Consider a square matrix $A$ and its $QR$ decomposition $A = QR$, where $Q$ is orthogonal and $R$ is upper triangular matrix. Now consider a change of basis. Let $A' = PAP^T$, where $P$ is an orthogonal matrix. Is there a way how to compute a $QR$ decomposition of $A'$ directly from $Q, R$ and $P$, without actually computing $A'$? In other words, if I know $Q$ and $R$, is there a way how to obtain $Q'$ and $R'$ without actually computing $A'$ and doing $QR$ decomposition?
2026-04-02 13:23:57.1775136237
QR decomposition and a change of basis
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If $A=QR,$ then $A'=PAP^T=PQRP^T=(PQP^T)(PRP^T).$
The matrix $PQP^T$ is orthogonal. Once you find a decomposition of $PRP^T=Q'R',$ where $Q'$ is orthogonal and $R'$ is upper triangular, you get $PAP^T=(PQP^T)Q'R'.$ The matrix $(PQP^T)Q'$ is orthogonal as the product of two orthogonal matrices. In this way knowing $Q$ and $R$ is sufficient for determining the decomposition of $A'.$