Quantum Country exercise 13

This is one in a series of posts working through the exercises in the Quantum Country online introduction to quantum computing and related topics. The exercises in the original document are not numbered; I have added my own numbers for convenience in referring to them.

Exercise 13. Show that the product UV of two unitary matrices U and V is itself a unitary matrix.

Answer: Since U and V are unitary matrices we have U^\dagger U = I and V^\dagger V = I.

Now consider the product UV. We have

\left( UV \right)^\dagger \left( UV \right) = \left( V^\dagger U^\dagger \right) \left( UV \right)

= V^\dagger \left( U^\dagger U \right) V = V^\dagger I V = V^\dagger V = I.

Since \left( UV \right)^\dagger \left( UV \right) = I the product matrix UV is unitary. Thus the product UV of two unitary matrices U and V is itself a unitary matrix.

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