A demo R script of [the paper EIGENVECTORS FROM EIGENVALUES](https://arxiv.org/pdf/1908.03795.pdf), modified from [Yu-Chen Shu](https://www.facebook.com/yuchen.shu/posts/10157876738839228) rewritten in R.

eigenvectors_from_eigenvaluies.R

# Generate random matrix with dim = N
N <- 3
A <- matrix(runif(N*N), N, N, byrow=TRUE)
A <- A+t(A)

# Setting up eigenvalues lambda, true eigenvector V and derived eigenvector U
ev <- eigen(A)
(lambda <- ev$values)
(V <- ev$vectors)
U <- matrix(0L, N, N)

# Calculate from lemma 2 in https://arxiv.org/pdf/1908.03795.pdf
for (i in 1:N) {
  lambda_D <- lambda; lambda_D[i] <- NA; lambda_D <- lambda_D[!is.na(lambda_D)]
  denominator <- prod(lambda[i]-lambda_D)
  for (j in 1:N) {
    M <- A; M[,j] = M[j,] <- NA; M <- matrix(M[!is.na(M)], N-1, N-1)
    lambda_M <- eigen(M)$values
    fraction <- prod(lambda[i]-lambda_M)
    U[j,i] <- fraction/denominator
  }
}

# Check the infinity norm of DIFF(U, V^2) shows that they are close enough
norm(U - V*V, type="I")

Thanks for sharing