A unitary transformation of a set of vectors to another set of vectors preserves the lengths of the vectors and the angles between the vectors.

In other words, a unitary transformation is a rotation of axes in the Hilbert space. This implies that if the transformation involves matrices of eigenvectors, the eigenvalues of the eigenvectors are preserved. Consider 2 complete sets of orthonormal bases:

where *I * is the identity matrix.

The relation between the two basis sets are

where are elements of the transformation matrix *U*.

We say that is transformed to by . Similarly, we have .

The reverse transformation of eq68 is:

###### Question

Show that .

###### Answer

Similarly,