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Householder's classic 1958 paper popularized a Hermitian unitary matrix now well-known as the Householder reflector. The reflector has a special structure $I - 2P$, where $P$ is a rank-1 orthogonal projector. The reflector plays a key role in several significant algorithms for solving least squares and eigenvalue problems. The influential paper, however, left a subtle oversight that still lingers in recent books and course materials. To correct this oversight, we show that the structure $I - 2P$ needs to be extended. In particular, the Hermitian property must be forsaken when the source of reflection and its target do not obey a certain symmetry condition. We introduce extended structures that fulfill the task of reflection without requiring this condition on symmetry. For reflecting a single vector, we extend the $I - 2P$ structure into $I - \eta P$ using a specific $\eta \in \mathbb{C}$. This is in the same spirit of a standard fix of the Householder reflector. For the more general scenario of reflecting multiple vectors simultaneously, we first study the norm-preserving linear targeting problem: we extend the $I - 2P$ structure into $I - (P_1+P_2)$, where only $P_1$ is required to be an orthogonal projector; the $I - (P_1+P_2)$ structure naturally leads to another structure $I-USU^*$, where $U$ has orthonormal columns and $S$ is a square matrix of a suitable size. We further show that the $I - (P_1+P_2)$ structure also arises in a linear targeting problem with a requirement on positive-definiteness instead of norm-preservation.