Householder reflection

A mirror-flip transformation, chosen to send one particular vector straight onto a coordinate axis while leaving lengths unchanged.

A Householder reflection is an orthogonal matrix $H=I-2\mathbf v\mathbf v^{\mathsf T}$ built from a unit vector $\mathbf v$ , geometrically a reflection across the hyperplane normal to $\mathbf v$ . For any nonzero $\mathbf x$ , a specific choice of $\mathbf v$ sends $H\mathbf x$ to a multiple of the first standard basis vector -- zeroing every other entry in one stable step, without ever dividing by a small pivot.

Examples

One Householder reflection can send $(3,4)$ to $(5,0)$ or $(-5,0)$ -- the same length, rotated onto the axis, in a single reflection rather than several row operations.