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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.

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  • $\begingroup$ Compute $UU^*$ from the "block construction" of $U$ $\endgroup$ – fgp Mar 19 '14 at 16:30
  • $\begingroup$ Is it correct to write: $$U \bigoplus V=\begin{pmatrix} U &0 \\ 0 &V \end{pmatrix}$$? @fgp $\endgroup$ – nam Mar 19 '14 at 16:33
  • $\begingroup$ Yes... $\phantom{}$ $\endgroup$ – draks ... Mar 19 '14 at 16:35
  • $\begingroup$ @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 $U\oplus V$") $\endgroup$ – fgp Mar 19 '14 at 16:37
  • $\begingroup$ thank you for your hints. it is good to start with. $\endgroup$ – nam Mar 19 '14 at 16:43
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As you already noticed " the direct sum is "block" construction of two matrices as diagonal elements":

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

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  • $\begingroup$ @name you're welcome... $\endgroup$ – draks ... Mar 19 '14 at 16:44

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