Linear Algebra and Its Applications
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.
Johnson, C. R., Marijuán, C., & Pisonero, M. (2012). Submatrix monotonicity of the Perron root. Linear Algebra and its Applications, 437(10), 2429-2435.