anti-diagonal matrix

Let $A$ be a square matrix (over any field $𝔽$). An entry in $A$is an anti-diagonal entry if it is on the line going from thelower left corner of $A$ to the upper right corner. If allentries in $A$ are zero except on the anti-diagonal, then $A$ is ananti-diagonal matrix.

If $a_1,\mathrm{\dots },a_n\in 𝔽$, let

Properties of anti-diagonal matrices

1. If $A$ and $D$ are $n\times n$ anti-diagonal anddiagonal matrices, respectively, then $AD,DA$ are anti-diagonal.

2. The product of two anti-diagonal matrices is an diagonal matrix.