Suppose \( \hat{A} \) is the triangular matrix we obtained after the Householder transformation, $$ \hat{A} = \hat{Q}\hat{U}, $$ and multiply from the left with \( \hat{Q}^{-1} \) resulting in $$ \hat{Q}^{-1}\hat{A} = \hat{U}. $$ Suppose that \( \hat{Q} \) consists of a series of planar Jacobi like rotations acting on sub blocks of \( \hat{A} \) so that all elements below the diagonal are zeroed out $$ \hat{Q}=\hat{R}_{12}\hat{R}_{23}\dots\hat{R}_{n-1,n}. $$