Some Extremal Problems for Strictly Totally Positive Matrices

For a strictly totally positive M × N matrix A we show that the ratio IlAxllp/llxllp has exactly R = min{ M, N ) nonzero critical values for each fixed p ~ (1, ~ ) . Letting ~ denote the i th critical value, and x ~ an associated critical vector, we show that ~l > " " " > An > 0 and x i (unique up to multiplication by a constant) has exactly i 1 sign changes. These critical values are generalizations to I p of the s-numbers of A and satisfy many of the same extremal properties enjoyed by the s-numbers, but with respect to the I p norm. 1. I N T R O D U C T I O N In the course of our inves t iga t ions in to n-widths of Sobolev spaces in L p, w e w e r e led to a cons ide ra t ion of the fo l lowing mat r ix problems. L e t A be an M × N matr ix , and p c [1, ~ ] . F o r x ~ R " , set Ilxllp = (E?llxil") x/p, 1 ~< p < ~ , and Ilxll~ = max{[xil:i = 1 . . . . . m } . Cons ide r the fo l lowing th ree quant i t ies : o2(p) = min max II(A P")x 11~ (A) eo ~ . o Ilxllp ' *Supported in part by a grant from the U.S. Army through its European Research Office under contract No. DAJA 37-81-C-0234, and in part by the Technion V.P.R. Fund--K.&M Bank Research Fund. LINEAR ALGEBRA AND ITS APPLICATIONS 64:141-156 (1985) 141 ~ Elsevier Science Publishing Co., Inc., 1985 52 Vanderbilt Ave., New York, NY 10017 0024-3795/85/$3.30 142 A. PINKUS where P,, ranges over all M × N matrices of rank u, t lAxt lp o,~( p ) = min max x ~ ~x, , llxlt,, (B) where X,, is any subspace of R '~ of dimension n; IIAxil , , o,~( p ) = max rain x ~ 0 t O ) where X,,+l is any subspace of R "~' of dimension n + 1. It is easily proven that o~(p)>/o ,7(p) -/o,~(p). For p = 2, it is well known that equality holds for any matrix A, i.e., a,~(p) = a ~ ( p ) = o,3(p) (by the Rayleigh-Ritz characterization of eigenvalues). The common value is the square root of the (n + 1)st eigenvalue of ATA (arranged in nonincreasing order of magnitude), i.e., the singular values or s-numbers of A. Appropria te opt imal P,, X , , and X n ~ t in (A), (B) and (C), respectively, may be obtained from the singular value decomposi t ion of ArA. For p ~ 2 very little is known except when A is a diagonal matrix (see [6]), or when p = i, ~ and A is strictly totally positive (STP) (see Micchelli and Pinkus [5] for p = ~ . and a duality a rgument gives p = 1). We prove the following result. THEOREM 1.1. Let A be an M × N STP matrix and p ~ [1, ~ ] . 1hen for each n, 0 ~< n < rank A = min ( M, N } = R, ~l ) o ~ ( p ) = ~ , ~ p ) = o,:?(p)o,,(p~ (,2) o 0 ( p ) > • • • > oi, ~(p ) > 0. We also identify an optimal rank n matrix in (A) and optimal subspaces in (B) and (C). For A as above it follows by a theorem of Gan tmacher and Krein [1] that ArA has simple, distinct, positive eigenvalues ~1 > " " " > ~ > 0, mad if x ' is the eigenvector of ArA associated with ~i, then S ~ ( x ~ ) = S ( x i ) = i l , i = 1 . . . . . R (see the next section for a definition of S + and S ). When /9 = 2, Melkman and Miechelli [4] constructed an optimal rank n matrix in (A) and an optimal subspaee in (B) which were not derived from the singular value decomposi t ion of A, but depended on the STP proper ty of A and the sign change proper ty of the eigenvectors, These are the results which are generalized here. The case p = 1, ~ has been solved, and we therefore consider STRICTLY TOTAL POSITIVE MATRICES 143 only p ~ (1, oo). Effectively, we set