When is the matrix $A^{\ast} A$ isometric?

Question

Are there conditions for a square matrix $A$ such that $A^{\ast} A$ is isometric, that is $\| A^{\ast} A x \| = \| x \|$ for all $x$?

Answer 1

Since $A^*A$ is a positive self-adjoint matrix, it is diagonalizable, and its eigenvalues are all positive. Requiring that $A^*A$ is an isometry will impose that all its eigenvalues are $1$, hence $A^*A =I$.

So $A^* A$ is isometric will imply that $A$ is unitary.