# construct a permutation matrix P from the permutation vector p
function permutation_matrix(p)
    P = zeros(Int, length(p),length(p))
    for i = 1:length(p)
        P[i,p[i]] = 1
    end
    return P
end

P =
permutation_matrix([2,4,1,5,3])

"ONE-HOT VECTOR"

I₅ = eye(5)

P * I₅

P

P'*P

P*P'

A =rand(3,5)

A'

function my_transpose(A)
    m,n=size(A)
    B = zeros(n,m)
    for i=1:m, j=1:n
        B[j,i]=A[i,j]
    end
    B
end

A

my_transpose(A)

A = rand(3,3)
B = rand(3,3)
(A*B)'

B' * A'

x = rand(3)

y  = rand(3)

x ⋅ (A*y) # The transpose of A is the unique matrix such that
# x ⋅ (A*y) = (A'x)⋅y for all x and y

(A'x)⋅y

C = rand(-9:9, 4,4)
D = rand(-9:9, 4,4)
(C*D)' == D'*C'

A = [4  -2  -7  -4  -8
     9  -6  -6  -1  -5
    -2  -9   3  -5   2
     9   7  -9   5  -8
    -1   6  -3   9   6]

inv(A')

inv(A)'

L,U,p = lu(A)

p

L'

U'

b = [4,2,1,-2,3] # "randomly" chosen right-hand side
A' \ b # correct solution to Aᵀx = b

c = U' \ b # forward-substitution
d = L' \ c # backsubstitution
permutation_matrix(p)' * d

LU = lufact(A)
LU' \ b

A

S = A' + A

S' == (A')'+ A' == A +A' == A' + A

S = A' * A

S' == (A' *A )' == A' * A

L, U = lu(S, Val{false}) # LU without pivoting

L

U

diag(U)

D = diagm(diag(U)) # diagm makes a diagonal matrix from a 1d array

inv(D)

inv(D) * U

L'

S

L
D = abs.(D)
S =  L*abs.(D)*L'
U = L'
L

K = L* diagm(sqrt.(diag(D)))

K*K' - S # kind of a matrix sqrt: cholesky factorization

chol(S)

S - S'

chol(Symmetric(S))

K'