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

Oct 25, 2015

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 $2 \times 2$ orthogonal matrix would take the form:

$\left(\begin{matrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{matrix}\right)$

The determinant of $A$ will be $\pm 1$

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