Vol. 36, No. 3, 1971

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On nonnegative matrices

Mordechai Lewin

Vol. 36 (1971), No. 3, 753–759

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.

Mathematical Subject Classification
Primary: 15.60
Received: 27 January 1970
Published: 1 March 1971
Mordechai Lewin