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

1 Answer
Oct 25, 2015

Answer:

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.