The following characterisation
of totally indecomposable nonnegative n-square matrices is introduced: A
nonnegative n-square matrix is totally indecomposable if and only if it diminishes
the number of zeros of every n-dimensional nonnegative vector which is
neither positive nor zero. From this characterisation it follows quite easily
that:
I. The class of totally indecomposable nonnegative n-square matrices is closed
with respect to matrix multiplication.
II. The (n − 1)-st power of a matrix of that class is positive.
A very short proof of two equivalent versions of the König-Frobenius duality
theorem on (0,1)-matrices is supplied at the end.
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