Putting those two facts together, you see that the (i,j)-element of AA* is the inner product of row i of A with row j of A. If that is equal to the (i,j)-element of the identity matrix then ... .

\(\displaystyle (AA^*)_{ij}= <x_i,x_j> \)

if \(\displaystyle i=j <x_i, x_i> must = 1\) therefore \(\displaystyle ||x_i||=1\) -> normal

if \(\displaystyle i \not=j <x_i, x_j> must = 0 \) therefore \(\displaystyle x_j\) is perpendicular to \(\displaystyle x_i \) -> orthogonal

=> orthonormal

Going the opposite direction:

if the rows of A are orthonormal

\(\displaystyle <x_i,x_j> = 0\)

except when i=j when \(\displaystyle <x_i, x_j>=1\)

therefore \(\displaystyle (AA^*)_{ij} = a_i \overline{a_j} = 1 \) if \(\displaystyle i=j \)

or = 0 if \(\displaystyle i \not=j \)which is the definition of the identity.

One more question: is this enough exposition? That is, did I prove it well enough?

Thank you!