Show that a product of unitary orthogonal matrices is unitar

Show that a product of unitary (orthogonal) matrices is unitary (orthogonal) as well.

Solution

A unitary matrix is a square matrix, U, where UU* = I. (* denotes the conjugate transpose or hermitian operator).

In order to prove that the product of two unitary matrices, A and B, is unitary you must show that AB(AB)* = I.
We have , (AB)* = B*A* and since A*A = B*B = I.

Assume A and B are unitary matrices,
=> AB(AB)* = AB(B*A*) = A(BB*)A* = AIA* = AA* = I
=> AB(AB)* = I
Therefore , AB is unitary.