How to define the graph of a square matrix $\mathbf{G}$ with real entries?
I know that given a graph $\Gamma(V, E)$, one can define its adjacency matrix $\mathbf{A}$. But given a matrix $\mathbf{G}$ how to define its graph?
P.S. I was reading in this book Matrix Analysis. There is a theorem that asserts that $\mathbf{A}$ is irreducible $\Leftrightarrow$ $\Gamma(\mathbf{A})$ is strongly connected (where $\Gamma(\mathbf{A})$is the graph of $\mathbf{A}$). I do not know what is the definiton of the graph of a square matrix.
In this context, if the matrix is $n\times n$ we define a directed graph with vertex set $\{1,\ldots,n\}$, where there is an arc from $i$ to $j$ if $A_{i,j}\ne0$. This directed graph may have loops, but they do not affect whether the directed graph is strongly connected. It is not weighted.