In 1912, G. Frobenius [6] provided the following upper bound for the spectral radius of nonnegative matrices.

Transcription

1 SIMPLIFICATIONS OF THE OSTROWSKI UPPER BOUNDS FOR THE SPECTRAL RADIUS OF NONNEGATIVE MATRICES CHAOQIAN LI, BAOHUA HU, AND YAOTANG LI Abstract AM Ostrowski in 1951 gave two well-known upper bounds for the spectral radius of nonnegative matrices However, the bounds are not of much practical use because they all involve a parameter α in the interval [0,1], and it is not easy to decide the optimum value of α In this paper, their equivalent forms which can be computed with the entries of matrix and without having to minimize the expressions of the bounds over all possible values of α [0,1], are given Key words Spectral radius, Ostrowski, Nonnegative matrices AMS subject classifications 15A18, 15A42, 15A48 1 Introduction AmatrixA = a ij ) R n n iscallednonnegativeifa ij 0for any i,j N = 1,2,,n The well-known Perron-Frobenius theorem [1, 6, 7, 15] states that the spectral radius ρa) of a nonnegative matrix A is the eigenvalue of A with a corresponding nonnegative eigenvector One important problem in nonnegative matrices is to estimate the spectral radius of a nonnegative matrix [2, 6, 8, 9, 10, 11, 12, 19] In 1912, G Frobenius [6] provided the following upper bound for the spectral radius of nonnegative matrices Theorem 11 [6, 18] Let A = a ij ) R n n be nonnegative Then 11) where R i A) = a ij j N ρa) max i N R ia), Since a matrix A and its transpose A T have the same eigenvalue [1], we have ρa) = ρa T ) Hence, for the nonnegative matrix A, we get from Theorem 11 that 12) ρa) max i N C ia), Received by the editors on September 3, 2013 Accepted for publication on March 10, 2014 Handling Editor: Daniel B Szyld School ofmathematics and Statistics, Yunnan University, Kunming, Yunnan, , PRChina lichaoqian@ynueducn, @qqcom, liyaotang@ynueducn) Supported by National Natural Science Foundations of China and ) and the Natural Science Foundations of Yunnan Province of China 2013FD002) 237

2 238 CQ Li, BH Hu, and YT Li where = R i A T ) Combining inequality 11) with inequality 12) gives 13) ρa) minmax i N R ia),max i N C ia) Here, we call the bound in 13) the Frobenius upper bound for ρa) To estimate ρa) more precisely, many researchers gave some upper bounds [8, 9, 10, 11, 12], which are smaller than the Frobenius upper bound Particularly, in 1951 AM Ostrowski [13] gave the following well-known upper bound; also see [11] Theorem 12 [11, 13] Let A = a ij ) R n n be nonnegative Then for any α [0,1], that is, ρa) max i N R ia)) α ) 1 α, 14) ρa) min α [0,1] max i N R ia)) α ) 1 α Moreover, from the generalized arithmetic-geometric mean inequality [4]: αa+1 α)b a α b 1 α, where a,b 0 and 0 α 1, another upper bound is obtained easily Theorem 13 [13] Let A = a ij ) R n n be nonnegative Then for any α [0,1], that is, ρa) max i N αr ia)+1 α), 15) ρa) min α [0,1] max i N αr ia)+1 α) Although Ostrowski gave many well-known results, such as the bounds in [14], we here call the bounds in 14) and 15) the Ostrowski upper bounds for ρa) Note that when α = 0, and α = 1, max R ia)) α ) 1 α = max αr ia)+1 α) = max C ia), i N i N i N max R ia)) α ) 1 α = max αr ia)+1 α) = max R ia) i N i N i N

3 Simplifications of the Ostrowski Upper Bounds for the Spectral Radius 239 Therefore, min maxr ia)) α ) 1 α min maxαr ia)+1 α) minmax i N R ia),max i N C ia), which implies that the Ostrowski upper bounds are smaller than the Frobenius upper bound However, they are not of much practical use because they all involve a parameter α and it is not easy to decide the optimum value of α Therefore, one often take some special α in practical, such as α = 1 2, 1 4, 3 4 and so on, but this leads to that the estimating is not good enough In this paper, we focus on the simplification problem of the Ostrowski upper bounds, and give their equivalent forms which do not include a minimization over all parameters α in the interval [0,1] Numerical examples are also given to verify the corresponding results 2 Main results In this section, we give equivalent forms of the Ostrowski upper bounds which do not include a minimization over all parameters α [0,1] First, we consider the Ostrowski upper bound min maxαr ia) + 1 α), and give a lemma as follows Lemma 21 Let A = a ij ) R n n be nonnegative Then there exists α [0,1] such that for all i N, 21) ρa) > αr i A)+1 α), if and only if the following two conditions hold: i) for any i N, 22) ρa) > minr i A),, ii) for any i Λ and any j, 23) ρa) R i A) > C ja) ρa) C j A) R j A), where Λ = i N : R i A) > and = j N : C j A) > R j A) Proof Let Ξ = i N : R i A) = Then N = Ξ Λ First, suppose that there exists α [0,1] such that inequality 21) holds for all i N, then for any i Λ, ρa) R i A) > α,

4 240 CQ Li, BH Hu, and YT Li and for any j, α > C ja) ρa) C j A) R j A) Therefore, for any i Λ and any j, inequality 23) holds Furthermore, from α [0,1] it is easy to get that inequality 22) holds for any i N Conversely, suppose that the conditions i) and ii) hold Obviously, inequality 21) always holds for each i Ξ Thus, it remains to prove that inequality 21) holds for all i Λ and all j 24) For each i Λ, we have R i A) > 0 and ρa) > 0 Therefore, ρa) R i A) > 0 And for each j, we have C j A) R j A) > 0, ρa) R j A) > 0, and which implies 25) C j A) R j A) > C j A) ρa), C j A) ρa) C j A) R j A) < 1 Combining inequality 23), inequality 24) with inequality 25) gives that there exists α [0,1] such that for all i Λ and all j, C j A) ρa) ρa) Ci A) 26) max 0, < α < min C j A) R j A) R i A),1 By inequality 26), we have that for any i Λ, α < ρa) C ia) R i A), that is, ρa) > αr i A)+1 α), and that for any j, C j A) ρa) C j A) R j A) < α, that is, ρa) > αr j A) +1 α)c j A) Therefore, there exists α [0,1] such that inequality 21) holds for all i N The proof is completed According to Lemma 21, we can obtain the equivalent form of the Ostrowski upper bound min maxαr ia)+1 α) Theorem 22 Let A = a ij ) R n n be nonnegative Then ρa) max i N minr ia),,

5 or That is, 27) ρa) max Simplifications of the Ostrowski Upper Bounds for the Spectral Radius 241 ρa) max i Λ, j R i A)C j A) R j A) R i A) +C j A) R j A) max minria),cia),max R ia)c ja) C ia)r ja) i N i Λ, R j ia) C ia)+c ja) R ja) Proof Suppose on the contrary that ρa) > max or equivalently, and max minr ia),,max i N i Λ, j ρa) > max i Λ, j ρa) > max i N minr ia), This implies respectively that for any i N, and that for any i Λ and any j, ρa) > R i A)C j A) R j A), R i A) +C j A) R j A) R i A)C j A) R j A) R i A) +C j A) R j A) ρa) > minr i A),, R i A)C j A) R j A) R i A) +C j A) R j A) Furthermore, by Lemma 21, there exists α [0,1] such that for all i N, that is, ρa) > αr i A)+1 α), ρa) > max i N αr ia)+1 α) which contradicts to Theorem 13 The conclusions follows Next, we prove that the bound in inequality 27) is equivalent to the Ostrowski upper bound min maxαr ia)+1 α) Theorem 23 Let A = a ij ) R n n be nonnegative Then =max min maxαr ia)+1 α) max minr R i A)C j A) R j A) ia),,max i N i Λ, R j i A) +C j A) R j A)

6 242 CQ Li, BH Hu, and YT Li Proof First we prove that if inequality 15) holds, then inequality 27) holds This implies that 28) max In fact, if ρa) > max min maxαr ia)+1 α) max minr R i A)C j A) R j A) ia),,max i N i Λ, R j i A) +C j A) R j A) max minr R i A)C j A) R j A) ia),,max, i N i Λ, R j i A) +C j A) R j A) then by the proof of Theorem 22, there exists α [0,1] such that for all i N, This gives ρa) > max i N αr ia)+1 α) ρa) > min α [0,1] max i N αr ia)+1 α) This is a contradiction to inequality 15) Hence, inequality 28) holds We now prove that if inequality 27) holds, then inequality 15) holds, which implies that max max minr R i A)C j A) R j A) 29) ia),,max i N i Λ, R j i A) +C j A) R j A) min maxαr ia)+1 α) In fact, if ρa) > min α [0,1] max i N αr ia)+1 α), that is, there exists α [0,1] such that for all i N, ρa) > max i N αr ia)+1 α) αr i A)+1 α), then by Lemma 21 and the proof of Theorem 22, we have ρa) > max max minr ia),,max i N i Λ, j R i A)C j A) R j A) R i A) +C j A) R j A)

7 Simplifications of the Ostrowski Upper Bounds for the Spectral Radius 243 This is a contradiction to inequality 27) Hence, inequality 29) holds The conclusion follows from inequality 28) and inequality 29) Remark 24 From Theorem 23, we know that Theorem 22 provides an equivalent form of the Ostrowski upper bound min maxαr ia) +1 α) Obviously, this form only relates to the entries of A and has nothing to do with α, and hence, it is much easier to estimate the spectral radius of nonnegative matrices Similarly, we can obtain easily the equivalent form of the Ostrowski upper bound min maxr ia)) α ) 1 α Lemma 25 Let A = a ij ) R n n be nonnegative Then there exists α [0,1] such that for all i N, ρa) > R i A)) α ) 1 α, if and only if the following two conditions hold: i) for any i N, ρa) > minr i A),, ii) for any i Λ, 0 and any j, R j A) 0, 210) log R i A) ρa) > log C j A) C j A) R j A) ρa) Proof Similar to the proof of Lemma 21, the conclusion follows easily Lemma 26 Let A = a ij ) R n n be nonnegative Then for any i Λ, 0 and any j, R j A) 0, inequality 210) holds if and only if ρa) > C j A) Cj A) R j A) ) logri A) ) 1 C C ia) j A) 1+log Ri A) R j A) that is, Proof Inequality 210) is equivalent to log R i A) ρa) log R i A) > log C j A) R j A) C j A) log C j A) ρa), R j A) 211) log R i A) ρa)+log C j A) R j A) ρa) > log R i A) +log C j A) C j A) R j A)

8 244 CQ Li, BH Hu, and YT Li Note that i Λ and j, then RiA) C > 1, C ja) ia) R > 1 and log C ja) ja) R i A) R > 0 ja) Therefore, inequality 211) holds if and only if or equivalently, log R i A) ρa) that is, log R i A) ρa)+ 1+log R i A) log R i A) log R i A) ρa) > 212) = log R i A) log R i A) ρa) C log ja) R i A) R ja) ) C j A) R j A) > log R i A) + log R i A) C j A), C log ja) R i A) R ja) C j A) > log R i A) log R i A) Cj A) = log R i A) = log R i A) C j A) 1+log R i A) C j A) CjA) R ja) R j A) C j A) ) log Ri A) C ) ia) C ja) R ja) Cj A) R j A) Since RiA) C ia) > 1, Then inequality 212) is equivalent to ρa) > The proof is completed C j A) Cj A) R j A) ) ) log Ri A) C ia) R j A) +log R i A) C j A) +log R i A) C j A) ) ) logri A) C Cj A) ia), R j A) ) logri A) C ) 1+log ia) Ri A) C i A) ) logri A) C ) 1+log ia) Ri A) C i A) ) 1 C j A) R j A) ) 1 C j A) R j A) Similar to the proof of Theorems 22 and 23, we can obtain easily the following theorems from Lemmas 25 and 26 or Theorem 27 Let A = a ij ) R n n be nonnegative Then ρa) ρ 1 = max i Λ, 0, j,r j A) 0 ρa) max i N minr ia),, C j A) Cj A) R j A) ) logri A) C ) 1+log ia) Ri A) C i A) ) 1 C j A) R j A)

9 Simplifications of the Ostrowski Upper Bounds for the Spectral Radius 245 That is, 213) ρa) max max minr ia),,ρ 1 i N Theorem 28 Let A = a ij ) R n n be nonnegative Then min maxr ia)) α ) 1 α = max max minr ia),,ρ 1, i N where ρ 1 is defined as in Theorem 27 Remark 29 Theorem 27 provides an equivalent form of the Ostrowski upper bound min maxr ia)) α ) 1 α However, it is determined with more diffi- cultly than that in Theorem 22 because of computing log R i A) or log R i A) C ja) R ja) difficultly So in general we estimate the spectral radius of nonnegative matrices by Theorem 22 3 Numerical comparisons Besides the Frobenius bound and the Ostrowski bounds, there are another results on upper bounds for the spectral radius of nonnegative matrices [2, 3, 5, 8, 11, 12, 16, 17] We now list some of the well-known bounds, and compare with the bound in Theorem 22 In 1964, Derzko and Pfeffer [5] provided an upper bound for the spectral radius of complex matrices, which is also used to estimate the spectral radius of a nonnegative matrix A = a ij ) R n n 31) ρa) ǫa) 2 )1 ) 2 2 max R i A), i N where ǫa) = 32) a 2 ij i,j N ρa) 1 2 max j i )1 ) )1 2 2, Ri A) = a 2 ij a 2 ii j N In 1974, Brauer and Gentry [2] derived the following bound: a ii +a jj + a ii a jj ) 2 +4R ia)r ja) where R i A) = R ia) a ii and = R i A T ) ) 1) 2 In 1994, Rojo and Jiménez [16] obtained the following decreasing sequence of upper bounds:, 33) ρa) v k A) v 2 A) v 1 A),

10 246 CQ Li, BH Hu, and YT Li where v p A) = tracea) n γ p A) = and MA) = A+AT 2 +γ p A), n 1) 2p 1 n 1) 2p 1 +1 trace MA) tracea) ) ) 1 2p 2p I n In 1998, Taşçi and Kirkland [17] obtained another sequence of upper bounds based on an arithmetic symmetrization of powers of A: 34) where σ k = )) 2 k ρ MA 2k ) ρa) σ k σ 2 σ 1, In 2006, Kolotilina [11] provided the following bound: 35) Ri ρa) max A) α R j A) 1 α) β Ci A) α C j A) 1 α) 1 β, i,j:a ij 0 where 0 α,β 1 In 2012, Melman [12] derived an upper bound for the spectral radius, that is, ρa) 1 2 maxmin a ii +a jj +R ija)+a ii a jj +R ija)) 2 +4a ij R ja)) 1 2, i N j i 36) where R ij A) = R i A) a ij = R i A) a ii a ij Very recently, Butler and Siegel [3] obtained the following upper bound for the spectral radius of nonnegative matrices with nonzero row sums 37) ρa) max i,j N Ri A K+P )R j A K+Q ) R i A K )R j A K ) ) 1 P+Q : a P) ij > 0, where a P) ij is the i,j) entry of A P, P > 0, Q 0 and K 0 We now give some numerical examples to compare the Ostrowski upper bound min maxαr ia)+1 α), or equivalently, the bound in Theorem 22 with the listed bounds Example 31 Let A = a ij ) R n n with n 2, where a ij = i+j n i+j

11 Simplifications of the Ostrowski Upper Bounds for the Spectral Radius 247 Obviously, A is nonnegative We compute by Matlab 70 the Frobenius upper bound, the bounds in 31), 32), 33), 34), 35), 36), 37), and the Ostrowski upper bound min maxαr ia)+1 α), ie, the bound in 27), which are showed in Table 1 n the Frobenius upper bound the bound in 31) the bound in 32) the bound in 33) the bound in 34) the bound in 35) the bound in 36) the bound in 37) the Ostrowski upper bound ρa) Table 1 Comparison of the bounds for the nonnegative matrix in Example 31 Example 32 Let A = The bounds are showed in Table 2 the Frobenius upper bound 5 the bound in 31) the bound in 32) the bound in 33) the bound in 34) the bound in 35) the bound in 36) the bound in 37) the Ostrowski upper bound ρa) Table 2 Comparison of the bounds for the nonnegative matrix in Example 32 Example 33 Let A =

12 248 CQ Li, BH Hu, and YT Li By computation, the Ostrowski upper bound is 5, and the bounds in 34) and 35) are and 54772, respectively In fact, ρa) = 5 Remark 34 I) In Examples 31, 32 and 33, the bounds in 33), 34) are givenbyv 1 A) andσ 1 A), respectively And the bound in 35) is givenforα = β = 1 For the bound in 37), we compute its value for K = P = Q = 1 in Example 31, and for P = 2, K = Q = 1 in Example 32 II) From Examples 31 and 32, we have that the Ostrowski upper bound is smaller than the Frobenius upper bound, smaller than the bounds in 31), 32), 33), 36), and 37) in some cases III) Example 33 shows that the Ostrowski upper bound is smaller than the bounds in 34) and 35), and that the Ostrowski upper bound is sharp Acknowledgment The authors are grateful to the referees for their useful and constructive suggestions REFERENCES [1] A Berman and RJ Plemmons Nonnegative Matrices in the Mathematical Sciences Academic Press, New York, 1979 [2] A Brauer and IC Gentry Bounds for the greatest characteristic root of an irreducible nonnegative matrix Linear Algebra Appl, 8: , 1974 [3] BK Butler and PH Siegel Sharp bounds on the spectral radius of nonnegative matrices and digraphs, Linear Algebra Appl, 439: , 2013 [4] LC Cvetković, V Kostić, R Bru, and F Pedroche A simple generalization of Geršgorin s theorem Adv Comput Math, 35: , 2011 [5] NA Derzko and AM Pfeffer Bounds for the spectral radius of a matrix Math Comp, 19:62 67, 1965 [6] G Frobenius über Matrizen aus nicht negativen Elementen Sitz ber Preuss AkadWiss Berl, , 1912 [7] RA Horn and CR Johnson Matrix Analysis Cambridge University Press, Cambridge, 1985 [8] LY Kolotilina Bounds and inequalities for the Perron root of a nonnegative matrix Zap Nauchn Semin POMI), 284:77 122, 2002 [9] LY Kolotilina Bounds and inequalities for the Perron root of a nonnegative matrix II Circuit bounds and inequalities Zap Nauchn Semin POMI), 296:60 88, 2003 [10] LY Kolotilina Bounds and inequalities for the Perron root of a nonnegative matrix III Bounds dependent on simple paths and circuits Zap Nauchn Semin POMI), 323:69 93, 2005 [11] LY Kolotilina Bounds for the Perron root, singularity/nonsingularity conditions, and eigenvalue inclusion sets Numer Algor, 42: , 2006 [12] A Melman Upper and lower bounds for the Perron root of a nonnegative matrix Linear Multilinear Algebra, 61: , 2013 [13] AM Ostrowski Über das nichtverschwinden einer klasse von determinanten und die lokalisierung der characterisrichen wurzeln von matrizen Compos Math, 9: , 1951

13 Simplifications of the Ostrowski Upper Bounds for the Spectral Radius 249 [14] AM Ostrowski Bounds for the greatest latent roots of a positive matrix J London Math Soc, 27: , 1952 [15] O Perron Zur theorie der matrizen Math Ann, 64: , 1907 [16] O Rojo and R Jiménez A decreasing sequence of upper bounds for the perron root Comput Math Appl, 28:9 15, 1994 [17] D Taşçi and S Kirkland A sequence of upper bounds for the perron root of a nonnegative matrix Linear Algebra Appl, 273:23 28, 1998 [18] RS Varga Geršgorin and His Circles Springer Series in Computational Mathematics, Springer, Berlin, 2004 [19] RJ Wood and MJ O Neill Finding the spectral radius of a large sparse non-negative matrix ANZIAM J, 48: , 2007