Accumulated QR factorization identity
= Accumulated QR factorization identity
{c}
Let
$$
\overline Q_k=Q_0Q_1\cdots Q_k,
\qquad
\overline R_k=R_kR_{k-1}\cdots R_0.
$$
Then
$$
A_{k+1}=\overline Q_k^TA\overline Q_k,
\qquad
A^{k+1}=\overline Q_k\overline R_k.
$$
Thus $\overline Q_k\overline R_k$ 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.