# anti-diagonal matrix

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

If $a_{1},\ldots,a_{n}\in\mathbbmss{F}$, let

 $\operatorname{adiag}(a_{1},\ldots,a_{n})=\begin{pmatrix}0&0&0&0&a_{1}\\ 0&0&0&a_{2}&0\\ 0&0&a_{3}&&0\\ 0&\cdot&0&&0\\ a_{n}&0&0&&0\end{pmatrix}.$

## Properties of anti-diagonal matrices

1. 1.

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

2. 2.

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

Title anti-diagonal matrix AntidiagonalMatrix 2013-03-22 15:12:20 2013-03-22 15:12:20 matte (1858) matte (1858) 8 matte (1858) Definition msc 15-00