Suppose $U$ and $V$ are two unitary matrices. Prove $U \oplus V$ is also unitary.

The question is given two unitary matrices $U$ and $V$ but their orders are not specified. Then, we are asked to prove matrix $U \oplus V$ which is a direct sum of two matrices is also unitary.

I know the direct sum is "block" construction of two matrices as diagonal elements of the new one. However, I do not know how to proceed by using the fact that $U^{H}U=UU^{H}=I$ to prove that direct sum matrix to be unitary too.

• Compute $UU^*$ from the "block construction" of $U$ – fgp Mar 19 '14 at 16:30
• Is it correct to write: $$U \bigoplus V=\begin{pmatrix} U &0 \\ 0 &V \end{pmatrix}$$? @fgp – nam Mar 19 '14 at 16:33
• Yes... $\phantom{}$ – draks ... Mar 19 '14 at 16:35
• @nam Yes. Now figure out how $(U\oplus V)^H$ looks like, and compute the product, using that $UU^H = VV^H = I$. Then do the same for the reversed product (U\oplus V)^H (U\oplus V)$. (Btw, my inital comment should obviously have read "... of$U\oplus V\$") – fgp Mar 19 '14 at 16:37
• thank you for your hints. it is good to start with. – nam Mar 19 '14 at 16:43

$$(U \oplus V)^H (U \oplus V)=\begin{pmatrix} U &0 \\ 0 &V \end{pmatrix}^H \begin{pmatrix} U &0 \\ 0 &V \end{pmatrix}=\begin{pmatrix} U^HU &0 \\ 0 &V^HV \end{pmatrix}=\begin{pmatrix} 1 &0 \\ 0 &1 \end{pmatrix}$$