Page 136 - Textos de Matemática Vol. 44

126 J. K. MERIKOSKI AND A. KOVACˇEC
4. Comparisons
For comparison, we recall certain other upper bounds for λ. Let A be non- negative. Then (e.g., [4, Theorem 8.3.1]) its spectral radius is its eigenvalue; denoteitbyλ(A)=λ.Letr1 ≥···≥rn betheorderedrowsumsofA.Due to Frobenius (e.g., [4, Theorem 8.1.22]),
λ≤r1 =:uF.
There are several improvements of uF but many of them reduce to it if A has zero diagonal. In [5, Chap. III, Sec. 3.1] the only exception is the bound by Ostrowski and Schneider. If A has at least one positive off-diagonal entry, then [7, Theorem 1]
λ≤r1 − m n−1(r1 −ρ)=:uOS. r1 − α
Here α is its smallest diagonal entry, m its smallest positive off-diagonal entry,
and
In the zero-diagonal case,
ρ= 1(r1 +···+rn). n
uOS =r1− m n−1(r1−ρ). r1
To provide another improvement of uF , let s1, . . . , sn be the row sums of A. By Brauer [1, Theorem 12],
1 1 λ≤max aii +ajj + (aii −ajj)2 +4(si −aii)(sj −ajj) 2 =:uB.
i̸=j 2
In the zero-diagonal case (as noted by Brauer and Gentry [2])
λ ≤ √r1r2 =: uBG.
The bound uB is ”of a different class” from the competing bounds because it requires much more calculation (to compute n(n−1)/2 expressions). To obtain in the nonzero-diagonal case a bound ”of the same class”, we apply to uBG the same trick as we applied to u in (3.1). We obtain
λ = λ(A) ≤ λ(MI + A0) = M + λ(A0) ≤M+uBG(A0)=M+max (si−aii)(sj −ajj)=:u′BG.
i̸=j
Wewantfirsttofavoru.Leta>1.Considerthen×nmatrixAn (n≥3) with diagonal zero, off-diagonal entries of the first and second row and column equal to a, and the remaining entries equal to 1.
Proposition 4.1. If A = An and n is large enough, then u < uBG.