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Accumulated QR factorization identity

Let

\overline{Q}_{k} = Q_{0} Q_{1} \dots Q_{k}, \overline{R}_{k} = R_{k} R_{k - 1} \dots R_{0} .

Then

A_{k + 1} = \overline{Q}_{k}^{T} A \overline{Q}_{k}, 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.

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