Unshifted QR algorithm

The unshifted QR algorithm factors $A_k=Q_k{R}_k$ and forms $A_{k+1}=R_k{Q}_k=Q_k^T{A}_k{Q}_k$, preserving eigenvalues while driving the matrix toward block diagonal form.

LetThenThus ${\overline{Q}}_k{\overline{R}}_k$ is a QR decomposition of the matrix power $A^{k+1}$. In particular, the first $r$ columns of ${\overline{Q}}_k$ span the same subspace as the first $r$ columns of $A^{k+1}$ whenever those columns are independent.

An unshifted QR step preserves the bandwidth of a real symmetric banded matrix. A Givens-rotation implementation exposes this as bulges created during triangularization and removed when the factors are multiplied in reverse order.

The accumulated orthogonal factor in QR iteration is the $Q$ factor of a matrix power, so its leading columns span the same spaces as simultaneous power iteration.

Let a real symmetric matrix have an orthonormal eigenbasis $w_1,\dots ,w_n$, withWrite two starting vectors asFor two-column subspace iteration to recover
$\mathrm{span}(w_{n-1},w_n)$, the two projections onto that dominant subspace must be independent. The exact condition isRequiring each of the four coefficients to be nonzero does not imply this determinant condition.

When an off-block subdiagonal entry tends to zero, QR iteration asymptotically separates the corresponding invariant spectral blocks even if eigenvalues within one block have equal modulus.