Householder QR: One Mirror per Column

Here are the three columns of a matrix A, in red, blue and green. The goal is an upper triangular R, using nothing but mirrors.

A Householder matrix, the identity minus two v hat v hat transposed, reflects every vector across the plane perpendicular to v. The part along v is reversed, and the part in the plane is kept.

The rest of the narration is in the book's Companion.