Approximate Representation of Uncertain Systems

APPROXIMATE REPRESENTATION OF UNCERTAIN SYSTEMS G. Menga* and A. Perdon** · CENS, Istituto di Elettrotecnica Generale, Politecnico di Torino, Torino, Italy •• Istituto d,' Matematica A pp l£cata, Universitil di Padova, Padova, Italy

Abstract. In the identification of l inear invariant systems the optimum choice of order and structure of a state variable model representation is strictly dependent on the level of uncertainty present in the system and on the identification experiment. This concept is formalized here by applying re cent results found in numerical analysis literature . An original approach to identification seen as a subspace approximation pro blem is proposed . Keywords . Multivariable system representation; canonical forms; system order reduction; least squares approximation; numerical linear algebra; identification .

INTRODUCTION A state variable representation of a linear time invariant dynamical system is not identifiable only from the knowledge of input output data without specifying a rule, henceforth indicated as the model structure, so that a unique element is determined in the class of equivalence of minimal order models defined by the data. This rule may be viewed as a canonical form for the action induced by a change of basis in the system state space. Different canonical forms have been proposed in the literature by Popov (1972), Rissanen (1974),Hazewinkel and Kalman (1975) etc. In particular the local continuous algebraic canonical forms studied by Hazewinkel and Kalman (1975) show the existence of a set of charts (structures) for the variety of systems of given dimensions within which a minimal number of parameters identifies a specified system. From this point of view the only difference between single input - sin gle output (5150) and multi input-multi out put (MIMO) systems is that in the first case there exists a globally defined continuous agebraic canonical form, i.e. a unique structure to represent the system, while in the second case it doesn't. It is well known that order and structure for these forms are defined by the dependence relations of the rows (or columns) of the

577

Hankel matrix of the system pulse response. Clearly detection of linear dependencies from the data is a quite ill posed problem be cause of noise, uncertainties and the high dimensionality of any real system . Nevertheless it has become very common to see the determination of algebraic canonical forms from input-output data proposed for the identification of MIMO systems. In spite of the large body of literature on this argument few approaches, in our opinion , have tackled the real problem behind structure choice for system identification, i.e. the accuracy of the identified model . We refer here to the work of Rissanen (1 975 , 1976) where concepts of entropy and complexity of a random variable are proposed to relate structure selection to the accuracy of the parameter estimates, and to the work of Akaike (1974, 1975)where information theoretic arguments are proposed for statistical model identification. In this paper those ideas are pursued further from a geometric instead of a statistical point of view . Precisely a model representation consists in both cases (5150 as well as MIMO systems) in the detection (approximation) of a linear subspace and in the choice of a basis for it. Different to commonly used canonical forms the choice we propose here is based on the singular value decomposition Cs.v.d.) of the Hankel matrix . Arguments of the numeri-

G. Menga and A. Perdon

578

cal linear algebra show that with this form order and structure determination is no longer an ill posed problem in the presence of uncertainty. What is more an exact representation problem has a natural extension to approximation ~ith lower order models. With the singular value decomposition a definition of measure of system state "observability/controllability" is introduced, by which the system state space can be divided into the direct sum of two subspaces, respectively, of strong and weak " observable/controllable" states. Accordingly a system representation exists where strong and weak dynamic components are discriminated. The level of uncertainty present in the system and the choice of the input for the identification has a critical role in that discrimina~ tion, as results on rank degeneracy and perturbation theory in eigenproblems show that below a certain level the weak (or weakly excited) dynamic components of the system are completely masked by uncertainty and only the strong dynamic components can be reliably detected and accurately identified from the data. As a related result the "ade~ quacy" of the input for identification assumes from this point of view a very transparent significance . The organization of the paper is as follows: Section 2 reviews some basic facts on canonical forms. In section 3 a representation obtained with singular value decomposition is proposed and the exact realization problem is linked to approximate lower order model approximation. In section 4 results on the sensitivity of the representation from uncertainty are proved and in section 5 an extension of the realization problem to identification from input-output data is ontlined.

= (s2

q2

+ 1). m. The product

matrix

Hsl ,S2 (A,B,e) = 0sl (A,e). Rs2 (A,B)

(2)

is the truncated Hankel matrix of the system pulse response. The system is said compleTely observable and completely controllable iff, for sl, s2 ~ n-l, rank 0sl (A,e) = rank Rs2 (A,B) = n, H (A,B,e) = n. sl,s2 Two triples are said to be state space equivalent Le. (A,B,e) - (A,B,C) Hf hence iff rank

" -

...;

(A,B,e)

= (SAS -1 ,SB,

es

-1

)

(3)

for s E G the linear group of n x n non singular real matrices. In particular, two pairs are state space equivalent, i.e. (A,e) -v

O:,C)

(A,e) = (SAS -1 ,es -1 ) (A,B,e) - (A,B,C) from (3) Hf

If

that T (A,e) n

--

°

~

=S

-

=S

R (A,B) n ~

H

~

it follows

o~ (A,e)

R

·n

(A,B)

S.:G

(4)

-J

(A,B,e)

n

-T

S€G.

= Hn

(A,B,e) •

All state space equivalent triples describing the same system ~ give rise to the same Hankel matrix, then we shall denote H = H (S sl, s2 (A,B,e) where (A,B,e) is any realization of ~. From (4) it results that the completely observable pair (A,e) determines a n vector space Vs of IRql, the column space of (A, e ), which coincides for completely ob s~rvable and contr~llable triples with ~(H~)l the column space ot Ho' Then Vc depends only on c and not on the chosen realization.

°

The one-to - one correspondence between (A,e) and (A,e) (Kalman 1969) indicates that a choige of a pair (A,e) in the equivalence class of realizations of 0 means a choice of a basis for V~ and viceversa. Then given (A,e),the matrix B which satisfies H... = 0 (A,e)· R (A,B) is unique.

°

eANONIeAL FORMS Let ~ be the linear time invariant n-dimensional discrete time dynamical system described by the triple of matrices (A,B,e) respectively n x n, n x m, p x n with entries in the field K (IR or e). Denote by T T T T slT T 0sl (A,e) = (e ,A e , .•. A e) R (A, B) s2

= (B, AB, •.• ,A s2 B)

respectively the truncated observability and controllability matrices of the system of dimension n x ql, ql = (sl + 1). p; n x q2,

..

n

n

Denote by M (or M ) the space of all comp 0 ' pletely observable and controllab le trlples (completely observable pairs), (3) means that the group G acts on Mn (or Mo) (Hazewinkel, 1977).

r

A complete l y observable and controllable system ~ is represented by an equivalence 1

GU X) indicates the column space of matrix X.

Approximate representation of uncertain systems class of triples on Mp and a canonical form for such an equivalence relation is , roughly speaking, the choice of a representative element in each equivalence class, i. e. a map 'f such that the following diagram commutes

REPRESENTATION BY SINGULAR VALUE DECOMPOSITION Let H be a ql x q2 real matrix where for simplicity we assume ql ~ q2 . It is known (Golub, 1969) that H can be decomposed as

•

=P ~

H (5 )

QT

(7)

-T - -T _T P P = P P = Q

where

i: = diag (6"'1'''' ,G"ql)' ~

is the canonical projection in the quotient space (Popov, 1972). n.ql . A be the aff~ne space of all n x reg ~, ql> n matrices with entries in field K (= IR for instance) of rank n. Let

The map An . ql reg

'f

M

0

•

(A,C)I

OT ( A,C) n

is one-to-one and

oT n

But

(SAS

-1

, CS

-1

-T T 0

n

(A,C)

VSfG

'f is

not surj ect i ve , i. e. not every elen ql ment in A . can be viewed as the observareg bility matrix of some pair (A,C) (for details see Hazewinkel and Kalman, 1975). The following lemma gives a characterization of Um I f M~ A~~il; let Ml and M2 be respectively the matrices consisting of the first and of the last sl blocks of P columns of M.

'f.

. n.ql Lemma 1.1: g~ven MCl: A , then M EO ::lm'f, T reg i.e. M = 0 (A,C) 3 (A,C) E M if and n 0 only if (6 )

Being 'f one-to-one and induces an imbedding

G-invariant then

'f

...

M /G ( o

'f. An . q2 /G = G reg

Q=

I

, and ql 6"'1 ~ ' 2 #

coql 1/

O.

The matrix P is orthogonal composed by the T ergenvectors of H H and the matrix Q consists of the ql orthonormal ergenvectors assotiated wit~ the ql largest eig~nvalues of H H. The d~agonal elements of ~ are the nonnegati¥e square roots of the eigenvalues of H H . They are called the singular values of H. Thus rank H = n iff ~ > 0 and -----n ~ = CO = ~ = O. The singular values n+l n+2 ql of H provide otner important information about the nature of H, for istance U H »2=6""1 the largest singular value is equal to the spectral norm of the matrix (Golub, Klema, Stewart, 1976).

G-invariant, i.e. ) = S

579

n,q2

The quotient space Mo/G turns out to be a quasi projective n.m-dimensional subvariety of the Grassman variety of the n-vector subspaces of Kq2(for details see Hazewinkel and Kalman 1975).

If H has rank n <. ql then the first n columns P of P and Q of Q are orthonormal bases for the n-dimensional column space of H,GL(H) and for the n-dimensional row space T of H,CK(H ), respectively. The singular values of H are unique and if they are distinct also the columns of P and Q are uniquely determined. If there is a multiple singular value ~. = ... b . the columns ~+l ~+k of P and Q are determined up to an orthogonal transformation of the form I.

~

T

T

=

where

kxk

T k is orthogonal and arbitrary . kx

If the matrix H of (7) is the Hankel matrix of a system 6', H = H~, then by writing H5" = 0 Sl (A,C)' Rs2 (A,B) (8)

H

(;-

= P £.

QT

with "L. = diag (1;'1' G"'2 ' ... , G"n)' the s .v. d. gives as realization of 6' the triple (A, B, C) where ° sl(A,C) = P

(9 )

G. Menga and A. Perdon

580 R (A,B) ::: ~QT s2 i.e. by applying lemma 1.1: A = p+ P C = PO' B = 1 2'

'""1 (12)

~ T

( 10)

Q O

where PO' Q are the first p and m row O blocks of P and Q respectively, PI and P the first and last sI blocks of p 2 rows of P and p+ is the pseudo inverse of P . Then the s.v.a: of H defines a canonIcal form on M, the space of all triples of matrices up tg the state space equivalen2 ce •

where PI: and Qt are the first r columns of P and Q. The columns of P are an orthonormal basis of Gt(H), then from (12) we can express
@..(P ) is called the e - section of ~(H) e (Golub and others, 197 6). Remark: ex. (Pi is also the column space of the matrix

APPROXIMATE REPRESENTATION In this section we investigate the situation where the Hankel matrix H of rank n is "near" to a matrix of lowe~ rank. From (8) the singular values Q , ... , Q of H can n · 1 b e assume d un1vocally as a measure of the degree of the "observability/controllability" of the corresponding state vector components in the representation (10). This measure is relative to the horizons sI and s2. It is fairly common in practice, especially with high dimensional systems , to find the last nonzero singular values being several digits smaller than the others. In this case the last state components in model (10) are scarcely "observable/controllable" and the matrix H~ is near according, for instance, to the spectral norm, to a matrix of lower rank. This situation is characterized in the following definition and lemma: Definition 3 .1 (Golub and others 1976): The ql x q2 matrix H of rank n has numerical rank (6, S, r) with respect to the spectral norm if 2 i) r = inf ~ rank K: ii) ~ = sup

{ "'L:

11

H - K

U H - K l\ 2 ~ ,

Ii 2 ~ t ='>

I

(11)

rank K); r

Lemma 3.1 (Golub and others, 1976): Let ~ ~ 1 b. ~ '1~n be the nonzero singular values . o the matr1x H, then H has numerical rank (e, S, r)2 iff

t

..

G"r ~ ~ 7 C"l/6'"r+l· If H has numerical rank (t:, ~,r) with . . t h e f20110w1ng r ~ n we can d ecompose H 1n way 2

If there are multiple singular values one of the possible representations is chosen a~bitrarily, as all of them share identical geometric properties.

1

K

=P

(14 )

o o

of rank r belonging to the So -neighbour =G'" ~l. of H i.e. liH-KiI 2 r+l Eventually the matrix K can be used instead of H, i.e. Gt(PE ) is assumed to approximate~(H). To interpret the €-section of G.(H) let us measure how well Gl(Pi), Pi are the first i columns of P, represents
liP.1 H ~

2

= G":1+ 1

(15)

In the presence of a distinct gap in the sequence of ~., for instance from ~ and 6"" , 1 . r r+l we see that the approx1mation becomes better and better as i varies from 1 to r then it improves only slightly if we add other columns. Moreover we will show in the next ection that Gl(Pt ) is also numerically stable to data perturbations. Let X be the n-dimensional state space of system E) and let H6" have (E,'; , r) rank. The decomposition (13) induces an a~alogous decomposition in X that hence can be represented as the direct sum of an r-dimensional subspace Xs and its orthogonal complement X w

X = X Q;) X s w spanned respectively by the first rand remaini~g n-r components of the natural basis e i defined by the realization (10). 3

e. n-dimensional vector with 1 in the i~th entry and zero in the others.

Approximate representation of uncertain systems Then the t~iple (A,B,C) in (10) which zes Ii is patitioned as All . .-

\2

- -.- _.

i

A21 whe~e

etc.

A

11

Bl

,

-

A 22 is

,

B 2

x

~

~,

A

22

~eali­

(Cl' C2 ) is

(16)

(n-~)

x

(n-~),

Definition 3.2: The t~iple (A , B , C ) in . . f 11 1 1 (16) 1S a ~ep~esentat10n 0 tne st~ongly !-obse~vable/cont~ollable dynamic components of 5 ~elative to the ho~izons sl, s2. The ~ep~esentation (A ,B ,C ) can be vie11 1 1 wed as an example of the design of non pe~fect agg~egation of dynamic systems (Aoki, 1978). Let us conside~ now the p~o­ blem of lowe~ o~de~ app~oximate ~ep~esenta­ tion given H. P~eviously we suggested to use the column space of the mat~ix K (14) in o~de~ to app~oximate the column span of H (12),i.e. to use as a basis fo~ Gt(H) P . d of P. We can obse~e that P is £ 1nstea an obse~vability mat~ix i.e. satisfies (6) but Pt in gene~al (unless in (16) A =0 . not. We p~ove, howeve~,21 with o~ C2 -- 0) 1S the next theo~em that if t is sufficiently small Pe also must be close to an obse~­ vability mat~ix. Using lemma 1.1 and defiping acco~dingly P and P a measu~e ~l .. 2' of the p~oximity of PE to a~ obse~vability mat~ix can be given, as fo~ (15), by ",T

RT 1 T 21i

whe~e T and

T

2

(17 )

a~e o~thono~al basis span-

ning ~tP~2) akd the o~thogonal complement of ~PE1)' ~espectively.

Theo~em

3 .1 : Let H = P 1:.. QT be a sen n n n quence of ~eal ql x q2 Hankel mat~ices of ~ank n, and nume~ical ~ank (c ,S ,~) n which conve~ges to the ~eal q£ x q2 ~ankel l' mat~ix H of ~ank ~. Let K = P ~' Q £ ' _ d' ,..n 0,0)I.l be nthe nse- n ' - 1ag ( f) nl' ... , ~~, quence of mat~1ces of ~ank ~ obtained as in (14) then Kn c oDve~ges to H. P~oof: in the appendix.

581

of the column space of a mat~ix H assume thei~ ~eal significance in the p~esence of unce~tainty. Let us assume that no H but

""H = H

~E

whe~e

+ E

~

t.

(18) 1 is available, and E rep~esents the unce~­ tainty on the data. In this case we cannot know eithe~ the t~ue singula~ values of H, no~ the t~ue o~thogonal basis P fo~ Gl(H). By using ~esults of the nume~ical algeb~a lite~atu~e we show, howeve~, that: 1) smalle~ singula~ values and ~elated eigenvecto~s a~e ~elatively mo~e sensitive to unl

2

ce~tainty.

2) if a distinct gap in the singula~ values of H is d~tected ve~y likely the e-section of &(H) is a ~eliable app~oximation of the E -section of (52(H). The need fo~ detecting f~om the data a lowe~ dimensional ~ealization of a system is indeed common in identification. Infact eve~y physical system is high dimensional and uncertainty, ~umeri­ cal errors or inadequecies of the input signal can fade the weak components of the system. Lemma 4.1 (Golub and others, 1976): If~., 1 r;., t. ~re respectively the singula~ valu~s of H, Hand E in (18) then

le;'. - r.l~e 1

1

1

=

~Eli ' 2*fi

(19)

Remarks: In the hypothesis of lemma 4.1 if the numerical rank of H is (c., S', r) where we have chosen t.. = l' ,$ = 'r" ~nd . ' 'I:' r+l r' 1f f 1 ~ £ = l' then H belongs to the E.-neighborhooa+ of H and rank H ,., r. Moreover if '( is the gap between '( and t' from (19) we get r r+l

r -2 t 1

~

f)

r

-

.~

S'r+l

'~{ + 2 E.

l

·

Hence if e is smalle~ than ~ /2 we can . d uce that 1 also 1n . the sequence of the sin1n gular values of H a gap between ~ and f) exists. We now p~ove the main~theorem r+l of the pape~.

H,

Theo~em 4.1 : Let given by (18), have distinct singular values 7 , ... , I (c 1 1" r) 2 rank, and hE Q 2 = t · Indicat~ with

'S

l

Algo~ithms

to compute app~oximate lowe~ o~­ de~ ~ep~esentations based on s.v .d. and lemma 1.1 a~e p~oposed in a companion pape~ to be p~esented at this confe~ence (Menga, G~eco, 1979).

REPRESENTATION IN THE PRESENCE OF UNCERTAINTY The concepts of

nume~ical ~ank

and

~ ~section

IV

H

= (U e

U)I/\c. t 0

H

= (Pe..;

Pt)

A

where

Af = d'1ag

, ... l'ql)' :£.t.

0

(v

At

-

C,

;

'"

vt.')

T

l~o t. i of I (Q. i "QE)T , I

/:.

A

(~ "1"'" .... It') ' . f"'t

= diag

= diagA

«()l'"'' Or)'

(,.,., 1'+1

~E... =

G. Menga and A. Perdon

582

diag (h , .•. ,C:;-, 0, .. ,0), the singular va~ 1'+1 n "" lue decomposition of Hand H, then for a sufficiently small value of e = Cl/i . l Pc is represented by a convergent power ser~­ es in 4i:

O(~

+

2

(20)

)

where .::i ij

ijlj,f ..

+ ,.+~. ]

~~

=1

(Xii

+ 2 t'.

~

~

"S ij i = l, .... ,ql-r, t:+1' . ] r+~ <. ~.7:

l'

I '\: "

<: ~J

l <

f- •'t'

l'

Proof: in the appendix. Theorem 4.1 shows that each column of ~ can be expressed as the sum of two components, one belonging toGl(U~) and the other (a perturbing term) to its orthogonal complement 6\.( u!). Examining the entries of the columns of G,which give the magnitude of these perturbing terms we see that perturbations grow with j the order of the column of P~ , and that, due to them, the last columns, related to the smallest singular values, loose any relationship with ~U.) i f t" and 1:" 1 <0 I' 1'+ are too close. This confirms theassert~on that with uncertainty only the (-section of 3JH) is a reliable approximation of &(H) and that to increase the size of the model is useless unless a distinct gap separates the singular values representing the system dynamics from those likely to build up due to uncertainty.

THE IDENTIFICATION PROBLEM Previous results on realization from Hare extended to the identification problem from input-output data. It can be eas ily shown (see for instance Menga and Greco,1979) that different identification approaches with least squares methods permit the determination of the order and the structure of the model and the estimation of the output pair (A,C) through the analysis of the following matrix equation H=O

sl

(A,C)·U+E

(21)

where U of dimension n x q2 is an unknown matrix which depends on the identification method and on the input signal

chosen for the experiment, and E is the matrix of perturbations. Let us note that, as before, 6G(ih (or its e -section) is representative of the system dynamics, however_the realization resulting by s.v .d . of H, and in particular the discrimination between strong and weak dynamic components, are no longer canonical but depends now on the experiment (because of U). This is, in our opinion, not a disadvantage; viceversa we consider it to be very significant and instructive to evidence those dynamic components of the system less effected by uncertainty and to link the choice of the model structure to the experiment. Conversely the approach offers useful information regarding the selection and the evaluation of test inputs for identification, i.e. the conditioning number ~l/~ (to be minimized) related to the accuracy of the parameter estimates of the model and the gap 1'" 11"' (to be to 1'+1 maximized) between reliable information on the system dynamics and disturbances. Experiments to implement these ideas with interactive approaches have been done by Menga and Greco (1979).

CONCLUSIONS In this paper we have shown with arguments drawn from numerical analysis literature that the concept of input-output equivalence of dynamical models is not appropriate for the representation or the identification of systems in the presence of uncertainty. Instead the choice of the system representation cannot ignore, with reference to the accuracy of the identified model, the presence of uncertainty, and we find it remarkable that the classical algebraic cano nical forms, proposed for identification, do not appear to play from this point of view any practical role. The results of this study are being used in the realization of numerically efficient algorithms for MIMO system identification and approximation.

REFERENCES Akaike, H (1974). A new look at the statistical model identification. IEEE Trans. Autom. Control, Vol. AC-19, No 6 , pp. 716-72'3.

Akaike, H (1975). Markovian representation of stochastic processes by canonical variables. SIAM J. Control , Vol. 13,

Approximate representation of uncertain systems No.l , pp. 162 - 173. Aoki, M. (1978) . Some approximation methods for estimation and control of large scale systems. IEEE Trans . Autom . Control , Vol . AC- 23, No. 2 , pp. 173-182 . Golub , G.M ., (1969) . Matrix decompositions and statistical calculations . In Milton R.C. and Nelder , J . A. (Ed . ) Statistical computation, Academic Press, New York , pp . 365 - 397. Golub , G., Klema , V., and Steward , G. W. , (1976) . Rank degeneracy and least squares problems . Rep . Stan-CS-76-559 Stanford Univ. Hazewinkel , M., and Kalman , R.E . (1975) . On invariants , canonical forms and moduli for linear , constant, finite dimensio nal , dynami c systems.ln Marchesini , G. and Mitter , S.K. (Ed . ) Mathematical Systems theory , Springer Verlag , New York. Hazewinkel , M. (1977) . Moduli and canonical forms for linear dynami cal systems . Rep . 750/M-7609/M-76 10/M Erasmus Univ. Rotterdam. Kalman , R.E . , Fal b , P.L. , and Arbib , M. A. (1969) . Topics in mathematical system theory . McGraw Hi l l, New York . Lawson , G. L., and Hanson , R. J ., (1974) . Sol ving least squares probl ems . Prent i ce Hall , Englewood Cliff s . Menga G. , and Greco, C. (1979) . Experiences with a modeling and i dentification pac kage: a power plant study . Proc. 5th IFAC Symp . on Identification and System Parameter Estimation , Darmstadt . Popov, V. M. ( 1972) . Invariant description forms for linear dynamical systems. SIAM J . on Control, Vol . 10; No . 2 Rissanen, J. (1974) . Basis of invariants and canonical forms for linear dynamical systems . Automatica , Vol . l0 Rissanen , J . , and Ljung , L. (1975) . Estimation of optimum structures and parameters for linear systems . In Marchesini , G. , and Miker , S . K. (Ed . ) Mathematical System Theory , Springer Verlag , New York . Rissanen , J . (1976) . Minmax entropy estimation of models for vector processes. In Mehra , R.K . and Lainiotis, D.G . (Ed.) System identification , advances and case studies , Academic Press , New York .

583

APPENDIX Poof of Lemma 1 . 1 : Let M = (A , .... ,A ), 2 1 n T T Ml = (A ,···· , A _ ) then if ~ (M2) c ~(Ml) a n 1 O matrix F n x n exists such that M2 = F Ml

= FA.1-1

and

Al'

Al

= FAO'

M~

i

= 1 , .. ... ,

A2 = F2 A0 " '"

1 mf '~

pair (A,C)

n-l or

. . M = OT(FT l.e n 'A 0)' MO exists such that

€

M = OT (A , C) then n

= (T A CT , . . . AnT CT) = (CT , .... An - iT CT)

M2

M 1 T M2::A Ml

(it T )

Proof of theorem 3.1: P /'. QT , ./. 1I

Hn - H 112 -

T C· G(M ,x. 1 )

. ( M2

~~

= diag

or .

H converges to

H

n

(r; , " " r.,'

r

1

0, 0

=

iff

for any n In ; from (19 ) it

E.,

implies (A . l)

n ..' n

Hn has numerical rank (:n ' ~ n ' r)2' then • = C:)n ... = (-i n . From -n . r +l' v n r n;> n and for any n /" n ( A . 1) I C n \ .... f .

we can choose

-r:

KN Hn il2 = Elr:l + H - H 112 H ;, 2 = liB ' B I n - Hn n '. n , H + le:. H .. .1 H .2-: .: B n .~ n n

11

~

¥

n > n.

Proof of theorem 4 . 1: Let us indicate the s •v . d. of if ( 18 ) as

H --

U

qlxql

(S

: 0

qlxql'

qlx(q2-ql)

) ,

(iq2xql : l q2x(q2-ql) )T and define the symmetric matrices

=

Tt

AJ

1

H'

0 0 . =1El' ·HT HI 0 ' C

O.

satisfying from ( 18 ) the relation j(

= /\. - C,

(A . 2)

It is proven (Lawson and Hanson, 1974) that

Wilkinson , J . H., (1965) . The algebraic eigenvalue problem . Clarendon Press , Oxford.

C'V

Tt = P where

l

(A.3)

and

S

S

o(q2 - ql)x(q2 - ql)

I

G. Menga and A. Perdon

584

p

U

-u qlxql

""1 V

~l

qlxql

=

oqlx( q2-ql)

V

q 2xql

I'

v2 q2x(q2-ql) ,

q2xql

-v -1 r1 and U = 1/ ~ 2 U, V = 1/ V 2 V. "Because K is obtained by perturbing }t. , we apply the results of Wilkinson (1965) to evaluate the eigenvectors of }( as~a perturbation of the eigenvectors P of~. In particular we are interested in their first ql components, i. e. the c~lumns 0::' U. , Let us note that the entries of ~ and ..:, because of ( A.3 ) are bounded respectively by

" .. / Ih lJ

I e ij

= t1

<.

1I )-tl~

<'

lie ii2 =

I

i:'

1

=A

where

lIt - 1/ t

1

·X . ~

,t:

1

A and

ql+q2 . f!, .. / (A. - }.. ) xi' i=l \ lJ J l

(' . ;i

I \J"

<

1

(A. 4)

i1j From (A.3) the eigenvalues of A are given by Aj+ql = - ~j' j = 1, ... , ql

A. .

= 0,

j''7 2 , ql,

therefore we obtain

J for (A.4)

1

+ ~. B

''V

A;

and

Then by dividing both sides of eq. (A. 2 ) by 't' 1 we obtain 1/'1 • h

With the notation of Wilkinson, indicating with x. and ;. . a generic eigenvector and eigenvalue of for sufficiently small values of e, the corresponding perturbed eigenvector x . ( ( ) can be represented by a convergent p6wer series of ~, where the constant term is x . and the first order perturbation is giv~n by

ql fi

ql

1"1 ( :~,~ 'ij (;.. / ( ,..( j - ~) 'i

x; ~

l=l q2-ql +L (?;, \2 ' ql+i j x 2'ql+il ( A. S ) i=l The proof is completed by noting that the vectors x . in (A. S ) are the columns of P

/t;

I a .. , lJ

< 1,

~ B and lb . .1 < 1, and " :: ':1/'(1' . lJ

+.:' /.. ('q'1+ ~;J. I ClJ. +'r;) ~ x ql + i + l=l

(A. 3).

J