AN INVERSE HOMOGENEOUS INTERPOLATION PROBLEM FOR V-ORTHOGONAL
RATIONAL MATRIX FUNCTIONS
JEONGOOK KIM
1. Introduction
For a scalar rational function, the spectral data consisting of zeros and poles with their respective multiplicities uniquely determines the function up to a nonzero multiplicative factor. But due to the richness of the spectral structure of a rational matrix function, reconstruction of a rational matrix function from a given spectral data is not that simple.
Our purpose here is to use the state space approach as elucidated in [BGR] to interpolation theory to recover a V-orthogonal rational matrix function from a given spectral data. An important basic idea to this approach is to represent a proper (i.e., analytic at infinity) rational matrix function W(z) by
W(z) = D+C(zI - A)-lB.
Then zero and pole data for W(z) is encoded in constant matrices A, B, C, D. This approach spurred by diverse applications in many en- gineering context has undergone rapid development in the past couple of decades. Here the spectral data is given by a certain quintuple of matrices T = (C7r ,A7r ; A C' BC; r) called a Sylvester data set and a rational matrix function 8(z) having T as its local null-pole triple is found in terms of given spectral data (see Section 2 for definitions).
Received January 13, 1995.
1991 AMS Subject Classification: 30E.
Key words: rational matrix function, null-pole triple, V-orthogonal.
This work is supported by Non-Directed Research Fund, Ministry of Education, 1995, BSRI-95-1426.
718 Jeongook Kim
This matrix 0(z) is the so called resolvent matrix for the nonhomo- geneous interpolation problem. For example, we are asked to find a rational matrix fuction F(z) for which
(1.1) (1.2)
XiF(z) =Yi, F(w·)u· - v·J J - J'
i = 1,··· ,n,
j=l,···,n1r
Un,.. ] Vn". '
where {Zi," . , zn<, Wll'" ,wn,..} are given distinct points in C, Xi, Uj are given 1 x M, N x 1 nonzero vectors respectively, Yi, Vj are given 1 x N, M x 1 vectors. Ifwe organize the data as
then 0(z) having T = (C1r,A1r ; A"B,; r) as its local null-pole triple provides the coefficients for a linear fractional map which parametrizes the set of all solutions of the nonhomogeneous interpolation problem (1.1)(1.2) in terms of free parameter matrix functions. The details are found in the literature, e.g., [ABKW), [BGR). Ifwe consider an extra constraints
F(z)T= F(z), Vz E C,
(correseponding to the transfer function of a reciprocal network in cir- cuit theory)to (1.1)(1.2), then the resolvent matrix 0(z) should satisfy the V-orthogonality condition,
(1.3) 0(Z)TV0(z) = V, Vz EC
This same state space approat:h was applied in the series of pa- pers [ABGR1]-[ABGR4] to a variety offactorization and interpolation problems involving other types of symmetries. Also the inverse prob- lem without V-orthogonality constraint is discussed in [GK][GKR1,2]
[BKGK].
This paper consists as follows: In section 2, some auxiliary notions and terminologies are introduced. In section 3, the zero-pole structure of rational matrix functions 8(z) which satisfy an identity (1.3) for some square matrix V for which V =aVT where a =±1 is developed.
The last section includes the existing results on the inverse problem without the extra constraint (1.3). By refining this results further, a local spectral structure of rational matrix functions satisfying (1.3) is understood and finally, an rational matrix function having prescribed local spectral data and satisfying (1.3) is found in terms of given data.
2. Preliminaries
By an M x N rational matrix function, we understand an M x N matrix with rational functions as its entries and shall regard it as a meromorphic matrix function over the extended complex plane Coo' For anM XN proper (i.e., analytic at infinity) rational matrix function W(z), we define a realization of W(z) to be a representation of the form (2.1) W(z) = D+C(zI - A)-l B, z ~ a(A)
where A, B,C,D are matrices of sizes n x n, n X N, M x n,M x N respectively, and a(A) refers to the spectrum of the matrix A. A realization (2.1) is said to be minimal if (C,A) is a null-kernel pair and (A, B) is a full-range pair, that is
n-l .
.n Ker CA) = {O}
)=0
n-lI:/m Aj B = cn.
j=O
If D is invertible in (2.1), then
(2.2) W-1(z) =D-1 - D-1C(zI - AX)-lBD-1
720 Jeongook Kim
with Ax = A - BD-lCis a minimal realization ofW-l(z) and (2.1) is minimal if and only if (2.2) is. Realizations for a rational matrix function always exist. In the rest of this section, we assume M = N and W(z) is regular (i.e., detW(z) =I- 0 for some zECC",). When (2.1) is a minimal realization forW(z), the pair(C, A)is said to be a (right) pole pair ofW(z) and (AX,B) is said to be a (left) null pair ofW(z),
r = PXllmP is a null-pole coupling matrix of W(z), where P(P X) represents the Riesz projection ofA(AX). By a global null-pole triple ofW(z), we mean a set of matrices T =(C, A; Ax,B;r).
A collection of matrices
is said to be an admissible Sylvester data set if
(C-rr, A,,) is a null-kernel pair of matrices of respective SIzes m x n-rr, n-rr x n"
(Ae;, Bc;)is a full-range pair of matrices of respective sizes ne; x ne;, ne; x m
and the ne; X n-rr matrix r satisfies the Sylvester equation
rA-rr - Ae;r = Be;C".
We note that a null-pole triple for a rational matrix function is an admissible Sylvester data set. From now on ,T denotes an admissible Sylvester data set
(2.3)
where the matricesC", A-rr, Ae;, Be;,r have respective sizes m xn", n1l"x n-rr, n( x ne;, ne; x rn, ne; x n".
THEOREM 2.1. For a given admissible Sylvester data set T as in (2.3), there exists a rational matrix function W(z) which has T asits global-null-pole-tripleifandonlyifr is invertible. In this case, such a W(z) is given by
and
Proof. See [GK].
Two Sylvester data sets
T = (C1l",A1I"iA"B,ir) , (C' A' 'A' B'·r')
T = 11"' 11"' " "
are said to be similar if there exist invertible matrices q>, '11 such that C~ =C1I"q>, A~ =q>-I A1I"q>
A' - '11-IA,- "'11 B' - '11-IB,- ,
r' = '!I-Irq>.
Let us denote the similarity ofT and Tt by T f'V T'. When we want to emphasize the matrices q> and '11 we say that T and T' are (q>, '11) - similar. The proof of the next theorem is found in [BGR].
THEOREM 2.2. Two admissible Sylvester data setsT andT' are the global null-pole triples of a rational matrix function E>(z) ifand only ifT f'V Tt.
3. The spectral structure of V -orthogonal rational matrix functions
LetV be an m x m invertible constant symmetric or anti-symmetric matrix, that is,
VT=aV
where either a = 1 or a = -1. Here, given an admissible Sylvester data set T as in (2.3), our goal is to find an m x m rational matrix function E>(z) for which
(3.1) (3.2)
E> has T as its global null-pole triple E>T(z)VE>(z)= V, Vz E C.
The :first step toward the solution is to understand the null and pole structure ofE>(z) satisfying (3.2). To a givenT as in (2.3), we associate another set of matrices
(3.3) TT _ ( V-IBT AT. AT cTv.rT)- - , , ' ,
11"' 1 1 " ' •
722 Jeongook Kim
It is easy to check that TT is an admissible Sylvester data set if and only if T is. The next theorem gives a characterization of a global null-pole triple ofe(z) satisfying (3.2).
THEOREM 3.1. Given is a Sylvester data set T=(CTr , ATr ; A" B,;r) ofsizes as in (2.3). H
(3.4) T is a global-nu1l-pole triple fore(z)
witb e(00) = D for an invertibleD satisfyingV-I D -Tv = D. Tben, eT(z)Ve(z) = V, Vz E C
ifand onlyifTI'V TT.
Proof. Since T is a global null-pole triple for e(z), by Theorem 2.1
r is invertible and
(3.5a) e(z) = D +CTr(zI - ATr)-lr-1B,D (3.6a) e-l(z) = D-I - D-ICTrr-l(zI - Ad-l B,.
Take the transpose of (3.5a) and (3.6a) premultiply by V-I, postmul- tiply by V and then substitute D in place ofV-I D-TV; the results (3.5b) v-le-Tv =D - V-I B[(zI - Af)-lr-Tc;VD and
(3.6b) v-IeTv = D-I +D-IV- l B[r-T(zI - A;)-IC;V.
If
(3.7) r TA[ -A;rT = C;V(-V-IBf), holds, from (3.5b), (3.6b), and (3.7) we know that
TT= (-V-IB T AT. AT CTV r T)
" " Tr' Tr ,
is a global null pole triple for v-le-Tv. But, (3.7) is equivalent to
rA Tr - A,r= B,CTr which is a part of our hypothesis. Hence (3.S) TT is a global-null-pole triple for v-le-Tv.
Applying Theorem 2.1 to (3.4) (3.S) we conclude that TI'V TT if and only ifV-Ie-T(z)V =e(z). D
The following theorem gives a canonical form ofT which is similar to TT.
THEOREM 3.2. Ha Sylvester datasetT asin (2.3) issimilartoTT,
then T issimilarto Tc which isin theform
Tc= (C1r ,A1r ;A;, -aC;V;rc) with
r~= -arc.
Proof. Suppose a given Sylvester data set
T = (C1r,A1r;A"B'jr) is similar to
TT = (_V-IB T AT. AT cTv·r T )
" " 1r' 1 r ' •
From the similarity, there exist invertible matrices ep and III such that (3.9) C1r=-V-IB[ep, A1r =ep-IA[ep
(3.10) A, =w- IA;IlI, B, =w-IC;V (3.11) r = w-IrT<I>.
Taking transpose in (3.9) (3.10) (3.11),we see that the same equalities are valid with III replaced by_aepTandepby -aIllT. By the uniqueness of the similarlity of two Sylvester data sets,
(3.12) W= _aq,T.
Let with
(3.13) r c= -arTq,.
Upon substituting (3.12) into (3.10) and (3.11), we get (3.14) A, =q,-TA;epT, B,= -a.p-TC;V
(3.15) r = _aep-TrTep.
ReplacingrTq, by -arc in (3.15), we can see from (3.13) (3.14) that
T is (I, epT) - similar to Tc • To complete the proof, the only thing left is to show that r~= -arc. But, the equality is straightforward from (3.13) and (3.14). 0
The following more detailed version of Theorem 3.2 is derived by substituting(3.12) in (3.9), (3.10) and (3.11).
724 Jeongook Kim
COROLLARY 3.3. Suppose an admissibile Sylvester data set T asin (2.3) is similar to TT. Then there exists an invertible matrix ep for which T andTT are (ep, -aepT) - similar.
We get the following theorem from Theorem2.1 and Theorem3.1.
THEOREM 3.4. Given is a eT-admissble Sylvester data set T. There existsa rational matrix functionW(z) satisfying(3.1)(3.2)ifandonly ifT I'.J TT and r is invertible. Inthis case, such a function is given by
W(z) = 1+ C1r(zI - A 1r)-lr-1B,.
4. An inverse homogeneous problem for V-orthogonal func- tions
Throughout Section 4, T denotes a eT-admissible Sylvester data set given by (2.3). By an inverse homogeneous interpolation problem, we mean a problem of finding a rational matrix function W(z) which has a given eT-Sylvester data set T as its eT-null-pole triple.
4.1. A minimal complement
The next result comes from [GK] (see also [GKR1], [GKR2]) and is also discussed in [BGR]. Amore refined version appears in [BKGK].
THEOREM 4.1.1. For a given eT-admissible Sylvester data set T as in (2.3), there always exists a rational matrix function W(z) which has
T as its eT-null-pole-triple.
Theorem 4.1.1 is obtained as a result of a completion problem of a Sylvester data set T in which T is augumented to a Sylvester data set f with matrices of larger size by adding extra zeros and poles in such a way that f has an invertible coupling matrix. A construction of such a f due to [GK] is following. Suppose a eT-admissible Sylvester data set
T as in (2.3) is given witheT cC. Let
TO:= (Co,A1rojA,o,Bojro)
be a E-admissible Sylvester data set for E, a subset of C, satisfying
EneT = 0. We callTO a complement to T if the matrix
r := [r:1~l:]
is square and invertible, where r 12 and r 21 are the unique solutions of
r12A1I'O - A,r12 = B,Go
r21A 1I' - A'Or21 = B oG1I"
The complement will be called minimal if and only ifamong all com- plements ofT, the size of the matrix f is as small as possible. IfTO is a complement to T, then the function
o ]f-1 [ B ] (zI - A1I'O)-1 B o has T as a IT-null-pole triple, and ifTO is a minimal complement, then 8(z) has the minimal possible McMillan degree among all rational matrix functions havingT as a IT-null-pole triple.
To describe such a minimal complement, first we need to introduce some notions. Let N be a complement of K er r in en.,.. and K be a complement ofI m r inen" i.e.,
en.,.. =Kerr+N
Let P1l' be the projection onto K er r along N and P, be the projec- tion onto K along Im r. Further, let 1711' be the embedding ofKer r
intoen.,.. and 17, be the embedding ofK into en,. The controllability indices of a full-range pair can be defined in many ways. Here the con- trollability indices of the pair (p,A,IK, p,B) are introduced through the following incoming subspaces. Let
Ho: = Im r
. 1
Hj := Im r +ImA,B, + ... +ImAr B" j = 1,2,··· . We define the incoming indicesW1 2: ... 2:Ws by
s := dim(Hd Ho) and
Wj :=#{k dim(Hk/Hk-1) 2: j}, j = 1,,·· ,so
726 Jeongook Kim
Then, the numbersWl ~ ••• ~Wsare the nonzero controllability indices of(p,AdK, p(Bc). Similarly, the observability indices of the null-kernel pair (C'/t"IKer r,p'/t"A'/t"!Kerr) are defined through outgoing subspaces
Ko: =Ker r
K j : =Ker rnKer C'/t"A'/t"n···nKer C'/t"A~-\ j = 1,2, ....
We also define outgoing indices al ~ ... ~atbyt = dim(Ko/ K 1)and aj =#{l : dim (K,-IIK,) ~j}, j =1, ... ,t.
Then al ~ '" ~ at are the nonzero observability indices of the pair (C'/t"IKer r,p'/t"A'/t"IKerr ).
Choose a .point € fj. (T. Let {djd~~l;=l be an outgoing basis for Ker r and {f;k}~~lj=lbe an incoming basis for K. This means (4.1.1 )
{!jdJ=1 forms a basis of a complement of1mrin1mr+1mB
(4.1.3)
(4.1.4)
(A'/t" - d)djk = dj,k+I, k = 1, ... ,aj - 1, j =1, ... ,t
Such a basis can be constructed (see [BGK]). The next theorem gives a minimal complement ofT. For the proof, see [GKR1].
THEOREM 4.1.2. Let T = (C'/t",A'/t"iA"B'ir) be a (T-admissible Sylvester data set. Then a minimal complement to T is given by
Here S : K er r ---+ K er r and T : K ---+ K aregiven by
(4.1.5)
(4.1.6)
Furtbermore, G: cm -+Ker r, F: K -+cm are chosen so tbat (4.1.7)
(4.1.8)
(For tbe details of tbe choices of G, F see [GK RI]). Finally,
(4.1.9)
(4.1.10)
"'1
X = I:(A1r - €)-vr+(AC+BCF)(T - €t-1
v=l
er1
y = I:(S - €)"-1 p1r(A1r - GC1r)r+(AC_ €)-v
v=l
wherer+ is a generalized inverseofr such thatrr+ =I - PC' r+r =
1-P1r' Ker r+ =K and Im r+ =N.
REMARK. In Theorem 4.1.2, a(TU{€}-admissible Sylvester data set
T EflTO is given by (4.1.11)
([C1r -C1rX -F], [A1r0 T '0]. [A0c S '0] [-YB,+GBC ] .-),r
where the null-pole coupling matrixris an (n 1r +nC-rankr) x(n 1r +
n, -rankr) invertible matrix given by
- [ r
r-- -yr+p1r
and
728 Jeongook Rim
In this case, an n Xn rational matrix function e(z) with e(00) = I and having T as its IT-nun-pole triple is given by
(4.1.12)
e(z) = 1+Cn:(zI - An:)-l{(r++Xp,)B, +1]n:G}
- (Cn:X+F)(zI - T)-l B, and
e-l(z) =1-{cn:(r+ +1]n:Y) - Fpd(zI - Ad-l B, - Cn:(zI - S)-l( -YB,+G).
4.2. An homogeneous problem with a symmetric spectral data
Assume now that IT is a proper subset of C. An important step to the main result Theorem 4.2.2 of this section is to construct a minimal complement
(4.2.1)
to a given IT-admissible Sylvester data set
similar to TT so that (4.2.2)
f:= T EBTO
:= ([Cn: Co]' [~n: A~o]; [~' A~o]' [~~] ;[r~l ~l02])
is similar to (TEBTo)T, where r12,r2l are the unique solutions of r12An:O - A,r12 = B,Co
r 2l A n: - A'Or2l = BoCn:.
To establish the existence of such a f, it is enough to show that we can construct a minimal complement TO ofTsuch that TOrvT;[.
THEOREM 4.2.1. HT as in (2.3) is a a-admissible Sylvester data set which is similar to TT, then there exists a minimalcomplement TO
toTwhich is similar toTt
Proof. SinceT is similar toTT,n2r= neand there exists an invertible matrix ep for which
(4.2.3) (4.2.4)
by Corollary 3.3. Let {djk}~~1:=1 be an incoming basis for Kerr cho- sen as in Theorem 4.1.2 and U be an n1l" Xn1l" invertible matrix having {djd~~l:=las the entries in column (a1 +.··aj-1+k). Postmulti- plying (4.2.4) by U and premultiplying by epT and taking transpose, we have
(rU)Tep = -aUTepTr.
By the choice of U, the first (a1 + ...at) rows of(rU)T are equal to zero. Thus
where {ej}f":;l is the standard orthonormal basis for en... Ifwe choose
fjkso that Ji ksatisfies the relation (4.2.5)
then, K := £({Jik}~~l:=l)is a complement of Im r, where £(A) is the linear span of the setA. Moreover{Ji k}~~1:=1 is an incoming basis for K. The details are checked in Lemma A at the end of this section.
Choose 77e, Pe,r/1l"' P1l" and define S, G, T as in Theorem 4.1.2. Then (4.2.6)
Indeed, for 1 :::; j :::;t, 1S k :::; aj,
730 Jeongook Kim
by (4.2.5). From (4.1.5), the right hand side of the above equality is the same as ~-TUj,aj-k ( UjO = 0) which turns out to be hk+1 by (4.2.5) (fi,aj+1 = 0). So, we have shown (4.2.6).
Ifwe put (4.2.7)
then the condition (4.1. 7) is satisfied.
Define X and Y as in (4.1.9) and (4.1.10) respectively. Then by Theorem 4.1.2,
(4.2.8) with (4.2.9)
TO = (-CrrX - F,T;S,-YB,+G; ro)
rO= yrX - Y7J' - PrrX is a minimal complement to 7-
Now, we show that TO '"T[. To this end, first we derive A,=~-TA;~T,
Crr = - V-IB[~,
Arr = ~-lAf~,
r+ = _a~-l(r+)T~T
from (4.2.3) and (4.2.4), and substitute the above equalities and (4.2.6) (4.2.7) in places ofA" A rr,Crr , r+, S, G of (4.1.10). Then
Wl
Y = ~-1L(TT _ dt- 1(Af+FTB[)( -a)r+T(A; _ d)-vipT
1/=1 Wl
= -aip-1{2)Arr - d)-vr+(A( +B,F)(T - dt- 1}TepT.
v=l
From (4.1.9), we see that the formula inside ofthe braces is X. Hence, we have
(4.2.10)
Now, we prove the similarity between TO in (4.2.8) and
On substituting (4.2.3), F = aV-1GTq;T1Kobtained from (4.2.6) and X =_aq;-lyTq;T1K from (4.2.10) into -C1rX - F, we have
(4.2.11)
Upon taking transpose of(4.2.7) and postmultiplying by V, premulti- plying by _q;-l , we get
(4.2.12)
Note that (4.2.6) is equivalent to (4.2.13)
From (4.2.11)"" (4.2.13) and (4.2.6), we see that if (4.2.14)
then TO and T l are (aq;T 1K,-q;IKerr)-similar. We replace (4.2.10) in (4.2.9) and take out common factors _aq;-l, q;TK to get
(4.2.15) ro= _aq;-l(XTq;TrXq;-T - X T +ap,q;Xq;-T)q;T,K' Upon taking the transpose of(4.2.9) and substitutingyT = -aq;Xq;-T obtained from (4.2.10) and rTq; = -aq;Tr obtained from (4.2.4), we get
rf = XTrTyTP1r - pc,yTP1r - X TP1r
= _aXTrTq;Xq;-TP1r +ap,q;Xq;-TP1r - X TP1r
= (XTq;TrXq;-T+ap,q;Xq;-T - X T )P1r'
Since the last line of the above equalities coincides with the formula in the parenthesis of(4.2.15), we conclude
r o= -aq;-lrfq;T 1K.
as we desired. This completes the proof. 0
The following theorem characterizes a local null-pole triple for a rational matrix function satisfying
8T(z)V8(z) = V.
732 Jeongook Kim
THEOREM 4.2.2. Suppose T asin (2.3) is a u-admissible Sylvester data set, wbere u c C. Then there exists an m x m rational matrix function B(z) for which
(i) BT(z)VB(z) = V for all z E C (ii) B has T as its u-null-pole triple
(iii) B±l is analytic at inHnity if and only ifT is similar to TT.
In this case such a function B(z) is given by (4.1.12) witb B in place ofEl wbere f+, X, p" 'T/7r' G, F, T are chosen as in tbeorem 4.2.1.
Proof. SupposeT is a given u-admissible Sylvester data set which is similar to TT. By Theorem 4.2.1, there exists a minimal complement TO given by (4.2.8) to T which is also similar to Tt According to Corollary 3.3, there exists an invertible matrix <p(<Po) such that T(TO) is (<p, -a<pT)-similar «<Po, -a<P5)-similar) to TT(Tl). Thus, T EEl TO is
([~ :0]' [_~<pT -a~oT ]) -similar to TT EEl Tt But it is easy to check that TT EEl Tl = (T EEl To)T. Hence if we let e(z) be an rational m xm matrix function which has TEElTo as its global null-pole triple with B(00) = I, thene(z) satisfies the conditions (i)-(iii) by its~onstruction
and such a function given by (4.1.12) with e in place of e. Applying Theorem 3.1, we conclude that eTVB=V. Conversely, suppose there exists an m x m rational matrix functione(z) which satisfies (i)- (iii).
Iff is a global null-pole triple for e(z), by Theorem 3.1, f is similar to fT. By applying Theorem 3.2 and Theorem 2.2 consecutively, we can assume that f is in form of
~ ~ ~ ~T ~T ~
T = (C7r,A7r;A7r,C7rV;f) with
~T ~
f = -af.
Ifwe represent the Riesz projection ofA.7rcorresponding to the eigen- values in u by p, then
- - - - -
T= (C7r ,A7r ;A" Bc,;r) is a u-null-pole triple for B(z) where
(C7r,A7r ) = (C7rllmp, A.7r11mp ),
- - ~T T~T - T~
(A"Bd =(A7r11mpT, -ap C7rV) and f = P f1Imp.
From the above equaIities, we observe that (X"Bd = (X;:" -aC;V) andrT = -ar. By applying Theorem 3.2 to the previous observation, we see T is similar to TT. On the other hand, two er-null-pole triples
T,Tfor 8(z) are similar by Theorem 2.2. IfT and Tare (q>, 'lJ')-similar, upon taking transpose of the similarity relations, we can see that TT andTT are(\liT,q>T)-similar. Since the similarity relation on Sylvester data sets is an equivalence relation, we can conclude that T is similar to TT. 0
LEMMA A. H we take {Jid;=~~=l as in (4.2.5), then {!jd;:tk=l is an incomingbasis for K.
Proof. To prove {lik};=~jk=l is an incoming basis for K, we need to show that the conditions (4.1.1) and (4.1.2) are fulfilled. To compute (A, - El)Jik, we replace A,,fjk by (4.2.3) and (4.2.5) respectively.
Then we have
(4.2.16) (A, - El)Jik= q>-T(A~- d)Uj,Oij-k+l
for j = 1, ...,t, k = 1, ...,aj, where Ujk represents the (al + ... +
aj-l +k)th column ofU-T. By (4.1.3) and the choice ofUjk, we have that
q>-T(A; - El)Ujk EJi,k+l +£({!jd}=l)
for 1 ::; j ::; t, 1 ::; k ::; Qj, where Ji,Oij+l = 0 and £(A) denotes the linear span of of the set A. This proves (4.1.2).
To check the condition (4.1.1), we note that (4.1.4) implies that the (al + ...aj-l +k)th column of C7rU is O. Since C7r = -VB[q> by (4.2.3), this observation implies that
the (al+··.aj_l+k)th ofUTq>TB,V is 0, 1::; k::; Qj-l, 1::; j::;t.
Thus,
the column space ofp,B,
= the column space of p,B,V
=£({(Ql +; .. +Qj)th of q>-TU-T}}=l)
= £({cP-TUjOij}}=l)
= £({!jk}}=l)·
The condition (4.1.1) now follows.D
734 Jeongook Kim
References
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Department of Mathematics Chonnam National University Kwangju 500-757, Korea
