Submatrix monotonicity of the Perron root
It is known that increasing an entry of a nonnegative matrix non-decreases (and generally increases) its Perron root. Motivated by a question raised by Jose Dias da Silva, we study the partial order on k-by-k nonnegative matrices in which A less than or similar to(DS) B if whenever A and B occur as submatrices in the same position in otherwise equal nonnegative matrices F and G, p (F) <= p (G). We find that this partial order is equivalent to the entry-wise partial order. This is proven with some asymptotic results about the Perron root that may be of independent interest. (C) 2012 Elsevier Inc. All rights reserved.