Notation for reversed rows and/or columns of a matrix?

Question

In this answer I am using a transformation of matrices, and I would like to know if there is a notation for this.

Given a matrix $A$, let $B$ be the same as $A$ with rows in reverse order, $C$ with the columns in reverse order, and $D$ with both rows and columns in reverse order. Is there a usual notation for any of $B,C$ or $D$ ?

The $D$ case has a nice property: if it's called $f$, then for any matrices $A,B$, $f(AB)=f(A)f(B)$.

I know one can write for instance $B=A[n:1,:]$ and it's not uncommon to see such MATLAB-like formulas in numerical analysis articles, but I wonder if there is a shorthand for this, or a usual name.

Answer 1

I don't know of anything like that in standard mathematical usage, but the APL programming language has operators $C = ⌽A$ and $B = \ominus A$.

Then your theorem is that $$⌽\ominus(XY) = (⌽\ominus X)(⌽\ominus Y)$$ or more briefly that $⌽\ominus$ is a homomorphism. (Clearly, $⌽\ominus = \ominus⌽$.)

Probably nobody would complain much if you used something like that.

Answer 2

The $n\times n$ permutation matrix $(J_n)$ with ones along counter-diagonal (often called the exchange matrix) can perform the indicated transformations. Given $A \in {\mathbb C}^{n\times m}\;$ the transformations are $$\eqalign{ J_nA &= A\big[n:1,\,:\big] \\ AJ_m &= A\big[:\,,\,m:1\big] \\ J_nAJ_m &= A\big[n:1,\,m:1\big] \\ }$$ And the "nice property" is a consequence of the involutory nature of the exchange matrix. $$\eqalign{ f(AB) &= J_n(AB)J_k \\ &= J_nA(J_mJ_m)BJ_k &\qquad \{J_m^2=I\} \\ &= (J_nAJ_m)\,(J_mBJ_k) \\ &= f(A)\,f(B) \\ }$$