The spectral norm of a nonnegative matrix

AbstractGiven a matrix A=[aij], define ‖A‖=[‖aij‖]. Let ⦀ · ⦀2 denote the spectral norm. We show that for any matrix A ⦀‖A‖⦀2=min{r1(B)c1(C):B∘C=A} and show that under mild conditions the minimizers in (1) are essentially unique and are related to the left and right singular vectors of A in a simple way. We also show that ⦀A⦀2⩽⦀‖A‖⦀2 and determine the case of equality