Directional stretch ($I+(\alpha -1)u{u}^T$)

For a unit vector $u$ in a real inner product space, stretching by a factor $\alpha$ along $u$ and leaving its perpendicular subspace fixed givesDecompose $x=(u^{T}x)u+(x-(u^{T}x)u)$. The first term is the orthogonal projection onto the line; the second is perpendicular to it. The displayed map multiplies the first by $\alpha$ and fixes the second. In the plane, if $u=(\cos \theta ,\sin \theta {)}^T$, then $T=R(\theta )\mathrm{diag}(\alpha ,1)R(-\theta )$.