Idempotent matrices in Julia

Let $A$ be a square matrix of order $n$ . Then $A$ is called an idempotent matrix if $AA = A$ .

If a matrix $A$ is idempotent, it follows that $A^n = A, \forall n \in \mathbb{N}$ . Idempotent matrices behave like identity matrices when raised to a power $n$ . All Idempotent matrices except identity matrices are singular matrices.

One way to make idempotent matrices is $A = I - u u^T$ , where $u$ is a vector satisfying $u^T u = 1$ . In this case $A$ is symmetric too.

#Idempotent matrices in Julia
using LinearAlgebra
u = rand(5)
u = u/norm(u) #Force u^T * u = 1
A = I - u*u'
isapprox(A^100,A) # true
isequal(A,A') # true (test for symmetricity)