Let $A$ be a $4 \times4$ matrix with complex entries with columns and rows are mutually orthogonal and row/column vectors have norm $\sqrt{2}$ where the the rows and columns are indexed by $(i,j)$, $i,j=1,2$ such that if you put any column/row of $A$ into a $2 \times 2$ matrix will be unitary. From $A$ I want to construct another matrix $B$ which has the following property.Columns /rows are mutually orthogonal and each has norm $\sqrt{2}$ and if we take any two columns indexed by (-,j). From these take the rows indexed by (i,-). The resultant $2 \times 2$ matrix should be unitary. How can we get $B$ from $A$? Please help me. By which matrix we should apply to $A$ to get $B$? That is explicitely I have the following data.
$A=\begin{bmatrix} a^{11}_{11} & a^{11}_{12} & a^{11}_{21} & a^{11}_{22} \\ a^{12}_{11} & a^{12}_{12} & a^{12}_{21} & a^{12}_{22} \\ a^{21}_{11} & a^{21}_{12} & a^{21}_{21} & a^{21}_{22} \\ a^{22}_{11} & a^{22}_{12} & a^{22}_{21} & a^{22}_{22} \end{bmatrix} $ and I have the following matrices are unitary.
$C_{11}=\begin{bmatrix} a^{11}_{11} & a^{21}_{11}\\ a^{12}_{11} & a^{22}_{11} \end{bmatrix}$,
$C_{12}=\begin{bmatrix} a^{11}_{12} & a^{21}_{12}\\ a^{12}_{12} & a^{22}_{12} \end{bmatrix}$,
$C_{21}=\begin{bmatrix} a^{11}_{21} & a^{21}_{21}\\ a^{12}_{21} & a^{22}_{21} \end{bmatrix}$,
$C_{22}=\begin{bmatrix} a^{11}_{22} & a^{21}_{22}\\ a^{12}_{22} & a^{22}_{22} \end{bmatrix}$
I need to construct a $B$ from $A$ having the following property. If $B=\begin{bmatrix} b^{11}_{11} & b^{11}_{12} & b^{11}_{21} & b^{11}_{22} \\ b^{12}_{11} & b^{12}_{12} & b^{12}_{21} & b^{12}_{22} \\ b^{21}_{11} & b^{21}_{12} & b^{21}_{21} & b^{21}_{22} \\ b^{22}_{11} & b^{22}_{12} & b^{22}_{21} & b^{22}_{22} \end{bmatrix} $ Then $B$ should satisfy the following conditions.
$B$ should have mutually orthogonal rows and columns with norm being $\sqrt{2}$
The following matrices formed from $B$ are unitary.
$D_{11}=\begin{bmatrix} b^{11}_{11} & b^{11}_{21}\\ b^{12}_{11} & b^{12}_{21} \end{bmatrix}$,
$D_{12}=\begin{bmatrix} b^{11}_{12} & b^{11}_{22}\\ b^{12}_{12} & b^{12}_{22} \end{bmatrix}$,
$D_{21}=\begin{bmatrix} b^{21}_{11} & b^{21}_{21}\\ b^{22}_{11} & b^{22}_{21} \end{bmatrix}$,
$D_{22}=\begin{bmatrix} b^{21}_{12} & b^{21}_{22}\\ b^{22}_{12} & b^{22}_{22} \end{bmatrix}$ Is there any way to construct $B$ using $A$?