ETNA Kent State University

Transcription

1 F N Electronic Transactions on Numerical Analysis Volume 8, 999, pp 5-20 Copyright 999, ISSN ON GERŠGORIN-TYPE PROBLEMS AND OVALS OF CASSINI RICHARD S VARGA AND ALAN RAUTSTENGL Abstract Recently, two Geršgorin-type matrix questions were raised These are answered here, using ovals of Cassini ey words Geršgorin circle theorem, ovals of Cassini by AMS subject classification 5A8 Introduction Given a matrix, its usual Geršgorin disks are given ) and, if 54 "! $# %'&) %+* &-, denotes the spectrum of, ie, 3 54 A@BC D B 0/ #;:<# = 4 is an eigenvalue of EGF the famous Geršgorin circle theorem [3] or Horn and Johnson [4, p 344] and [7, p 6]) is that 2) 3 54H ) M6 543N At this moment, only = P the entries 6 2 Q4 F / and the row sums F of ) determine the Geršgorin disks F 6 54 and the inclusion set, of 2), in the complex plane Then, with the notation R A@S92T$0U0UVUW = these T = pieces of information are used to define the following set of matrices: 3) XY It is evident that notation Z[ \ J]2 C ]2P ^ _ P / 23ZG4) / Q4 : R and for all Z`4 Z_ X, for each, must also satisfy the inclusion of 2), and, with the 4) acbmd Z`4 e Received March 30, 998 Accepted for publication August 3, 998 Recommended by V Mehrmann Institute for Computational Mathematics, Department of Mathematics and Computer Science,, ent, Ohio USA) varga@msckentedu) This research supported by National Science Foundation Grant DMS Department of Mathematics, Case Western Reserve University 0900 Euclid Ave, Cleveland, OH 4406 USA) axk66@pocwruedu) 5

2 p F F N 6 On a Geršgorin Problem and Ovals of Cassini we have 5) H N As we shall see, the inclusion in 5) is not always one of equality Recently, the following questions were asked of one of us: Question Can a precise description of be given Question 2 When is it the case that is a single closed disk with a nonempty interior) in the complex plane The point of this note is to answer the above questions We remark that the set of 4) =gf T is, for, in general larger than the minimal Geršgorin set cf [8]) for any matrix, simply _ h J P5D because the minimal Geršgorin set uses more specific information about the matrix in determining its minimal Geršgorin set ; A J PiC Specifically, the associated set of matrices, for the minimal Geršgorin set of a given matrix, is Xkj`l Z[ \ J] $$ ] _ ] : mg R 6) and for all ie, = in = P addition to knowing the diagonal entries F, one is also given the moduli of all! P nondiagonal entries F,on%'&),o* &$% =p, for a total of pieces of information Though it is not essential for this paper, we remark that it is evident from 3) and 6) that 7) X j`l H X so that X j`l 4qH rc for any =sf T$ where t it can be = f shown T that the equalities = rt of containment in 7) cannot in general hold for all, but in the case, there holds 8) X j`l X so that X j`l 4- uc p p N for any 2 On Question We need some additional notations and results Given the T vwt matrix 2) Zt ]MW x]0ip ] p x] p p p p its associated oval of Cassini is defined as 22) Then, for a matrix y 3ZG4 A@ C z k! ]0i {4+! ] p p 4 $#u ]MWp U] p C =;} T, we denote its ovals of Cassini by y M3 54 \@ k! P 24+! W 4 #;/ Q4^U / Q4 :~ tm R N 23) for all in J ƒ There are p y 7 Q4 ovals of Cassini associated with the matrix, and from a well-known result of A Brauer [] see also [4, p 380]), we have that if

3 Richard S Varga and Alan rautstengl 7 24) y Q4, n% &^,o* &$% y P then 25) Q4 H y Q4N But, as the ovals of Cassini in 23) use only the information given P F it is also true from 3) that / F Ic, H y Q4N 26) y :ˆ~ tm We remark from 23) that each oval of Cassini, is a bounded and closed hence, compact) y Q4 set in the complex plane, as is their finite union in 24) Next, if is a set in, then denote its closure and Š <Œ denotes its complement The boundary of is defined, as usual, by O Š y 3 54 Ž A J $ = } T THEOREM 2 Given any, then Our first result, which sharpens the inclusion in 26), shows exactly how fills out 27) ) y 54- y Wp 543 when = T$ and 28) ) y 3 54 when =;}Ž N = rt Z X Proof For, each matrix in is, from 3), necessarily of the form Z[ W / ' / p Q4 ' O p p with p N 29) arbitrary real numbers B Z Z If is any eigenvalue of, then det! BO 4^ u, so that from 29), i! B^4+3 p p! B^4) / 03 54^U / p N Hence, 20)! B U p p! B / M Q4^U / p 3 54N Bš As 20) corresponds to the case of equality in 23), we see that y ip Q4 Since this is true = for T B Z X y i any eigenvalue of any in and since, from 24), p 3 54c y 54 in this case, then ch y Wp 3 54œ y 54 Conversely, it is easily seen that each point of y ip 3 54 is, for suitable choices of real and p Z, an eigenvalue of some in 29), so ^ y ip Q4) y Q4 that, the desired result of 27)

4 @ ž 8 On a Geršgorin Problem and Ovals of Cassini =Y}_ Z h J] To establish 28), first assume that, and consider a matrix in, which has the partitioned form 2) where 22) with Ẑ W i ' Ÿ7 ' p p Z[ Z i Z ip Z p p with # Ÿ #s/ œ # # / p Q4 and Ÿ p arbitrary Ÿ s real ]2Wc A W numbers, / V Q4 9 # m #;= and with s / for all Now, for any choices of and with and p Q4 Ẑ i, the entries of the block p can be chosen so that the row =}[ / 0Z`4 / sums, and p 3ZG4 Z, in the first two rows of, equal those of Similarly, because, the row sums of the Z matrix p p of 2) can be chosen to be Z the same as those in the remaining row sums of Thus, by our construction, the matrix of 2) is /M an 54^ u X = element of We remark that this construction fails to work in the case, unless ) But from the partitioned form in 2), it is evident that 23) 3ZG4) Ẑ W M4 3Z p p 4N ŸM i However, from the parameters p Ẑ i in in 22), it can be seen from 23), that, for Ẑ each W y Wp 54, there are choices for these parameters such that is an eigenvalue of # In # / Ẑ W Ÿ # Ÿ # / M 54 other words, the eigenvalues of, on varying and with and p 54 y W, fill out p 54 Z, where we note that the remaining eigenvalues of namely, y Z those M3 54 of p p y 3 54 ) must still lie, from 26), in As this applies to all ovals of Cassini :D~ m Z, for, upon suitable permutations of the rows and columns of of 2), then ) y 54, = =s}d for all Z X For the remaining case of 28), any matrix in can be expressed as W ŸM ' / 03 54! Ÿ4 ' Z[ p p / p 3 54! 4 ' 24) / 3 54!š 4 where 25) # Ÿ #s/ œ # # / p Q4 Fi are arbitrary real numbers and # #;/ Q4 Now, choose any complex number in the oval of Cassini y i p Q4 satisfy 26) k! i U! p p #;/ 54U / p Q4N and, ie, cf 23)), let The above inequality implies that either Q! i œ#r/ M 54 or ˆ! p p ª#r/ p Q4 Assuming, without loss of generality, that ˆ! i ª#[/ such that "! i AŸ and "! i œ Ÿ7 ' is valid Ÿ, choose the parameters and, and similarly, choose the parameters and

5 X X Richard S Varga and Alan rautstengl 9 such that k! p p and! p p Ÿ<U #Ž/ Q4 U /, where, from 26), p 3 54 '9M9+ Then with the vector «Z u in, it can be verified that «, where œ A i ±ŽŸM ' p ' ±; p p 27) ' ±z / 3 54!š 4 N Then, on setting / 3 543²VT and ³ ±;, we have h Z Thus, ««, and Z is an eigenvalue of Hence, each y W in p 54 Z X is an eigenvalue of some in Consequently, y Q4 = as this :~ construction tm for can be applied to any point of any oval of Cassini with, then ) y 3 54, which X completes the proof of 28) We remark that there is a way to enlarge the set, in the spirit of what has been done in the treatment =r} T of minimal Geršgorin sets cf [8]), which gives = \T a unified =u}r inclusion result for any, which does not distinguish between the cases and, as in Theorem 2 To this end, set XY µ 28) X H X µ so that 29) Zt A J]2 ]2P ^, and this of course implies H But from the definitions in 23), we also have 220) 22) THEOREM 22 Given any and #;/ Z`4 #s/ 3 54 for all : R µ N X µ 4H y Q4N Ž A J $ =;} T with, then µ ) y Q4N Proof From Theorem = 2 AT and the inclusions = T of 29) and 220), it is only µ necessary to establish 22) in the case, ie, for, it suffices to show that q y 3 54œ y ip 3 54 The Z proof µ = T of this is similar to the proof in the first part of Theorem 2 For, each matrix in is, from 28), of the form Z[ W Ÿ7 ' ' O with p p # Ÿ #\/ 3 54k # # / p Q4, and 222) B Z Z p arbitrary real numbers If is any eigenvalue of, then det! BO 4^ u implies that 3 W! B^4+ p p! B)4) Ÿ so that 3 i! B)42 p p! B^4 _Ÿ i for all real p N Ÿ / V Q4 / But as and run, respectively, through the intervals and p Q4, and as p Z µ run through all real y numbers, i it is evident that the eigenvalues of the matrices in out cf 23)) the set p Q4^ y 3 54 = gt µ, ie, in this case ^ y ip Q4 y 54 and fill

6 20 On a Geršgorin Problem and Ovals of Cassini t 3 On Question 2 The answer to Question 2 is straight-forward, based on the =} results T of Section 2 and the following observations Given a matrix, then, with the definitions of of ) and y 54 of 23), it is easy to verify that 3) y Q4¹H : mº R :»~ [m for all with with equality ) _ W / 3 54) / 3 54) u $ holding/ W Q4- only if / Q4 f N or if and The set inclusions in 3) imply, with the definitions of 6 Q4 of 2) and y Q4 of 24), that one always has 32) y 3 54qH N ¼ J Pˆ So, to answer Question 2, it can be seen from Theorems 2 and 22 that, if $ = } T, = then ut a) for = ½T$ is never a single closed disk with a nonempty interior µ b) for W œ is a single closed disk with a nonempty interior only if p p / 0 54U / and p Q4 f ) X µ 4 is a simple closed disk with nonempty interior only = f T c) for Ÿ7 R Ÿ ~ if there are in with such y P3 54 ¾0¾c + / ¾ Q4U / i Q4 f that y ¾0 Q4 and, and all remaining ovals of Cassini lie in 4 Final Remarks It was a surprise for us to see that the determination of, in Theorem 2, was completely in terms of ovals of Cassini, a topic rarely seen in the current research literature in linear algebra In fact, on consulting many 6) books on linear algebra, we could find only three books where ovals of Cassini are even mentioned: in an exercise in Varga [7, p 22]), in a lengthier discussion in Horn and Johnson [4, pp ], and in Marcus and Minc [5, p 49] But, in none of these books was it stated that the ovals of Cassini are Q4 in general at least as good cf 32)) as the Geršgorin disks, in estimating the spectrum $ of a given matrix in This appears in Brauer s paper [] of 947 which curiously does not mention Geršgorin s earlier paper [3] of 93) What apparently was of more interest in the literature is the fact that Brauer s use of two rows at a time, to determine an inclusion set for the eigenvalues of a given matrix, cannot in general be extended in the same manner to À rows at a time, for À }g, and explicit counterexamples to this are given in [4, p 382] and [5, p 49] However, a correct generalization of Brauer s results, using graph theory, is nicely given in Brualdi [2] REFERENCES [] A BRAUER, Limits for the characteristic roots of a matrix II, Duke Math J, 4 947), pp 2 26 [2] R BRUALDI, Matrices, eigenvalues, and directed graphs, Lin Multilin Alg 982), pp [3] S GERSCHGORIN, Über die Abgrenzung der Eigenwerte einer Matrix, Izv Akad Nauk SSSR Ser Mat, 7 93), pp [4] R A HORN AND C R JOHNSON, Matrix Analysis, Cambridge University Press, Cambridge, 985 [5] M MARCUS AND M MINC, A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston, 964 [6] O TAUSSY, Bounds for characteristic roots of matrices, Duke Math J 5 948), pp [7] R S VARGA, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 962 [8] R S VARGA, Minimal Gerschgorin Sets, Pacific J Math, 5 965), pp [9] X ZHANG AND D GU, A note on Brauer s Theorem, Lin Algebra Appl ), pp 63-74