Quantum Country exercise 13

Posted on October 19, 2021 by hecker

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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