Orthogonal Matrix

Let be an orthonormal basis for . Then

where (see piecewise function, dot product).

If we define by , then

(see transpose, identity matrix, matrix multiplication)

so (see matrix inverse).

Definition

Let .

is called an orthogonal matrix if

(see identity matrix, matrix inverse)

Theorem

Let . The following are equivalent:

  1. is an orthogonal matrix
  2. The columns of form an orthonormal basis for
  3. The rows of form an orthonormal basis for

Emphasis on orthonormal.