fully indecomposable matrix

An $n\times n$ matrix $A$ that contains an $s\times (n-s)$ zero submatrix for some positive integer $s$ is said to be partly decomposable. If no such submatrix exists then $A$is said to be it fully indecomposable.By convention, a $1\times 1$ matrix is fully indecomposable if it is nonzero.$A$ is nearly decomposable if it fully indecomposable but whenever a nonzero entry is changed to 0 the resulting matrix is partly decomposable.