Miasnikov, A.

Roček, J.

Zezula, R.

English

Institute of Mathematics AS CR, v. v. i.

Acta Universitatis Carolinae. Mathematica et PhysicaR. Zezula; J. Roček; A. MiasnikovOn relations between right and left eigenvectors of nonselfadjoint matrix pencilsActa Universitatis Carolinae. Mathematica et Physica, Vol. 18 (1977), No. 1, 37--45Persistent URL: http://dml.cz/dmlcz/142394Terms of use:© Univerzita Karlova v Praze, 1977Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.cz1977 ACTA UNIVERSITATIS CAROLINAE - MATHEMATICA ET PHYSICA VOL. 18, NO. 1, Pag. 37-45 On Relations Between Right and Left Eigenvectors of Nonselfadjoint Matr ix Pencils R. ZEZULA*) Mathematical Institute of the Charles University, Prague J. ROCEK and A. MIASNIKOV**) Institute of Nuclear Research, ftez Received 30 October 1975 The authors consider (from the reactor physicist viewpoint) some useful relations between right and left eigenvectors of nonselfadjoint generalized matrix eigenvalue problems via their equivalence to properly constructed ordinary eigenvalue problems., and their use for the de­termination of the perturbed dominant eigenvalue. ABTOpbl 3aHHMaK)TCH3 C TO*IKH 3peHHH (})H3HKH peaKTOpOB, HCKOTOpblMH âCTO HCTIOJIb3ye-MblMH COOTHOmeHHHMH MCH<Ay npaBblMH H JieBblMH C06CTBCHHbIMH BeKTOpaMH B HecaMOCO-npH>KeHHbix o6o6ureHHbix MaTpHHHbix sa^anax Ha co6cTBeHHbie 3HaneHHH c noMomio HX 3KBHBa-jieHTHOCTH c HaAJiewamHM oSpasoM nocTpoeHbiMH o6biKHOBeHHbiMH 3â aHaMH Ha coScTBeHUbie 3HaneHHH H HX Hcnojib30BaHHeM ÍIJIH BbiHHCJieHHH B03Mym;eHHoro jjoMHHaHTHoro co6cTBeHHoro 3HaneHHH. Autoři studují, s hlediska reaktorové fyziky, některé užitečné vztahy mezi pravými a levými vlastními vektory zobecněného nesamoadjungovaného maticového problému vlastních hodnot pomocí vhodně sestrojeného ekvivalentního obyčejného problému vlastních hodnot, a jejich užití při výpočtu porušeného dominantního vlastního čísla. Introduction In this note, there are summarized and discussed in detail the problems con­cerning the equivalence question of the generalized nonselfadjoint eigenvalue problem Px -= XQx for the right eigenvector and also of the corresponding eigen­value problem y*P = Xy*Q for the left eigenvector in the EN space. There is accepted a reactor physicist's notation and point of view, i.e., the matrix Q is assumed to be nonsingular and the matrix M = Q~1P e S#(EN) of Perron -Frobenius type i.e., with A as positive dominant eigenvalue and x, z* resp. as the corresponding positive neutron flux vector or neutron importance vector resp. [1]). *) Mathematical Institute of the Charles University, Sokolovská 83, 186 00 Praha 8 **) Institute of Nuclear Research, 250 68 Řež u Prahy 37 There are also discussed the relations between these eigenvectors and their use for the determination of the perturbed dominant eigenvalue., needed in some reactor criticality calculations [2]. I. Equivalence problems for matrix pencils Let EN be a real jY-dimensional Euclidean space with <x, j!>, x, y e EN as scalar product. Let <V?(EN) denote the algebra of linear endomorphisms on EN with the involution A*+A*, A, A* es/(EN), A* transposed. Let A be the neutron absorbtion matrix, P the neutron production matrix (both in the sense of the heterogeneous method of reactor physics) and let I be the identity matrix in EN. Further, let us suppose that the matrix (1.1) Q = I + Aes/(EN) is nonsingular, i.e. (Via) Qr1es/(EN) It is well known [2] that the matrices P and A> and therefore also Q = I + A are nonselfadjoint, i.e. (1.2) P*P\ Q^ Q; Q1 # (Q"1)* = (p.*)-1 so that we have in general (1.3) M = QrxP -£ AT, N = PQr1 i=. N* and the matrix N is similar to the matrix M (1.3a) QMQ1 = Q(Qr1P)Qr1 - PQ"1 = N Now, let us make following Assumption I for have a reactor criticality mathe-matical model in EN (N denoting the number of fuel, safety and control rods in the reactor core): (I) The matrix M = Q_1P is of Perron - Frobenius type, with X as dominant simple positive eigenvalue. Clearly, X is also a dominant simple positive eigenvalue of N, N being similar to M in virtue of (1.3a). We shall consider both the following eigenvalue problems for the nonself-ad joint matrix pencil P — XQ e S#(EN) and its adjoint pencil P* — XQ* e S&(EN) : (1.4) Px = XQx (1.5) P'y = lQ'y Clearly, in virtue of the assumption (1.1a)., the generalized eigenvalue problem (1.4) is equivalent to the following non-selfadjoint matrix eigenvalue problem for the matrix M = QrxP (1.6) Mx^Xx 38 and the corresponding adjoint matrix eigenvalue problem (1.5) is equivalent to the following nonselfadjoint matrix eigenvalue problem for the matrix N' = [PQ-iy = (Q-iyp- = (Q'yip-(17) N'y = ly Because the matrices M, N e S/(EN) are in general nonselfadjoint, i.e., M jz M* = P*(Q~1)* = P\Q*)-\ N** = N = PQr1 ^ N\ we have, for using the perturbation theory, to solve both the adjoint eigenvalue problems for (1.6) (1.6a) M*z = Xz and for (1.7) (1.7a) Nw = Xw Using the well known relations [3] (1.8) (Q-iY = (Q')- i , QQ-i = Q-iQ = I, Q\Q')-^ = {Q'Y^Q' = I we obtain from (1.6a) and (1.5) the following useful relation between the left eigen-vectors z,y resp. of M, N E^(EN) resp.: (1.9) AQ'KQT 1 *] = P'KQ'yM => OC0 = Q'y Similarly, from (1.7a) and (1.4) using (1.8), there follows the following useful relation between the right eigenvectors x, w resp. of Af, N e S/(EN) resp.: (1.10) XQlQ^w] = P[Q-%] => pro = Qx Remark 1.1: a, /3 ^ 0 resp. in (1.9), (V10) resp. are suitable real numbers (norming constants). 2.Biorthogonality relations, perturbed dominant eigenvalue It is well known [3], that, if assuming both the matrices Af, N* e S#(EN)> with eigenvalues jui e a(M) <-- C, v* e a(N*) <-= C resp., i = 1, 2, . . . , N have the complete (i.e. forming a base in EN) system of normalized eigenvectors Xi = x(fit) 7^ 0, yi = y(v*) 9-= 0, \\xi\\ = \\yt\\ = 1, then the vectors zu Wi of the corresponding biorthogonal (with respect to the usual scalar product <(x,y} in EN) bases / 0, i 4= j (2.1) Zi = z(ii*) e EN, (ZU *j} = <% = \ \ 1,1 =j (v*, [A* resp. denotes the complex conjugate to Pi e a(M) <-= C, vte a(N) c C resp., i,j = 1, 2, . . . , AT) (2.2) «a< = w(v<) 6 IJN, <^ , w;> = dij = <̂  \ 1 , 1 = 7 39 are just the normalized eigenvectors of the adjoint matrices M*, N e,tf(EN) resp., i.e., they fulfil the relations (1.6a), (1.7a) resp. Using this fact, we obtain by help of the usual perturbation theory [4] the following explicite expressions for the perturbations A MX, A NX of the simple positive dominant eigenvalue X of AT, jV, dues to the perturbation zJAf, AN e s/(EN) resp. of Af, N e s/(EN) resp.: (2.3) AMX = <AMx>*> ( = <AMx, zy if ||x|| = \\z\\ = 1) <*5 z; (2.4) ANX = <ANw>y> ( = <ZJA7^3;> if ||W|| = M = 1) <&>, jy > Theorem 2.1: Let P, P0 e *o/(EN). Let Q = (I + A) e <tf(EN) and Q0 = (I + AQ) e s/(EN) be nonsingular. Let X > 0 be a simple dominant eigenvalue of MQ = Q^-Po* jY0 = = PoQj1 e <&(EN)3 with the corresponding right eigenvectors x, MQX = Xx, w, NQW = Xw and left eigenvectors zy M^z = Xz, y, N$y = Xy resp. Let M = Q--P, jV = PQ1. Then for the first order perturbations AMX,ANX resp. of A, dues to the perturbations AM = M — jM0, AN = N—NQ resp. of Af, jY resp. the following relation is valid: (2.5) AKX = AMX + «M*-Q-11W>*>*>. {xyzy where AMX,ANX resp. are given by (2.3), (2.4) resp. Proof: Clearly, we have, in virtue of the relations (1.9), (1.10) (2.6) j8 0 , j ! > = <Qx,j!> = <*, Q*yy = oc<x, zy so that the denominators in both the expressions (2.3), (2.4) resp. for AMX, ANX resp. are proportional. For the numerator in (2.4), we have obviously (2.7) -£ <ANw, yy = <(jV - jY0) w, (Q"1)* s> = = <Q-KPQ'1-PoQ^)Qx,zy = <(M-Q-iN0Q)x,zy = (AMx, zy + (MQ-Q^NQQ) x, zy Q.E.D. Corollary 2.1: If Q = l2o> i-e., -4 -= ylo> (no change in neutron absorbtion during the reactor perturbation), then we have A NX = A MX . Proof: For Q = Qo we have Q-WoQ -= QfrKPoQtf) Qo = QQXPQ = AT0, Q.E.D. Remark 2.1: Physical interpretation of the relations between the eigenvector pairs x, z and w, y is the following: Because the matrix Af = Q~XP is assumed 40 to be of Frobenius type with X e a(M) as dominant simple positive eigenvalue, M* is also of Frobenius type and both the corresponding left and right eigen-vectors x = x(X) and z = z(X) of M are positive. Therefore, x has the physical meaning of neutron flux vector (giving the neutron fluxes on the rods in the reactor core) and (2.3) implies, that the positive left eigenvector z of M is the weighting vector for the neutron flux vector x, giving the importance of the individual components of x with respect to the perturbation A MX of X. Therefore, x is called the neutron importance vector. Clearly, the physical interpretation of the equivalence between the equations (1.4) and (1.6) is the following: The neutron absorbtion A is so little, i.e. \\A\\ <^ 1, that P' = Q~lP & (I—A) P can again be interpreted as production matrix, to which there corresponds in (1.6) a zero absorbtion matrix A' = 0. The matrix jY, being similar to M, need not be of Perron - Frobenius type, so that the right and left eigenvectors w = w(X) and y = y(X), corresponding to the dominant simple positive eigenvalue X e a(N) of N have in general no direct physical interpretation in reactor physics. But if jY is also of the Perron - Frobenius type, the same physical interpretation of w,y can be applied. If Q is a matrix with positive entries, then Q* has the same property, so that both Q and Q* in this case leave invariant the cone of positive vectors in HN. But, Q being nonsingular, (Q*)_1 = (Q -1)* needs not leave the cone of the positive vectors in EN invariant. Therefore, if M = Q_1P is of Perron - Fro-benius type, and both the matrices Q, (Q~1)*P(Q~1)* leave invariant the cone of positive vectors in EN, then the right and left eigenvectors (with a > 0, ($ > 0) w(X) = p~1Qx(X) and y(X) = ~ ( Q T ^ X Q T 1 ^ ) of jY, corresponding to its X dominant simple positive eigenvalue X e a(N), are both positive and therefore can again be interpreted as the neutron flux w and neutron importance y in a reactor with production matrix P" = PQr1 ^ P(I—A) and absorbtion matrix A" = 0. Now, we shall generalize the biorthogonality property [3] to the nonselfadjoint matrix pencil P — XQe S#(EN) : Theorem 2.2 Let P —XQe S4(EN) be a given matrix pencil, with Q e S4(EN) nonsingular, Q-1 e S&(EN). Let (x, y) be the (complex type) scalar product of x, y e EN <=• UN . Let the set {xk}, (P — XkQ) xk = 0, xk 4= 0, k = 1, 2, . . . , jY of right eigenvectors xk = x(Xk) of the matrix pencil P —- XQ, corresponding to its eigenvalues Xk, span EN • Let {zi}, i = 1, 2, . . . , jY be the basis of EN biorthonormal to {xk} with respect to the scalar product <z, x}, z, x e EN ^ UN > [3], i.e., , 0 if i =t= k (2.8) <zu xky = dik=<f \ 1 if i = k 41 Then (2.9) yi=yW) = (Q-rzi are left eigenvectors of the matrix pencil A — AI, corresponding to its eigenvalues A*, i.e., we have for V i' = 1, 2, . . . , jY (2.9a) (P* - A * Q * ) y > - 0 , yi ^ 0 Proof: We have clearly, using the definition (2.9) of yu and the relation (Q-1)* = (QT1 (2,9b) Zi = Q*yi Let us compute, using the definition (2.8) of zi and the relation (2.9b) between Zi and yu the scalar product <P* v*, **> = <y*> Pxk) = <-y<3 foQxky = A* <Q*3/«, x&> = = A*. <#$, x#> = A J O M = A*<5a = = *7 <Q*yu **> = <m*yi> **> Thus, we have, for V Xk e {xk}, (2.10) <(P* -tiQ*)yi,xk> = Q anf therefore, in virtue of the assumption span {xk} = I}N , we have (2.10a) P*yi = m*yi where (2.10b) y* = (Q*)"1^ = (Q"1)*^ * 0 for V i = 1, 2, . . . , jY because we have, by (2.8), (,10c) <si, **> = a** => s« ?-= 0 for V i = 1, 2, . . . , N Q.E.D. Corollary 2.2 Let the adjoint matrix pencil P* — A*Q* ES^(EN) with nonsingular Q* E^(NN, (Q*)"1 = (Q -1)* EJ^(EN) have a complete system of right eigen-vectors 3!* = y(A*) ^ 0, (P* — A*Q*)j!A; = 0, so that we have (2.11) span {yk}Li = EN Let {wi}, i = 1, 2. . . . , be the basis of EN, biorthonormal to {yk} with respect to the scalar product <«;, j;>, w, y e EN, i.e. x 0 if i * k (2.12) <«*<->>.*> = <5** = : 1 if i = k Then (2.13) x* = *(A«) = Qriwi are the left eigenvectors of the adjoint matrix pencil P* —X*Q*, corresponding to its eigenvalues* A*, i.e., we have for V i• = 1> 2, . . . , jY (2.13a) (P - XiQ) xt = 0 , *i 4= 0 42 Proof. Because the adjoint matrix pencil P* — A*Q* fulfills in this case all assumptions of Theorem 2.2., we obtain both the assertions (2.13), (2.13a) by applying this theorem and the involutory properties of the involution operation * in s/(EN). Q.E.D. There is another possible way of reducing the generalized eigenvalue problems (1.4), (1.5) resp. to equivalent ordinary eigenvalue problems: via inverse eigenvalue problems instead inverting Q = I + A. We can consider X, X* resp. in (1.4), (1.5) as a physical parameter (the so called criticality parameter) instead as a mathe-matical eigenvalue of the matrix pencils P = AQ, P* —-A*2* resp. and introduce formally a new eigenvalue // = //(A), v = V(Q) resp., setting (2.14) A # 0, A* = Q * 0 and (2.14a) R(X) x = jux , where R(X) = ( y P — A\ (neutron balance condition) (2.14b) fji = ju (A) = 1 (criticality condition) instead (1.4), as an inverse eigenvalue problem for the corresponding right eigen-vector x > 0 of P(A), (2.15) S(Q)y = vy, where S(Q) = I — P* — A* J (neutron balance condition) \ Q i (2.15a) v — V(Q) = 1 (criticality condition) instead (1.5), as an inverse eigenvalue problem for the corresponding right eigen-vector y}0 of S(Q). Let us assume that, for A e <Amin, Amax>, Q e <omin5 omax>5 the operators R(X) and S(Q) are of Perron-Frobenius type, and both the dominant eigenvalue func-tions /u(X), v (Q) are monotone, with 1 — ei < ft(X)9 V(Q) < 1 + e^ £i> £2 > 0. We denote Acrit.> Ocrit. resp. the critical values of the criticality parameters A, O, for which the criticality conditions (2.14b), (2.15a) resp. are just fulfilled for the dominant eigenvalues, i.e., (2.16) /ia = MAcrit.) = 1 > (2.17) Va = KDcrit.) = 1 where jUd, Vd resp. denote the simple positive dominant eigenvalue of Iv(ACrit)> S(QCTU.) resp. By xd = *(ACrit.)> yd = y(^crit.)5 \\xa\\ = WyaW = 1 let us denote the corresponding normalized positive eigenvectors. Clearly, we have, denoting f* the complex conjugate of £ e C , and by R* the adjoint operator to R e J/(EN) (2.18) [R(X)r =-^P* -A* = S(Ji*) In general, we shall have obviously (2.19) A*rit. = [A?r i t.']• * Q 43 so that, despite the validity of the relation (2.18), the positive normalized eigen-vectors x, y shall not be, in general, elements of biorthonormal bases of right and left ЄІgЄПVЄCtOГS Of jR(Яcrit.) in EN . But if, in a special case, the relation (2.20) Я*rit. = [Acrit.]* = ^crit. shall be valid, and if the matrix K(ЯCГit.) = -= P — A shall have a complete system of normalized right eigenvectors {xk Ф 0}£LX, Я(A)crit. xk = fikXki span {xk} = EN , then the vectors yi, ì = 1, 2, . . . , jY, <-yг-, Xk) = Ь%k of the corresponding bi-orthonormal base in EN shall be normalized left eigenvectors of -R(ЯCГit.), i-e., we shall have [iv(ЯCГit.)]* УІ = lAyь У% Ф 0, i = 1, 2, ... , jY, џ*d = 1. 3. An illustrating numerical example For numerical illustration of the above mentioned results in the Euclidean space Lз y let us take [5] / 2,2 1 1,8 \ (3.1) P= 1,1 2 0,9 \ 1,1 1 1,8 / as the neutron production matrix, and (3.1) as the neutron absorbtion matrix, so that the matrix Q = I + A is nonsingular (det( Q) = 3 ф 0) and has positive entries. The matrices P and A were chosen so, that the matrix M = Q - 1 P is symmetric with nonnegative entries and has three simple real eigenvalues 0,9, 1 and 1,1, with Xd = 1,1 as simple positive dominant eigenvalue, and Xd = Zd = -= (1, 0, 0) as the corresponding nonnegative right and left eigenvectors, so that M is of Perron-Frobenius type. The matrix N = PQ'1 is a nonsymmetric one. Because AГ is similar to .M, its eigenvalues shall be the same as for Лí, i.e., also real and simple, again with Xd = 1,1 as simple positive dominant eigenvalue, to which there correspond in this case the right eigenvector Wd = (b-y>- (r-) and the left eigencvetor yd -= _ / l M — lӯӯj 0, — ïӯӯl 5 so that Л7 is not of Peггon-Frobenius type. kk We have clearly (3.3) /2,2 1,1 1,1\ 12 1 2\ P' = 1 2 1 , Q = 1 2 1 , Q' \l,8 0,9 1,8/ \ l 1 2 / so that (3.4 M = Q-ip '1,1 0 0 \ 0 1 0 . 0 0 0,9/ 38 2 -12 4 3 1 - 6 5 2 21 (3.5) N = PQ-i = ± j and (3.6) Qxd = (2 1 1), Pxd = (2,2 1,1 1,1) = 1,1 Qxd (3.7) <2><* = / - I , 0, o) , P>* = / M , 0, o) = 1,1 Qmyd so that xd,yd resp. are the right (left) eigenvector resp. of the matrix pencil P—^Q> corresponding to its dominant eigenvalue ld = 1,1. Verifying the relations (1.9) and (1.10), we compute (3.8) Q ^ = - 1 ( 1 , 0 , 0 ) = - 1 ^ and (3.9) Qxd = (2 1 1) = 2 ( l , l , l ) =2w* We see that, using the relations (1.9) and (1.10), we need not compute zd and wd from the homogeneous systems (Af * — Xd) zd = 0, (N — Xd) wd = 0, the norming constants for zd and wd being arbitrary. References [1] VARGA R. S.: Matrix iterative analysis, Prentice Hall Inc., Englewood Cliffs, N. Jersey (1962). [2] ROCEK J., MIASNIKOV A.: Report tJJV Rez 2196-R, P (in Czech). See also: Heterogeneous method and reactor criticality calculations (in Czech). Acta Polytechnica 7, IV, 151—158 (1973). [3] PEASE M. G. I l l : Methods of Matrix Algebra, Academic Press, N. York, London (1965). [4] GELFAND I. M.: Linear algebra (in Czech). Ed. CSAV, Prague (1953). [5] ROCEK J., MIASNIKOV A., ZEZULA R.: Numerical comparison of two methods for solving generalized nonselfadjoint matrix eigenvalue problems. (To appear). 45 

On relations between right and left eigenvectors of nonselfadjoint matrix pencils

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On relations between right and left eigenvectors of nonselfadjoint matrix pencils

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