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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?
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  • $\begingroup$ Notice that since $B$ in general will have rank $m-1=2n-2$, larger than the typical rank of $A$, which is $n$, you cannot generally find matrices $C$ and $D$ such that $CAD=B$; and then clearly never with $C$ and $D$ independent of $A$. So you are looking at just a linear embedding of ${\mathbb F}^{n\times n} \rightarrow {\mathbb F}^{m\times m}$ without immediately obvious further algebraic properties. $\endgroup$
    – Yaakov Baruch
    Commented Aug 8, 2017 at 12:46
  • $\begingroup$ You should first extend the matrix $A$ to a matrix $A'$ of size $m$ and attach to each element $a'_{i,j}$ a point $P=P(i,j)$ of coordinates $(x,y)$ in the plane, rotate those points to get $P'(i,j)$ of coordinates $(x',y')$ and get $(k,l)=P^{-1}(x',y')$. $\endgroup$
    – Sylvain JULIEN
    Commented Aug 8, 2017 at 18:57
  • $\begingroup$ Note that the transposition of $A'$ corresponds to the map $(i,j)\mapsto(m+1-i,j)$ in its image. This corresponds to an automorphism of the diedral group with $8$ elements. $\endgroup$
    – Sylvain JULIEN
    Commented Aug 8, 2017 at 19:13
  • $\begingroup$ Ad 3: I think straightforward group theoretic implications are rather unlikely already since your transformation does not preserve invertibility of a matrix. That said, of course a definitive negative answer to the question as stated cannot be given. $\endgroup$
    – Stefan Kohl ♦
    Commented Aug 8, 2017 at 20:21

1 Answer 1

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A way to rotate a 2D array by 45$^\circ$ by means of a shear mapping is worked out in this Stack Overvlow posting.

The 45$^\circ$ shear mapping transforms a diagonal in the original array (black points) into a vertical column of the sheared array (red).

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    $\begingroup$ According to the linked Wikipedia article, the shear map fixes one coordinate (depending on whether the shear is parallel to the x-axis or the y-axis ) which is not the case in my example. $\endgroup$
    – Mohammad Al-Turkistany
    Commented Aug 8, 2017 at 16:37

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