# Linear Algebra

## diag - Create a diagonal array or extract the diagonal elements of an array

Experimental

### Description

Create a diagonal array or extract the diagonal elements of an array

### Syntax

d = diag(a [, k])

### Arguments

a: Shall be a rank-1 or or rank-2 array. If a is a rank-1 array (i.e. a vector) then diag returns a rank-2 array with the elements of a on the diagonal. If a is a rank-2 array (i.e. a matrix) then diag returns a rank-1 array of the diagonal elements.

k (optional): Shall be a scalar of type integer and specifies the diagonal. The default k = 0 represents the main diagonal, k > 0 are diagonals above the main diagonal, k < 0 are diagonals below the main diagonal.

### Return value

Returns a diagonal array or a vector with the extracted diagonal elements.

### Example

program example_diag1
use stdlib_linalg, only: diag
implicit none
real, allocatable :: A(:, :)
integer :: i
A = diag([(1, i=1, 10)]) ! creates a 10 by 10 identity matrix
end program example_diag1

program example_diag2
use stdlib_linalg, only: diag
implicit none
real, allocatable :: v(:)
real, allocatable :: A(:, :)
v = [1, 2, 3, 4, 5]
A = diag(v) ! creates a 5 by 5 matrix with elements of v on the diagonal
end program example_diag2

program example_diag3
use stdlib_linalg, only: diag
implicit none
integer, parameter :: n = 10
real :: c(n), ul(n - 1)
real :: A(n, n)
c = 2
ul = -1
A = diag(ul, -1) + diag(c) + diag(ul, 1) ! Gil Strang's favorite matrix
end program example_diag3

program example_diag4
use stdlib_linalg, only: diag
implicit none
integer, parameter :: n = 12
real :: A(n, n)
real :: v(n)
call random_number(A)
v = diag(A) ! v contains diagonal elements of A
end program example_diag4

program example_diag5
use stdlib_linalg, only: diag
implicit none
integer, parameter :: n = 3
real :: A(n, n)
real, allocatable :: v(:)
A = reshape([1, 2, 3, 4, 5, 6, 7, 8, 9], [n, n])
v = diag(A, -1) ! v is [2,6]
v = diag(A, 1)  ! v is [4,8]
end program example_diag5


## eye - Construct the identity matrix

Experimental

Pure function.

### Description

Construct the identity matrix.

### Syntax

I = eye(dim1 [, dim2])

### Arguments

dim1: Shall be a scalar of default type integer. This is an intent(in) argument.

dim2: Shall be a scalar of default type integer. This is an intent(in) and optional argument.

### Return value

Return the identity matrix, i.e. a matrix with ones on the main diagonal and zeros elsewhere. The return value is of type integer(int8). The use of int8 was suggested to save storage.

#### Warning

Since the result of eye is of integer(int8) type, one should be careful about using it in arithmetic expressions. For example:

!> Be careful
A = eye(2,2)/2     !! A == 0.0
!> Recommend
A = eye(2,2)/2.0   !! A == diag([0.5, 0.5])


### Example

program example_eye1
use stdlib_linalg, only: eye
implicit none
integer :: i(2, 2)
real :: a(3, 3)
real :: b(2, 3)  !! Matrix is non-square.
complex :: c(2, 2)
I = eye(2)              !! [1,0; 0,1]
A = eye(3)              !! [1.0,0.0,0.0; 0.0,1.0,0.0; 0.0,0.0,1.0]
A = eye(3, 3)            !! [1.0,0.0,0.0; 0.0,1.0,0.0; 0.0,0.0,1.0]
B = eye(2, 3)            !! [1.0,0.0,0.0; 0.0,1.0,0.0]
C = eye(2, 2)            !! [(1.0,0.0),(0.0,0.0); (0.0,0.0),(1.0,0.0)]
C = (1.0, 1.0)*eye(2, 2)  !! [(1.0,1.0),(0.0,0.0); (0.0,0.0),(1.0,1.0)]
end program example_eye1

program example_eye2
use stdlib_linalg, only: eye, diag
implicit none
print *, all(eye(4) == diag([1, 1, 1, 1])) ! prints .true.
end program example_eye2


## trace - Trace of a matrix

Experimental

### Description

Trace of a matrix (rank-2 array)

### Syntax

result = trace(A)

### Arguments

A: Shall be a rank-2 array. If A is not square, then trace(A) will return the sum of diagonal values from the square sub-section of A.

### Return value

Returns the trace of the matrix, i.e. the sum of diagonal elements.

### Example

program example_trace
use stdlib_linalg, only: trace
implicit none
real :: A(3, 3)
A = reshape([1, 2, 3, 4, 5, 6, 7, 8, 9], [3, 3])
print *, trace(A) ! 1 + 5 + 9
end program example_trace


## outer_product - Computes the outer product of two vectors

Experimental

### Description

Computes the outer product of two vectors

### Syntax

d = outer_product(u, v)

### Arguments

u: Shall be a rank-1 array

v: Shall be a rank-1 array

### Return value

Returns a rank-2 array equal to u v^T (where u, v are considered column vectors). The shape of the returned array is [size(u), size(v)].

### Example

program example_outer_product
use stdlib_linalg, only: outer_product
implicit none
real, allocatable :: A(:, :), u(:), v(:)
u = [1., 2., 3.]
v = [3., 4.]
A = outer_product(u, v)
!A = reshape([3., 6., 9., 4., 8., 12.], [3,2])
end program example_outer_product


## kronecker_product - Computes the Kronecker product of two rank-2 arrays

Experimental

### Description

Computes the Kronecker product of two rank-2 arrays

### Syntax

C = kronecker_product(A, B)

### Arguments

A: Shall be a rank-2 array with dimensions M1, N1

B: Shall be a rank-2 array with dimensions M2, N2

### Return value

Returns a rank-2 array equal to A \otimes B. The shape of the returned array is [M1*M2, N1*N2].

### Example

program example_kronecker_product
use stdlib_linalg, only: kronecker_product
implicit none
integer, parameter :: m1 = 1, n1 = 2, m2 = 2, n2 = 3
integer :: i, j
real :: A(m1, n1), B(m2,n2)
real, allocatable :: C(:,:)

do j = 1, n1
do i = 1, m1
A(i,j) = i*j ! A = [1, 2]
end do
end do

do j = 1, n2
do i = 1, m2      ! B = [ 1, 2, 3 ]
B(i,j) = i*j !     [ 2, 4, 6 ]
end do
end do

C = kronecker_product(A, B)
! C =     [ a(1,1) * B(:,:) | a(1,2) * B(:,:) ]
! or in other words,
! C =     [  1.00      2.00      3.00      2.00      4.00      6.00  ]
!         [  2.00      4.00      6.00      4.00      8.00     12.00  ]
end program example_kronecker_product


## cross_product - Computes the cross product of two vectors

Experimental

### Description

Computes the cross product of two vectors

### Syntax

c = cross_product(a, b)

### Arguments

a: Shall be a rank-1 and size-3 array

b: Shall be a rank-1 and size-3 array

### Return value

Returns a rank-1 and size-3 array which is perpendicular to both a and b.

### Example

program demo_cross_product
use stdlib_linalg, only: cross_product
implicit none
real :: a(3), b(3), c(3)
a = [1., 0., 0.]
b = [0., 1., 0.]
c = cross_product(a, b)
!c = [0., 0., 1.]
end program demo_cross_product


## is_square - Checks if a matrix is square

Experimental

### Description

Checks if a matrix is square

### Syntax

d = is_square(A)

### Arguments

A: Shall be a rank-2 array

### Return value

Returns a logical scalar that is .true. if the input matrix is square, and .false. otherwise.

### Example

program example_is_square
use stdlib_linalg, only: is_square
implicit none
real :: A(2, 2), B(3, 2)
logical :: res
A = reshape([1., 2., 3., 4.], shape(A))
B = reshape([1., 2., 3., 4., 5., 6.], shape(B))
res = is_square(A) ! returns .true.
res = is_square(B) ! returns .false.
end program example_is_square


## is_diagonal - Checks if a matrix is diagonal

Experimental

### Description

Checks if a matrix is diagonal

### Syntax

d = is_diagonal(A)

### Arguments

A: Shall be a rank-2 array

### Return value

Returns a logical scalar that is .true. if the input matrix is diagonal, and .false. otherwise. Note that nonsquare matrices may be diagonal, so long as a_ij = 0 when i /= j.

### Example

program example_is_diagonal
use stdlib_linalg, only: is_diagonal
implicit none
real :: A(2, 2), B(2, 2)
logical :: res
A = reshape([1., 0., 0., 4.], shape(A))
B = reshape([1., 0., 3., 4.], shape(B))
res = is_diagonal(A) ! returns .true.
res = is_diagonal(B) ! returns .false.
end program example_is_diagonal


## is_symmetric - Checks if a matrix is symmetric

Experimental

### Description

Checks if a matrix is symmetric

### Syntax

d = is_symmetric(A)

### Arguments

A: Shall be a rank-2 array

### Return value

Returns a logical scalar that is .true. if the input matrix is symmetric, and .false. otherwise.

### Example

program example_is_symmetric
use stdlib_linalg, only: is_symmetric
implicit none
real :: A(2, 2), B(2, 2)
logical :: res
A = reshape([1., 3., 3., 4.], shape(A))
B = reshape([1., 0., 3., 4.], shape(B))
res = is_symmetric(A) ! returns .true.
res = is_symmetric(B) ! returns .false.
end program example_is_symmetric


## is_skew_symmetric - Checks if a matrix is skew-symmetric

Experimental

### Description

Checks if a matrix is skew-symmetric

### Syntax

d = is_skew_symmetric(A)

### Arguments

A: Shall be a rank-2 array

### Return value

Returns a logical scalar that is .true. if the input matrix is skew-symmetric, and .false. otherwise.

### Example

program example_is_skew_symmetric
use stdlib_linalg, only: is_skew_symmetric
implicit none
real :: A(2, 2), B(2, 2)
logical :: res
A = reshape([0., -3., 3., 0.], shape(A))
B = reshape([0., 3., 3., 0.], shape(B))
res = is_skew_symmetric(A) ! returns .true.
res = is_skew_symmetric(B) ! returns .false.
end program example_is_skew_symmetric


## is_hermitian - Checks if a matrix is Hermitian

Experimental

### Description

Checks if a matrix is Hermitian

### Syntax

d = is_hermitian(A)

### Arguments

A: Shall be a rank-2 array

### Return value

Returns a logical scalar that is .true. if the input matrix is Hermitian, and .false. otherwise.

### Example

program example_is_hermitian
use stdlib_linalg, only: is_hermitian
implicit none
complex :: A(2, 2), B(2, 2)
logical :: res
A = reshape([cmplx(1., 0.), cmplx(3., -1.), cmplx(3., 1.), cmplx(4., 0.)], shape(A))
B = reshape([cmplx(1., 0.), cmplx(3., 1.), cmplx(3., 1.), cmplx(4., 0.)], shape(B))
res = is_hermitian(A) ! returns .true.
res = is_hermitian(B) ! returns .false.
end program example_is_hermitian


## is_triangular - Checks if a matrix is triangular

Experimental

### Description

Checks if a matrix is triangular

### Syntax

d = is_triangular(A,uplo)

### Arguments

A: Shall be a rank-2 array

uplo: Shall be a single character from {'u','U','l','L'}

### Return value

Returns a logical scalar that is .true. if the input matrix is the type of triangular specified by uplo (upper or lower), and .false. otherwise. Note that the definition of triangular used in this implementation allows nonsquare matrices to be triangular. Specifically, upper triangular matrices satisfy a_ij = 0 when j < i, and lower triangular matrices satisfy a_ij = 0 when j > i.

### Example

program example_is_triangular
use stdlib_linalg, only: is_triangular
implicit none
real :: A(3, 3), B(3, 3)
logical :: res
A = reshape([1., 0., 0., 4., 5., 0., 7., 8., 9.], shape(A))
B = reshape([1., 0., 3., 4., 5., 0., 7., 8., 9.], shape(B))
res = is_triangular(A, 'u') ! returns .true.
res = is_triangular(B, 'u') ! returns .false.
end program example_is_triangular


## is_hessenberg - Checks if a matrix is hessenberg

Experimental

### Description

Checks if a matrix is Hessenberg

### Syntax

d = is_hessenberg(A,uplo)

### Arguments

A: Shall be a rank-2 array

uplo: Shall be a single character from {'u','U','l','L'}

### Return value

Returns a logical scalar that is .true. if the input matrix is the type of Hessenberg specified by uplo (upper or lower), and .false. otherwise. Note that the definition of Hessenberg used in this implementation allows nonsquare matrices to be Hessenberg. Specifically, upper Hessenberg matrices satisfy a_ij = 0 when j < i-1, and lower Hessenberg matrices satisfy a_ij = 0 when j > i+1.

### Example

program example_is_hessenberg
use stdlib_linalg, only: is_hessenberg
implicit none
real :: A(3, 3), B(3, 3)
logical :: res
A = reshape([1., 2., 0., 4., 5., 6., 7., 8., 9.], shape(A))
B = reshape([1., 2., 3., 4., 5., 6., 7., 8., 9.], shape(B))
res = is_hessenberg(A, 'u') ! returns .true.
res = is_hessenberg(B, 'u') ! returns .false.
end program example_is_hessenberg