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

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Answer 1 · 2015-10-25T20:57:49

It means $A$ is an orthogonal matrix.

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.

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