I have $A_{n \times n}$ integer matrix. I want to obtain a matrix $B_{m \times m}$ such that the rows of $A$ are the diagonals of $B$ and the columns of $A$ are the anti-diagonals of $B$ ($m \ge n$). Equivalently, $B$ is the result of rotating $A$ clockwise $45°$ . Here is an example: $$\left( \begin{array}{cc} 1& 2 & 3\\ 4& 5 & 6 \\ 7& 8& 9 \end{array} \right)$$

is transferred to

$$\left( \begin{array}{cc} 0& 0& 1 & 0 &0 \\ 0& 4& 0& 2 &0 \\ 7& 0& 5& 0& 3 \\ 0 & 8& 0& 6 & 0 \\ 0& 0& 9& 0& 0 \end{array} \right)$$

Has anyone studied the properties of such transformation? Which algebraic operation does achieve such transformation? Is there any interesting group-theoretic uses of such transformation?

I have an $n \times n$ integer matrix $A$. I want to obtain an $m \times m$ matrix $B$, where $m \ge n$, such that the rows of $A$ are the diagonals of $B$ and the columns of $A$ are the anti-diagonals of $B$. Equivalently, $B$ is the result of "rotating" matrix $A$ a total of $45^\circ$ clockwise.

For example, the image of

$$\left( \begin{array}{cc} 1& 2 & 3\\ 4& 5 & 6 \\ 7& 8& 9 \end{array} \right)$$

is the following

$$\left( \begin{array}{cc} 0& 0& 1 & 0 &0 \\ 0& 4& 0& 2 &0 \\ 7& 0& 5& 0& 3 \\ 0 & 8& 0& 6 & 0 \\ 0& 0& 9& 0& 0 \end{array} \right)$$

My questions:

1. Has anyone studied the properties of such transformation?
2. Which algebraic operation does achieve such transformation?
3. Is there any interesting group-theoretic uses of such transformation?