -1; # operator alias
say $i:
-2 1
1.5 -0.5
```

### reduced-row-echelon-form

Return the reduced row echelon form of a matrix, a.k.a. row canonical form

```
my $rref = $matrix.reduced-row-echelon-form();
my $rref = $matrix.rref(); # short alias
```

## Decompositions

Methods that return a list of matrices, which can be recombined into the original matrix (mostly by dot product).

LU, LU-Crout, LDU, LUP, Cholesky LDL

### LU-decomposition

Decompose an invertible, matrix into a lower triangular (called L) and an upper triangular matrix (called U). The algorithm works along the lines of what is called Gaussian elimination.

my ($L, $U) = $matrix.LU-decomposition(); $L dot $U eq $matrix; # True $U.is-triangular(:upper); # True $L.is-triangular(:lower, :unit); # True

#### LU-Crout-decomposition

L will also be unit triangular matrix, but not U - meaning the diagonal elements of L are all equal 1 and the diagonal elements of U might differ from 1. If you prefer this to be the other way around, use the optional, boolean parameter *:Crout*.

my ($L, $U) = $matrix.LU-decomposition(:Crout); $L dot $U eq $matrix; # True $L.is-triangular(:lower); # True $U.is-triangular(:upper, :unit); # True

#### LDU-decomposition

If you want the diagonal elements separated into its own matrix, use the optional, boolean parameter *:diagonal*. Note that you can not use this and *:Crout* at once.

my ($L, $D, $U) = $matrix.LU-decomposition( :diagonal); $L dot $D dot $U eq $matrix; # True $L.is-triangular(:lower, :unit); # True $U.is-triangular(:upper, :unit); # True $D.is-diagonal(); # True

#### LUP-decomposition

If you choose the optional, boolean parameter *:pivot*, you get additionally a permutation matrix, that contains the information which rows and columns were swapped, in order to achieve a decomposition. That is why the matrix no longer has to be invertible, but only square to run a LU decomposition with permutation. This option might be combined with *:Crout* or *:diagonal*, but not both.

```
my ($L, $U, $P) = $matrix.LU-decomposition( :pivot );
$L dot $U eq $matrix dot $P; # True
```

### Cholesky-decomposition

This decomposition does roughly the same as LU and is faster. But it works only on matrices that are symmetric and positive-definite. The result will be an orthogonal, lower triangular matrix matrix (here called G) that multiplied with its transposed gives you the original matrix.

```
my $G = $matrix.Cholesky-decomposition( ); # $G is a left triangular matrix
my $G = $matrix.Cholesky-decomposition(:!diagonal); # same as before
$G dot $G.T == $matrix; # True
```

#### LDL-decomposition

When the optional, boolean parameter :diagonal is positive (negative by default) you get two matrices (L and D) as a result. L is then a lower unit triangular matrix, (with ones in its main diagonal). D is a diagonal matrix ( G = L * sqrt(D)), containing the values formerly on the diagonal of G, but squared. This output format is also known as *LDL decomposition*. You get the second L matrix easily by transposing the L you got.

```
my ($L, $D) = $matrix.Cholesky-decomposition(:diagonal);
$L dot $D dot $L.T == $matrix; # True
```

## Mathematical Operations

Math methods that work on a whole matrix or just some parts of it. The operands will not be changed but a the result matrix will be returned.

equal, **add**: (matrix, vector, scalar), **multiply**: (matrix, vector, scalar), dot-product, tensor-product.

### equal

Checks two matrices for equality. They have to be of same size and every element of the first matrix on a particular position has to be numerically equal (as checked by *==*) to the element (on the same position) of the second matrix.

```
if $matrixa.equal( $matrixb ) { # method variant
if $matrixa == $matrixb { # operator alias
if $matrixa ~~ $matrixb { # smart match redirects to ==
```

### add

Adding a matrix, vector or scalar. Named arguments *:row* and *:column* are only used to add a vector (one) or scalar (both).

#### add matrix

When adding two matrices, they have to be of the same size. Instead of a Math::Matrix object you can also provide the content of a matrix as new [], new () or new "" would accept it.

```
$matrix.add( $matrix2 );
$matrix.add( [[2,3],[4,5]] ); # data alias
$matrix + $matrix2 # operator alias
Example: 1 2 + 2 3 = 3 5
3 4 4 5 7 9
```

#### add vector

To add a vector you have to specify to which row or column it should be added. The other argument is a list or array (which have to fit the row or column size).

```
$matrix.add( row => 1, [2,3] );
Example: 1 2 + = 1 2
3 4 2 3 5 7
$matrix.add( column => 1, (2,3) );
Example: 1 2 + 2 = 1 4
3 4 3 3 7
```

#### add scalar

When adding a single number to the matrix (providing one argument), the number will be added to every element of the matrix. If you additionally provide a row or column number, only in that row or column, every element gets added to. In case you provide row and column numbber (three args), only a single element of the result matrix might be different.

```
$matrix.add( $number ); # adds number to every element
$matrix + $number; # works too
Example: 1 2 + 5 = 6 7
3 4 8 9
$matrix.add( row => 1, 3 ): [[1,2],[6,7]]
$matrix.add( row => 1, column=> 0, 2 ): [[1,2],[5,4]]
```

### multiply

Unlike the dot-product and tensor-product, this operation is the simple, scalar multiplication applied to each ore some elements. The second factor might come from the cells of another matrix, a vector or a single value. That is why this method works analogous to add.

#### multiply matrix

When a matrix of same size is given, the result will be a matrix of that size again. Each element will be the product of two the two elements of the operands with the same indices (position). Instead of a Math::Matrix object you can also provide the content of a matrix as new [], new () or new "" would accept it.

```
my $product = $matrix.multiply( $matrix2 ); # element wise multiplication of same size matrices
my $p = $matrix * $matrix2; # works too
Example: 1 2 * 2 3 = 2 6
3 4 4 5 12 20
```

#### multiply vector

Takes a vector (list of numbers) and a Pair thst specifies an existing row or column. This row or column will than multiplied with the vector (element one by vector element one, and so on).

```
my $product = $matrix.multiply( row => 0, (2, 3) ); # multiply every element with number
Example: 1 2 * 2 3 = 2 6
3 4 3 4
```

#### multiply scalar

If you provide just one number, every matrix element will be multiplied by this number. When you additionally add a valid row or column index, only the cells in that row or column get multiplied. (This is especially useful, when doing Gaussian elimination.) If both are provided, only one cell gets multiplied.

```
my $product = $matrix.multiply( $number ); # multiply every element with number
my $p = $matrix * $number; # does the same
Example: 1 2 * 5 = 5 10
3 4 15 20
$matrix.multiply( row => 0, 2);
Example: 1 2 * 2 = 2 4
3 4 3 4
$matrix.multiply(2, column => 0 );
Example: 1 2 = 2 2
3 4 6 4
*2
$matrix.multiply(row => 1, column => 1, 3) : [[1,2],[3,12]]
```

### dot-product

Matrix multiplication of two fitting matrices (colums left == rows right).

```
Math::Matrix.new( [[1,2],[3,4]] ).dot-product( Math::Matrix.new([[2,3],[4,5]]) );
Example: 2 3
4 5
*
1 2 10 13 = 1*2+2*4 1*3+2*5
3 4 22 29 3*2+4*4 3*3+4*5
my $product = $matrix1.dot-product( $matrix2 )
my $c = $a dot $b; # works too as operator alias
my $c = $a ⋅ $b; # unicode operator alias
A shortcut for multiplication is the power - operator **
my $c = $a ** 3; # same as $a dot $a dot $a
my $c = $a ** -3; # same as ($a dot $a dot $a).inverted
my $c = $a ** 0; # created an right sized identity matrix
```

### tensor-product

The *tensor product* (a.k.a *Kronecker product*) between a matrix A of *size|#size* (m,n) and matrix B of size (p,q) is a matrix C of size (m*p,n*q). C is a concatination of matrices you get if you take every element of A and do a scalar multiplication with B as in $B.multiply($A.element(..,..)).

```
Example: 1 2 X* 2 3 = 1*[2 3] 2*[2 3] = 2 3 4 6
3 4 4 5 [4 5] [4 5] 4 5 8 10
3*[2 3] 4*[2 3] 6 9 8 12
[4 5] [4 5] 8 15 16 20
my $c = $matrixa.tensor-product( $matrixb );
my $c = $a X* $b; # works too as operator alias
my $c = $a ⊗ $b; # unicode operator alias
```

## List Like Operations

Methods (or extensions thereof) that usually are provided by Lists and Arrays, but make also sense in context of matrices: elems, elem, cont, map, map-with-index, reduce, reduce-rows, reduce-columns

### elems

Number (count) of elements = rows * columns (see size).

```
say $matrix.elems();
```

### elem

Asks if all element of Matrix (cell) values are an element of the given set or range.

```
Math::Matrix.new([[1,2],[3,4]]).elem(1..4) : True
Math::Matrix.new([[1,2],[3,4]]).elem(2..5) : False, 1 is not in 2..5
```

### cont

Asks if the matrix contains a value equal to the only argument of the method. If a range is provided as argument, at least one value has to be within this range to make the result true.

```
Math::Matrix.new([[1,2],[3,4]]).cont(1) : True
Math::Matrix.new([[1,2],[3,4]]).cont(5) : False
Math::Matrix.new([[1,2],[3,4]]).cont(3..7): True
MM [[1,2],[3,4]] (cont) 1 # True too
```

### map

Creates a new matrix of same size by iterating over all or some elements. For every chosen element with the indices (m,n), a provided code block (required argument) will be run once. That block will be given the elements(m,n) value as an argument. The return value of the block will be the content of the element(m,n) of the resulting matrix.

```
say $matrix.map(* + 1) :
2 3
4 5
```

By provding values (Ranges) to the named arguments *rows* and *columns* (no special order required), only a subset of rows or columns will be mapped - the rest will be just copied.

```
say $matrix.map( rows => (0..0), {$_ * 2}) :
2 4
3 4
say $matrix.map( rows => (1..*), columns => (1..1), {$_ ** 2}) :
1 2
3 16
```

### map-with-index

Works just like map with the only difference that the given block can recieve one to three arguments: (row index, column index and cell value).

```
say $matrix.map-with-index: {$^m == $^n ?? $^value !! 0 } :
1 0
0 4
```

### reduce

Like the built in reduce method, it iterates over all elements and joins them into one value, by applying the given operator or method to the previous result and the next element. I starts with the element [0][0] and moving from left to right in the first row and continue with the first element of the next row.

```
Math::Matrix.new([[1,2],[3,4]]).reduce(&[+]): 10 = 1 + 2 + 3 + 4
Math::Matrix.new([[1,2],[3,4]]).reduce(&[*]): 10 = 1 * 2 * 3 * 4
```

### reduce-rows

Reduces (as described above) every row into one value, so the overall result will be a list. In this example we calculate the sum of all elements in a row:

```
say Math::Matrix.new([[1,2],[3,4]]).reduce-rows(&[+]): (3, 7)
```

### reduce-columns

Similar to reduce-rows, this method reduces each column to one value in the resulting list:

```
say Math::Matrix.new([[1,2],[3,4]]).reduce-columns(&[*]): (3, 8)
```

## Structural Operations

Methods that reorder rows and columns, delete some or even add new. The accessor submatrix is also useful for that purpose. move-row, move-column, swap-rows, swap-rows, splice-rows, splice-columns

### move-row

```
Math::Matrix.new([[1,2,3],[4,5,6],[7,8,9]]).move-row(0,1); # move row 0 to 1
Math::Matrix.new([[1,2,3],[4,5,6],[7,8,9]]).move-row(0=>1); # same
1 2 3 4 5 6
4 5 6 ==> 1 2 3
7 8 9 7 8 9
```

### move-column

```
Math::Matrix.new([[1,2,3],[4,5,6],[7,8,9]]).move-column(2,1);
Math::Matrix.new([[1,2,3],[4,5,6],[7,8,9]]).move-column(2=>1); # same
1 2 3 1 3 2
4 5 6 ==> 4 6 5
7 8 9 7 9 8
```

### swap-rows

```
Math::Matrix.new([[1,2,3],[4,5,6],[7,8,9]]).swap-rows(2,0);
1 2 3 7 8 9
4 5 6 ==> 4 5 6
7 8 9 1 2 3
```

### swap-columns

```
Math::Matrix.new([[1,2,3],[4,5,6],[7,8,9]]).swap-columns(0,2);
1 2 3 3 2 1
4 5 6 ==> 6 5 4
7 8 9 9 8 7
```

### splice-rows

Like the splice for lists: the first two parameter are position and amount (optional) of rows to be deleted. The third and alos optional parameter will be an array of arrays (line .new would accept), that fitting row lengths. These rows will be inserted before the row with the number of first parameter. The third parameter can also be a fitting Math::Matrix.

```
$matrix.splice-rows(0,0, Math::Matrix.new([[5,6],[7,8]]) ); # aka prepend
$matrix.splice-rows(0,0, [[5,6],[7,8]] ); # same result
5 6
7 8
1 2
3 4
$matrix.splice-rows(1,0, Math::Matrix.new([[5,6],[7,8]]) ); # aka insert
$matrix.splice-rows(1,0, [[5,6],[7,8]] ); # same result
1 2
5 6
7 8
3 4
$matrix.splice-rows(1,1, Math::Matrix.new([[5,6],[7,8]]) ); # aka replace
$matrix.splice-rows(1,1, [[5,6],[7,8]] ); # same result
1 2
5 6
7 8
$matrix.splice-rows(2,0, Math::Matrix.new([[5,6],[7,8]]) ); # aka append
$matrix.splice-rows(2,0, [[5,6],[7,8]] ); # same result
$matrix.splice-rows(-1,0, [[5,6],[7,8]] ); # with negative index
1 2
3 4
5 6
7 8
```

### splice-columns

Same as splice-rows, just horizontally.

```
$matrix.splice-columns(2,0, Math::Matrix.new([[5,6],[7,8]]) ); # aka append
$matrix.splice-columns(2,0, [[5,6],[7,8]] ); # same result
$matrix.splice-columns(-1,0, [[5,6],[7,8]] ); # same result with negative index
1 2 ~ 5 6 = 1 2 5 6
3 4 7 8 3 4 7 8
```

## Shortcuts

Summary of all shortcut aliases (left) and their long form (right).

T --> transposed

conj --> conjugated

det --> determinant

rref --> reduced-row-echelon-form

## Operator Methods

Operators (left) with the methods they refer to (right). (Most ops are just aliases.) For more explanations of the ops (with examples) see the next chapter:

prefix ? --> Bool

prefix + --> Numeric

prefix - --> negated

prefix ~ --> Str

prefix | --> list

prefix @ --> Array

prefix % --> Hash

prefix MM --> new

infix == --> equal

infix ~~ --> equal ACCEPTS

infix + --> add

infix - --> add

infix * --> multiply

infix ⋅ dot --> dot-product

infix ÷ --> dot-product inverted

infix ** --> dot-product inverted

infix ⊗ X* --> tensor-product

circumfix ｜..｜ --> determinant

circumfix ‖..‖ --> norm

postcircumfix [..] --> AT-POS

# Operators

The Module overloads or introduces a range of well and lesser known ops, which are almost all aliases.

==, +, * are commutative, -, ⋅, dot, ÷, x, ⊗ and ** are not. All ops have same precedence as its multi method siblings - unless stated otherwise.

They are exported when using no flag (same as :DEFAULT) or :ALL, but not under :MANDATORY or :MM). The only exception is MM operator, a shortcut to create a matrix. That has to be importet explicitly with the tag :MM or :ALL. The postcircumfix [] - op will always work.

```
my $a = +$matrix # Num context, Euclidean norm
my $b = ?$matrix # Bool context, True if any element has a none zero value
my $str = ~$matrix # String context, matrix content, space and new line separated as table
my $l = |$matrix # list context, list of all elements, row-wise
my $a = @ $matrix # same thing, but as Array
my $h = % $matrix # hash context, similar to .kv, so that %$matrix{0}{0} is first element
$matrixa == $matrixb # check if both have same size and they are element wise equal
$matrixa ~~ $matrixb # same thing
my $sum = $matrixa + $matrixb; # element wise sum of two same sized matrices
my $sum = $matrix + $number; # add number to every element
my $dif = $matrixa - $matrixb; # element wise difference of two same sized matrices
my $dif = $matrix - $number; # subtract number from every element
my $neg = -$matrix # negate value of every element
my $p = $matrixa * $matrixb; # element wise product of two same sized matrices
my $sp = $matrix * $number; # multiply number to every element
my $dp = $a dot $b; # dot product of two fitting matrices (cols a = rows b)
my $dp = $a ⋅ $b; # dot product, unicode (U+022C5)
my $dp = $a ÷ $b; # alias to $a dot $b.inverted, (U+000F7)
my $c = $a ** 3; # $a to the power of 3, same as $a dot $a dot $a
my $c = $a ** -3; # alias to ($a dot $a dot $a).inverted
my $c = $a ** 0; # creats an right sized identity matrix
my $tp = $a X* $b; # tensor product, same precedence as infix: x (category Replication)
my $tp = $a ⊗ $b; # tensor product, unicode (U+02297)
｜$matrix ｜ # determinant, unicode (U+0FF5C)
‖ $matrix ‖ # L2 norm (euclidean p=2 to the square), (U+02016)
$matrix[1][2] # 2nd row, 3rd column element - works even under :MANDATORY tag
MM [[1]] # a new matrix, has higher precedence than postcircumfix:[]
MM '1' # string alias
```

# Export Tags

:MANDATORY (nothing is exported)

:DEFAULT (same as no tag, most ops will be exported)

:MM (only MM op exported)

:ALL

# Authors

Pierre VIGIER

Herbert Breunung

# Contributors

Patrick Spek

Juan Julián Merelo Guervós

# License

Artistic License 2.0 (GPL and BSD at the same time)

# See Also

- Math::Libgsl::Matrix