A alternating matrix is a matrix A such that

$$A^{t}=-A $$

where \(A^{t}\) is a transposed matrix of A. Such a matrix is necessarily square. Its main diagonal entries are 0s, and its other entries occur in negative pairs on opposite sides of the main diagonal.

---

Suppose that A is a \(3\times 3\) alternating matrix:

$$A=\begin{bmatrix}a_{11} &a_{12}&a_{13}\\a_{21} &a_{22}&a_{23}\\a_{31} &a_{32}&a_{33}\\\end{bmatrix}$$

Since \(A^{t}=-A\), we have

$$\begin{bmatrix}a_{11} &a_{12}&a_{13}\\a_{21} &a_{22}&a_{23}\\a_{31} &a_{32}&a_{33}\\\end{bmatrix}=-\begin{bmatrix}a_{11} &a_{21}&a_{31}\\a_{12} &a_{22}&a_{32}\\a_{13} &a_{32}&a_{33}\\\end{bmatrix}$$

Therefore, an alternating matrix is given as below:

$$A=\begin{bmatrix}0 &a_{12}&a_{13}\\-a_{12} &0&a_{23}\\-a_{13} &-a_{23}&0\\\end{bmatrix}$$

In general, an alternating matrix is given as below:

$$A^{t}=-A\ \ \Longleftrightarrow\ \ A=\begin{bmatrix}0&a_{12}&a_{13}&\cdots&a_{1n}\\-a_{12} &0&a_{23}&\cdots&a_{2n}\\-a_{13} &-a_{23}&0&\cdots&a_{3n}\\\vdots &\vdots &\vdots& \ddots & \vdots\\-a_{1n} &-a_{2n}&-a_{3n}&\cdots&0\\\end{bmatrix}$$

That is

\begin{eqnarray*}&&a_{ii}=0 \ (\ i= j\ ) \\&&a_{ij}=-a_{ji} \ (\ i\not= j\ )\end{eqnarray*}

where \(a_{ij}\) is \((i, j)\)-entry in a matrix A.

Example

\begin{eqnarray*}\text{Alternating } \ &:& \ \begin{bmatrix}&1\\-1 &0\end{bmatrix}, \ \ \begin{bmatrix}0 &1&2\\-1 &0&3\\3 &-2&0\\\end{bmatrix}, \ \ \ \begin{bmatrix}0&1 &2&3\\-1&0&4&5\\-2& -3&0&6\\-3&-5&-6&0\\\end{bmatrix}\\\text{Nonalternating } \ &:& \ \begin{bmatrix}1&2\\3 &4\end{bmatrix}, \ \ \begin{bmatrix}1 &2&3\\4&5&6\\7 &8&9\\\end{bmatrix}, \ \ \ \begin{bmatrix}1 &2&3&4\\5&6&7&8\\9& 10&11&12\\13&14&15&16\\\end{bmatrix}\end{eqnarray*}