Which matrices preserve length and angle?

Source scope: §3.9. Worked numerical examples, experiments, and practice are teaching supplements unless identified as source results.

Turn without stretching

A quarter turn sends $(2,1)$ to $(-1,2)$. Both lengths are $\sqrt5$. Its matrix $Q=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$ has orthogonal columns of unit length.

A square matrix $Q\in\mathbb R^{n\times n}$ is orthogonal when its columns are pairwise orthonormal. Entry $(i,j)$ of $Q^TQ$ is the inner product of columns $i,j$, so

The square assumption matters. A rectangular matrix with orthonormal columns satisfies $U^TU=I$, but $UU^T$ can be a projection rather than the identity.

Source derivation: preserve the product first

For all $x,y\in\mathbb R^n$,

Setting $y=x$ and taking square roots gives

For nonzero inputs, both the dot product and lengths are unchanged, so the cosine of their angle is unchanged. Orthogonal maps may rotate or reflect about the origin. A reflection $\operatorname{diag}(1,-1)$ has determinant negative one and still preserves length. A translation preserves distances too, but it is not multiplication by an orthogonal matrix because it moves zero.

Explore in three dimensions

Transfer

Is $2I$ orthogonal because its columns are perpendicular?