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I am given a matrix having blocks of unitary matrices along the main diagonal eg. $$M = \begin{pmatrix}A & 0\\ 0 & D\end{pmatrix}$$ Here $A$ and $D$ are $3$x$3$ and $2$x$2$ unitary matrices repsectively.
Are matrices of type $M$, having block unitary matrices ( of different sizes ) along the main diagonal unitary in general ?

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Recall that $$\left (\begin{matrix}A & 0 \\ 0 & B\end{matrix} \right )\left (\begin{matrix}C & 0 \\ 0 & D\end{matrix} \right )=\left (\begin{matrix}AC & 0 \\ 0 & BD\end{matrix} \right )$$

If the blocks $A$ and $C$ have the same size, and the blocks $B$ and $D$ have the same size.

Use this, the fact that the blocks are unitary, and that $\left (\begin{matrix}A & 0 \\ 0 & B\end{matrix} \right )^\ast = \left (\begin{matrix}A^\ast & 0 \\ 0 & B^\ast\end{matrix} \right )$ to show that $$\left (\begin{matrix}A & 0 \\ 0 & B\end{matrix} \right )\left (\begin{matrix}A^\ast & 0 \\ 0 & B^\ast\end{matrix} \right )=\left (\begin{matrix}A^\ast & 0 \\ 0 & B^\ast\end{matrix} \right )\left (\begin{matrix}A & 0 \\ 0 & B\end{matrix} \right )=I$$

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  • $\begingroup$ Is it only true same block sizes then ? $\endgroup$
    – sashas
    Apr 22 '15 at 23:12
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    $\begingroup$ Not all the blocks have to have the same size; the upper blocks can have a different (common) size than the lower blocks. $\endgroup$
    – Brian Tung
    Apr 22 '15 at 23:27
  • $\begingroup$ @BrianTung I mean can $A$ and $C$ have different sizes ? $\endgroup$
    – sashas
    Apr 22 '15 at 23:34
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    $\begingroup$ If they do, then the first equation provided by Reveillark is not true. But in the problem you ask, they are the same size, since $C$ is just $A^*$. $\endgroup$
    – Brian Tung
    Apr 22 '15 at 23:35
  • $\begingroup$ @Reveillark thanks for explaining. $\endgroup$
    – sashas
    Apr 22 '15 at 23:52
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With $T$ representing the appropriate transpose for your field:

$$(Mx)^T Mx = x^T M^TM x = x^T \begin{pmatrix} A^T & 0 \\ 0 & D^T\end{pmatrix} \begin{pmatrix} A & 0 \\ 0 & D\end{pmatrix} x = x^T\begin{pmatrix} A^TA & 0 \\ 0 & D^TD \end{pmatrix}x.$$

Write $x = \begin{pmatrix} x_A \\ x_D\end{pmatrix}$ and note that since $A, D$ are unitary, $x_A^T A^TA x_A = x_A^T x_A$ and $x_D^T D^TDx_D = x_D^T x_D$.

Then,

$$x^T\begin{pmatrix} A^TA & 0 \\ 0 & D^TD \end{pmatrix}x = \begin{pmatrix} x_A^t & x_D^T\end{pmatrix}\begin{pmatrix} A^TA & 0 \\ 0 & D^TD \end{pmatrix}\begin{pmatrix} x_A \\ x_D\end{pmatrix} = x_A^Tx_A + x_D^Tx_D = x^Tx.$$

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  • $\begingroup$ @Aramis is it true in general for matrices have unitary block matrices along main diagonal with different sizes ? $\endgroup$
    – sashas
    Apr 22 '15 at 23:38
  • $\begingroup$ @sasha Think a little on this and you should be able to answer that question for yourself. $\endgroup$
    – Emily
    Apr 22 '15 at 23:42
  • $\begingroup$ got it thanks :) $\endgroup$
    – sashas
    Apr 22 '15 at 23:57

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