A Rank-One Change of the Identity: Identity, Projection, Mirror

Take the identity and subtract beta times u u transposed, with u a unit vector. Watch what it does to the unit sphere, whose top cap is colored orange.

Along u, the matrix multiplies by one minus beta. Every vector in the plane perpendicular to u, the grey disk, is left exactly as it is. Only one eigenvalue changes.

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