Question

What does it mean about matrix A if `A^TA=I`?

Answer 1

It means `A` is an orthogonal matrix.

Explanation:

The rows of `A` form an orthogonal set of unit vectors.

Similarly, the columns of `A` form an orthogonal set of unit vectors.

`A` is essentially a rotation about the origin and possible reflection. It preserves distances and angles.

A typical `2 xx 2` orthogonal matrix would take the form:

`((cos theta, sin theta), (-sin theta, cos theta))`

The determinant of `A` will be `+-1`

If the determinant of `A` is `1`, then `A` is called a special orthogonal matrix. It is essentially a rotation matrix.