Bounds for the Extreme Eigenvalues Using the Trace and Determinant

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1 ISSN , Eglad, UK Joural of Iformatio ad Computig Sciece Vol 4, No, 9, pp Bouds for the Etreme Eigevalues Usig the Trace ad Determiat Qi Zhog, +, Tig-Zhu Huag School of pplied Mathematics, Uiversity of Electroic Sciece ad Techology of Chia, Chegdu, Sichua, 654, P R Chia (Received July 4, 7, accepted October, 8 bstract Bouds for the etreme eigevalues ivolvig trace ad determiat are preseted lso, we give the upper bouds for the Perro root of a oegative symmetric matri uder certai coditios Keywords: Eigevalue, Trace, Determiat, Noegative symmetric matri, Perro root Itroductio Let be a comple matri with sigular values σ σ σ The properties are well ow, where ( σ + σ + + σ, F σ σ σ det ad det deote the Frobeius orm of ad the determiat of, F : respectively I [], Rojo presets mootoic sequeces of bouds for σ ( where α ad β are the positive roots of the equatio d if, ( { } ( { } ( α σ β, + F det is a icreasig sequece of lower bouds for σ is a decreasig sequece of upper bouds for σ where { } ( det + F ( F Similarly, Let be a comple matri with real ad positive eigevalues The properties > tr, det ; if (, F is a sequece defied by motivate oe to estimate the bouds for eigevalues of where tr deotes the trace of I [, p - are preseted usig the same techique i [] as ], mootoic sequeces of bouds for + address: bbs3_zq@6com Published by World cademic Press, World cademic Uio

2 5 Qi Zhog, et al: Bouds for the Etreme Eigevalues Usig the Trace ad Determiat follows: where α ad β are the positive roots of the equatio d if, { } α β, tr+ det ( is a icreasig sequece of lower bouds for if tr where { } is a sequece defied by decreasig sequece of upper bouds for I this paper, let C ( ( det + tr tr, { } is a ( be the matrices with real ad positive eigevalues > We use this symbol throughout We give bouds for the etreme eigevalues usig trace ad determiat The paper is orgaized as follows I Sectio, two short proofs of lower boud for the smallest eigevalue of are give I Sectio 3, we obtai aother lower boud for the smallest eigevalue, which is sharper tha the result i Sectio Eamples are preseted i Sectio 4 which give comparisos with results i the related literatures Fially, i Sectio 5, we cosider the upper bouds for a oegative symmetric matri uder certai coditios Two simple lower bouds for the smallest eigevalue I [3], Yu ad Gu give lower bouds for the smallest sigular value usig arithmetic-geometric-mea iequality Here we utilize the similar techique ad give the followig theorems First, we prove the followig weaer versio of lower boud for Theorem Let C ( sequece ad tr The be a matri with real ad positive eigevalues ordered i decreasig > det (3 Proof : Cosiderig the fact that the geometric mea of positive umber doer ot eceed their arithmetic mea ad the idetity We have tr + + +, < we obtai the result Multiply this iequality by ad solve for Theorem Let C ( be a matri with real ad positive eigevalues ordered i decreasig sequece ad tr The > det + det (4 Proof : The arithmetic-geometric-mea iequality ad the idetity give tr JIC for cotributio: editor@jicorgu

3 Joural of Iformatio ad Computig Sciece, 4 (9, pp Multiplyig both sides of this iequality by Hece, Because > + + +, we have det, we obtai + ( det > ( det (5 Solvig this iequality for Sice, it follows that ( det ( det > + gives > det + det > det + det i (4 is a improvemet of the Remar From the above proof we ow that the lower boud for result i (3 3 Further lower boud for the smallest eigevalue First, we give the followig corollary which follows from ( immediately Corollary 3 Let B C ( be a matri with real ad positive eigevalues ordered i decreasig sequece ad tr B Let { } be a sequece defied by The, if, { } if Let, { } Hece, det B + is a icreasig sequece of lower bouds for ( B is a decreasig sequece of upper boud for (6 ; B B C ( be a matri with trb Let The, from (6, det B det B< B This is the result of Theorem JIC for subscriptio: publishig@wuorgu

4 5 Qi Zhog, et al: Bouds for the Etreme Eigevalues Usig the Trace ad Determiat gai from (6, we have We ow that > B det B ( det ( det B ( det B B + B ( det B det B ( det B is a lower boud for The, det B + det B ( > det B + det B ( det B Thus, we have preseted the followig lemma Lemma 3 Let B C ( sequece ad tr B The where, det B be a matri with real ad positive eigevalues ordered i decreasig det det B, (7 ( B > B + θ( B θ ( B (8 detb With the Lemma 3 we may ow establish the followig theorem Theorem 33 Let C ( be a matri with real ad positive eigevalues ordered i decreasig sequece The > det + θ det, (9 tr tr where θ ( tr det Proof : pplyig Lemma 3 to matri B ( / tr This theorem cotais Lemma 3 as a special case 4 Compariso with related results Estimatio of etreme eigevalues is importat i theory ad practice Bouds for eigevlaues have bee C be a matri with real ad positive eigevalues obtaied by may authors Let ad let l > Bouds for l ad + + l, ivolvig ltr,,,, ad det oly, are preseted as follows Theorem 4 [4, Theorem ] Let l The JIC for cotributio: editor@jicorgu

5 Joural of Iformatio ad Computig Sciece, 4 (9, pp tr det ( + ( l l l + l l tr l det l l tr ( Theorem 4 [4, Theorem 3] Let l The + l + tr tr + det ( l ( l l+ + l+ tr ( l + ( det l + Let us recall aother possible eigevalue bouds usig ltr,,,, ad tr oly Wolowicz ad Stya derive the followig theorem It is worth otig that the eigevalues are real; their positivity is ot eeded Theorem 43 [5, Theorem ] Let l The ( tr tr tr tr l tr l + tr l + l (3 s special cases, bouds for idividual eigevalues, especially for the smallest eigevalue, ca be obtaied by the above theorems We coclude the sectio with two eamples to compare the lower bouds for the smallest eigevalue ad give some remars Eample Let 3 This matri was used i [4] to compare the lower bouds for 3 (, by (3 ad 3 45, oly prior to the result ( To further illustrate our bouds we cosider the followig eample Eample Let For this matri with ad they were 3 98 by 3 74 by ( I this ote, the boud (9 gives B 5 5 3, we have the compariso results of lower bouds for B i Table trb However, the eact smallest eigevalue is ( B JIC for subscriptio: publishig@wuorgu

6 54 Qi Zhog, et al: Bouds for the Etreme Eigevalues Usig the Trace ad Determiat Table : Lower bouds for ( B 3 ( B > 3 ( B (3 (4 ( ( ( l ( ( l (3 ( l Remar From the Eample we see that the lower boud for ( B [4] fails to provide otrivial lower boud for ( B i Eample is accurate by (7 Theorem 4 i Remar 3 The boud (7 is always at least as large as the boud (4 Sice for ay matri B with trb Hece, ( B i i trb det B i ( B i det B < This implies θ ( B > ad thus the lower boud for ( B i (4 has bee improved by Lemma 3 5 Upper bouds for the Perro root of oegative symmetric matrices Noegative matri has applicatios i may areas [6] Let be a matri with all etries oegative By the Perro-Frobeius theorem, has a characteristic root equal to its spectral radius, which is called the ρ Bouds for the Perro root have bee surveyed by may Perro root of ad is usually deoted by authors I this sectio, we give upper bouds for the Perro root of a oegative symmetric matri satisfied some certai coditios We have the followig result Theorem 5 Let be a oegative symmetric matri which is strictly diagoally domiat Let { } be is a decreasig sequece of upper bouds for ρ if the sequece defied by ( The { } tr The result is obvious ad hece its proof is omitted Eample 3 Let Clearly, is positive defiite ad ρ For this matri equatio ( is The applicatio of Theorem 5 gives the followig upper bouds for ρ i Table, JIC for cotributio: editor@jicorgu

7 Joural of Iformatio ad Computig Sciece, 4 (9, pp Table : Upper bouds for ρ Remar 4 I Theorem 5, the strictly diagoal domiace is sufficiet to guaratee oegative symmetric matri is positive defiite See the followig matri 4 5, which is ot strictly diagoally domiat i the last row ad we ca also apply Theorem 5 to estimate the upper bouds for the Perro root of ctually the above matri is positive defiite with eigevalues 97, 3474, 3 ( 536, Refereces [] O Rojo Further Bouds for the Smallest Sigular Value ad the Spectral Coditio Number J Computers Math pplic 999, 38: 5-8 [] Limig Liu Estimatio for the Sigular values ad Eigevalues of matrices, Dissertatio of Master, Uiv ESTC, 6 [3] Y -S Yu ad D-H Gu ote o a lower boud for the smallest sigular value J Liear lgebra ppl 997, 53: 5-38 [4] Jorma Kaarlo Meriosi ad ri Virtae Bouds for the Eegevalues Usig the Trace ad Determiat J Liear lgebra ppl 997, 64: -8 [5] H Wolowicz ad G H P Stya Bouds for eigevalues usig traces J Liear lgebra ppl 98, 9: [6] Berma, RJ Plemmos Noegative Matrices i the Mathematical Scieces Philadelphia: SIM Press, P, 994 JIC for subscriptio: publishig@wuorgu