Orthonormal Matrices: Lengths Survive, Even through Many Layers
An interactive live film: Q turns or mirrors a cube while lengths and angles stay; a gradient climbs ten orthonormal layers keeping its length, and scaled layers make it explode or vanish. Find the scale that keeps it.
Narration
A square matrix Q with orthonormal columns: each column has length one, and each pair is perpendicular, so Q transposed Q is the identity. Here it turns a cube and two vectors.
It preserves inner products. The two vectors keep their lengths and the angle between them, and the cube keeps its shape. Its inverse is simply its transpose.
The rest of the narration is in the book's Companion.
