Normal tropical $(0,-1)$-matrices and their orthogonal sets

Square matrices $A$ and $B$ are orthogonal if $A\odot B=Z=B\odot A$, where $Z$ is the matrix with all entries equal to $0$, and $\odot$ is the tropical matrix multiplication. We study orthogonality for normal matrices over the set $\{0,-1\}$, endowed with tropical addition and multiplication. To do this, we investigate the orthogonal set of a matrix $A$, i.e., the set of all matrices orthogonal to $A$. In particular, we study the family of minimal elements inside the orthogonal set, called a basis. Orthogonal sets and bases are computed for various matrices and matrix sets. Matrices whose bases are singletons are characterized. Orthogonality and minimal orthogonality are described in the language of graphs. The geometric interpretation of the results obtained is discussed.