orthogonal iteration

also called subspace iteration or simultaneous iteration
it let us compute n dimensional rather one by each time

equivalent to QR iteration

Steps

start $U_0=I$ and repeat following for $k=1,2,...$
$V_k=A{U}_{k-1}$
$U_k{R}_k=V_k$ by QR factorization
when $U_k-U_{k-1}<tol$, we say it converge and stop

After convergence:

1. Compute the Rayleigh quotient matrix: $H=U^{T}AU$ where $U$ is last $U_k$
2. Find eigen-decomposition of H: $HQ=QD$, where D is diagonal
3. Compute eigenvectors of original matrix A: $X=UQ$
4. The diagonal entries of D are the corresponding eigenvalues