Vandermonde matrices on the circle: Spectral properties ...Vandermonde Matrices on the Circle: Spectral Properties and Conditioning Antonio C6rdova 1' *, Walter Gautschi 2'**, and

Numer. Math. 57, 577-591 (1990) Numerische Mathematik �9 Springer-Verlag 1990

Vandermonde Matrices on the Circle: Spectral Properties and Conditioning

Antonio C6rdova 1' *, Walter Gautschi 2'**, and Stephan Ruscheweyh 3'***

Math. Institut, Universit~it Wiirzburg, D-8700 Wiirzburg, Federal Republic of Germany 2 Department of Computer Sciences, Purdue University, West Lafayette, tN 47907, USA 3 Departamento de Matem~tica, Universidad T6cnica Federico Santa Maria, Valparaiso, Chile, and Math. Institut, Universitiit Wiirzburg, D-8700 Wiirzburg, Federal Republic of Germany

Received May 29, 1989

Dedicated to R. S. Varga on the occasion of his sixtieth birthday

Summary. We study Vandermonde matrices whose nodes are given by a Van der Corput sequence on the unit circle. Our primary interest is in the singular values of these matrices and the respective (spectral) condition numbers. Detailed information about multiplicities and eigenvectors, however, is also obtained. Two applications are given to the theory of polynomials.

Subject Classifications: AMS(MOS): 15A12, 15A18; CR: G1.3.

1 Introduction

Vandermonde matrices have a reputation of being ill-conditioned. This reputation is well deserved for Vandermonde matrices whose nodes are all real, in which case the condition number is expected to grow exponentially with the order of the matrix. (Exponential growth of the D-condit ion number has recently been proved [2] in the case of positive nodes, as well as for nodes located symmetrically with respect to the origin. The same is likely to hold for arbitrary real nodes, since the nodes of optimally conditioned Vandermonde matrices are conjectured to have the above symmetry property.) The situation changes drastically, however, if one allows complex nodes. For example, taking the nth roots of unity as nodes in an (n • n)- Vandermonde matrix yields optimal (spectral) condition number 1 for each n [1,

* Research of A.C. supported by the Fundaci6n Andes, Chile, and by the German Academic Exchange Service (DAAD), Federal Republic of Germany

** Research of W.G. supported, in part, by the National Science Foundation, USA, (Grant CCR-8704404) *** Research of S.R. supported by the Fondo Nacional de Desarollo Cientflfico y Tecnol6gico (FONDECYT), Chile, (Grant 237/89), by the Universidad T~cnica F. Santa MarBa, ValparaBso, Chile, (Grant 89.12.06), and by the German Academic Exchange Service (DAAD), Federal Republic of Germany

578 A. C6rdova et al.

Example 6.4]. On the other hand, the roots of unity form a triangular array of points, which in applications - for example, polynomial interpolation or quadra- ture - may be inconvenient inasmuch as it requires that the function to be inter- polated, resp. integrated, be evaluated on a two-dimensional array of points. Re- stricting oneself to linear sequences of points, and placing them all on the unit circle, gives rise to the following interesting question: To what extent is the well-conditioning of the respective Vandermonde matrices maintained ? Intuitively, one expects well-conditioning if each segment of the sequence is as equally distributed on the circle as possible. A well-known sequence having such a property is the Van der Corput sequence; see e.g. [3, p. 127]. Our objective, then, is to study Vandermonde matrices whose nodes are the initial points of a Van der Corput sequence on the unit circle. The strikingly regular pattern of the singular values of such matrices, as exemplified by Fig. 4.1.1 in [4], has greatly stimulated our interest in this problem.

Given the integer v > 0 in its binary representation

V = ~ Vj2 j , ViE{O, 1}, j=O

define the fraction

C v = ~ , V j2 - j - 1 . j=O

The sequence {C~}v~ o is called Van der Corput sequence. For n e N define the n x n Vandermonde matrix

1 1 ... 1 t e 2 nico e 2 nic ~ . . . e 2 nlc~ -

Vn ~ . . ". "

\ e 2rd(n-1)c~ e 2n i (n - l ) c l . . i e 2ra(n "-l) . . . . /

We are interested in the (spectral) condition number of V.,

_

cond ,-L2mi~ ,

where 2m.x(S,) and 2ml n (S,) denote the largest and smallest eigenvalue of the matrix

In fact, our main result, Theorem 1, does much more: it describes completely all eigenvalues of S,, and in the proof we even find explicitly all eigenspaces. The matrices S, have a very interesting structure and may be worth further study. They turn out to be (Hermitian) Toeplitz matrices, namely

S, = Toep (Sto "), ~") . . . ~ O n - l l , where

n--1 (1.1) stu ")= ~,, e -2"I~' , # = 0 , 1 . . . . .

vmO

To simplify the statement of Theorem 1, we introduce the following alternatino binary representation of natural numbers n: let

(1.2) 2k-* <n<Zk , ke{0 ,1 .... }.

Vandermonde Matrices on the Circle: Spectral Properties and Conditioning 579

Then there exist a minimal integer l ( n )>0 and integers

k = no > n ~ > . . . > nt(n) _~ O ,

all uniquely determined, such that

l(n)

(1.3) n = Y~ (-1)J2", . j=O

Theorem 1. Let n ~ IN have the alternating binary representation (1.3). Then the eigenvalues o f S . are

2"J, j = 0 . . . . . l(n),

and the eigenvalue 2"J has the multiplicity

12"J-2(n mod 2"01.

It is readily seen that the sum of the given multiplicities is indeed n. This follows even easier f rom the following obviously equivalent but less explicit corollary.

Corollary 2. Let n~ iN satisfy (1.2). l f n = 2 k, then S . has the only eigenvalue 2 k. I f n < 2 k, set n' = 2 k - n. Then the following holds ."

i) All eigenvalues o f S , are < 2 k. ii) 2 k is eigenvalue o f multiplicity n - n ' o f S. .

iii) All eigenvalues o f S,, are also eigenvalues o f S . , with the same multiplicities.

Another immediate consequence of Theorem 1 is

Corollary 3. For each n ~ 1N we have cond V. < V~n-

We are proving Theorem 1 not directly for S. but for a similar real Toeplitz matrix T,, using an explicit inductive construct ion of all eigenspaces. The vectors of a basis of these eigenspaces are given through the coefficients of certain polynomials with integer coefficients which come from an interesting unconvent ional three-term recurrence relation. These polynomials (of even degree) may deserve some independent interest because of the proper ty described in Theorem 4, which is a consequence o f our p roo f of Theorem 1. They are given as follows: let e 1 (z) = 1, e3(z)= 1--~z--~-z 2, and for k , l ~ l N , / o d d , let

z ~{'(1 --2k)el(z) (1 +Z2k)+ZleZk_z(Z), 1 < l < 2 k-1 , (1.4) e2k+/( ) = [el(z)(1 +z2~)+zle2~_z(z), 2 k-1 < l < 2 k.

Theorem 4. The polynomials e. (o f degree n - 1), n odd, have all their zeros on Iz} = 1.

Finally, we mention another application of the results in Theorem 1 to the theory of polynomials.

Theorem 5. Let P be a polynomial o f degree m, m even. Then we have

max j=~--0 P(e-Z"iCJz)-P(O) <=m max [P(z)]. lzl__<l tzl_-<a

This estimate is sharp fo r P = const.


2 A Generating Function for the s~ "~

We start with some elementary observations concerning the sequence c~

Lemma 2.1. The c~ satisfy

C 2 k - , + v = 2 - - k J t - C v , O ~ V < 2 k - 1 , k e N , ( 2 . 1 )

and {2kc,,v=O ..... 2k -X- -1}={2j :O<j<2t -X- -1} ,

{2kc,, V=2 t-x . . . . . 2 t - - 1 } = { 2 j + l : 0 < j < 2 t - X - - 1 } .

We omit the simple details of the proof. For k, lelN, 0__<l<2 k, let

2k-1

(2.2) Pzka(z):= 1-I (1--e-2=/C'z), j= 2k - - l

with the convention P2k,o- 1.

Lemma 2.2. Let k, l be as above and set k-X

(2.3) 1= Z 4 2a, 4 e { 0 , 1 } �9 j=O

Then k - I

(2.4) P2~,,(e2"'c=k-'-'z) = 1-[ (1 +diz2') . j=o

Proof. We first prove the following recursion:

(2.5) P2k,,(z)=P2~-, ,_a~_ ,2~-,(ze -2"m-d~- ')2-k) (1 +dk_xZ2~-').

If d k _ x = 0, we use the index transformation j + 2 k- x + j in (2.2), and (2.1), to derive (2.5). If dk_ x = 1, then

2 k - t - x 2k--i

.P2k3(Z) = 1--[ (1-e-2'ac'z) l'-I (1 -e-2'ac'z) j=2k - - I j = 2 k - I

2 k - X

= P 2 k - x,l--2k- 1 (2') 1-I (1 - - e - 2 ' a ~ ' z ) - j = 2 k - 1

To identify the second factor, we use the last formula of Lemma 2.1, and obtain

1--[ (1 -e -2 '~eJz)= l-I 1 - e (eei'~2-kz) j = 2 k - I j = l

= 1 - ( e 2 i=2- ~z ) 2~-~

= 1 -k -dk_122k- ' "

Repeated application o f (2.5) yields ((11"z)) k - 1 --2rti ~ ~ 2/

P2k3(z)= I-I l + d i e ,=j+l j=O


Hence, with

we get

the assertion. []

, ~ - 1 1 - d , 2hi ~ 2 ~+1

w ~ _ e s=0 =e2~tiC2k-l-S

i l - d ,

~o 2 '+x Z k - 1

Pz%l(WZ) = l-[ 1 q-d~ j=o k-1

= I-[ (1 +die i"(1-nj)z2j), j = O

In the sequel we shall use the following nota t ion: if f , g are analytic functions in z = 0 , w e shall write

n

f " ~ g if f - g has an n-fold zero in z =0 .

Lemma 2.3. Let n ~ N satisfy n < 2 k, k ~ N. I f

then

(2.6)

Proof We set

so that

k - 1

2 k - - n = E dJ 2/, dj~{0 ,1} , / = 0

. -1 s~"~ (e ~., . . . . z)J " ~-1 E - - ~ E l o g ( l + d / z 2 J ) . j=l J /=o

/7. (z) = log P2k, 2 k _. (e 2"i . . . . z),

2 k - 1 _e -2~ i ( c j - c . - 1)

r ; ( z ) = j=nY 1--e-2~"c~-c~ and

2 k - 1 - 1 (2.7) zF'(z) -2k+n= E l_e-2~i(cj-c.-1)2"

j=n

Now using the second par t o f Lemma 2.1, we see that

2k 2k -1 1 2 k ,.~ ~

j = O 1 - -e -2~i(c~-c"- DZ

and thus 2 k n - 1 1

z r ; ( z ) + n "~ ~ _e_2•i(cj_c._a)z 1=o 1

' ~ E e - 2~i~cj e2~ic"- l Z) p" p = O \ / = o

This shows that

(2.8)

and by integration, using Lemma 2.1, we arrive at (2.6).

n - 1 n - 1

F.(z) ~ E s} "~e2"'~"-lzj-1, j = l

[]


We define the following sequences:

a~")=s~")e z"uc"-I , j = 0 , 1 . . . . . n - l , n ~ i N ,

and note f rom (2.6) that a~")~Z for a l l j , n. It is obvious that the matrices S. and

S* : = Toep t,,~,) ~ . ) \ v O , " " " ~ V n - 1 !

are diagonally similar, and have the same eigenvalues with the same multiplicities (later we shall s tudy a still slightly different set o f matrices). For easier reference, we note down a consequence of (2.7), (2.8):

n - 1 n 2 k - - 1 1 (2.9) 2 k - Z a} ")zj ~ E l _ e - Z , , c , - c . _ , ) z"

j = O j=n

Corollary 2.4. L e t n ~ IN sat is fy n < 2 k, k E IN. I f 1 < j < n - 1 has the representation j = 2" (2 m - 1), where p = O, 1,. . . and m E IN, then a~o ") = n and

(2.10) a~ "~ = [(2 k - n) mod 2" + 1 ] _ 2 [(2 k - n) m o d 2"].

Proof . F r o m L e m m a 2.3 we find that

a}" '= - E ( - d 0 j2- '2" , 2"lj

which yields (2.10) after a simple computa t ion . []

3 Reduction to the Case n Odd

The following is an immedia te consequence of Corol lary 2.4:

,~2.) , 0 < j < n - 1. v 2 j + l

Hence, if (Xo,Xl . . . . . x . _ l ) '

is an eigenvector for S* and the eigenvalue 2, then the two vectors

( xo ,O ,x~ ,O . . . . , x . _ l , 0 ) ' , (0 ,x0,0, xl .. . . , 0 , x . _ 0 '

are eigenvectors for S~'. and the eigenvalue 22. It is now easily seen that Theorem 1 holds for 2n if it holds for n. In view of this fact, we can restrict ourselves to the case n odd, which f rom now on will be always assumed.

4 The Largest Eigenvalue 2 k

The r ight-hand side of (2.9) can be writ ten as

2k-, 1 _ ~ " d~(4,) E i - -e -2ni(ci-cn- l)Z i --ei4*z ' j=n 0


with a positive (discrete) measure #. Hence,

2kI , - -S * = ( i ~ e itz-m'g'dl~(qb))o=<t,,.=<._ 1 '

which readily implies that 2 k I , - S * is positive semidefinite. This shows that all eigenvalues of S* are < 2 k if

(4.1) 2 k - l < n < 2 k , k~IN.

This proves assertion i) of Corollary 2. From now on we shall assume that n, k satisfy the relation (4.1). We shall use the following general result.

Lemma 4.1. Let m, n e lN, m < n. Let ~j~ ]R, ej~ ~ with l e j l = l , j = l , . . . , m , and set

as z s = ~ 2j s=O j = l 1--ejz

Then 2 = 0 is an eigenvalue of multiplicity at least n - m for the (Hermitian) Toeplitz matrix

E = Toep (a o, a 1 ... . . a,_ 1).

Note that the following proof contains a description of n - m linearly independent eigenvectors of E. We adopt the following terminology: a polynomial

n- - I

p(z) = y~ bjzJ j = O

is said to represent the vector

e= (bo ,b l ... . . b,_l) t~C" .

Sometimes, if we say this, the actual degree of the polynomial may be less than n - 1. In this case we assume that the missing high components of the vector are set to zero.

Proof o f Lemma 4.1. We write m

/'(z)= H (~j-z) , j = l

and we want to show that the polynomials

P~(z):=zSP(z), s=O ... . . n - m - 1,

represent (obviously linearly independent) eigenvectors for E and the eigenvalue zero. We can write

E = ~ 2 jToep(1 ,e~ ,~ ... . . e7-1). j = l

Hence, it suffices to show that the vectors es represented by P~ are annihilated by every term in this sum, for instance by

Toep (1, ea, e~, ..., e~- 1). We have

ra--1

es(z)=(~l - z ) E rJ Zj+s- j=0


Hence, the vectors e s can be written as

m - - I

e~= ~ rj(O . . . . . 0, e l , - 1 , 0 ... . . 0) t, j = 0

T r o w j + s

from which the assertion becomes obvious. []

It is clear from (2.9) and Lemma 4.1 that Sn has 2 k as an eigenvalue of multiplicity at least n -n ' (in the notation of Corollary 2). It should also be noted that

k - 1 k - 1

(4.2) z~ l-I ( l+diz2~), s=O .. . . . n - n ' - l , n'= ~ dj2 j, j = O j = O

represent eigenvectors of S* and the eigenvalue 2 k, as can be deduced from the proof of Lemma 4.1, formula (2.9), and Lemma 2.2.

To complete the proof of Corollary 2 (and Theorem 1), we only need to prove part iii) of the corollary for n odd. This will be done in the following two sections.

5 The Matrices T.

Let n > 3 be odd, 2 k- x < n < 2 k, and write

k - 2

2t -n= Z diZJ, dj~{O, 1}, j = O

so that

k- -2 k - 2

n = 2 k + ~ (1--dj)V--(2k-X--1)=l+2k-X+~ ( 1 - d j ) 2 j ( s i ncedo= l ) . j = O 1=1

This implies

(5.1) (n rood 2~) +((2k--n) rood 2~) = 2 s, 1 < s < k - 1.

Hence, for nodd and /~> 1, we obtain from (2.10) the relation

a~2(2m_~)=2(n mod 2~) - (n mod 2~+1), 2<2~(2m - 1 ) < n - 1,

while tr~ n) = 1 f o r j odd. For reasons of simplicity in the subsequent proofs, we now introduce the matrices

where T, = Toep (to t") , t~ ") , --.,'.-lJ,'~") a n o d d ,

t•o")=n, r n + l

.~.) ~ ( - 1 ) 2 , # = 0 , t2"(2m-l)=~2(nmod2~)-(nmod2U+l), p > 0 .

It is here always understood that the index is in the proper range, and that m e N. It is immediately clear that the eigenvalues of T. are identical with those of S, and S.*. We shall now exhibit a number of important properties of the matrices T..


As before , we are re la t ing k = k ( n ) E IN to the o d d na tu ra l n u m b e r n by

2k-1 < n < 2 k,

and we are a s soc ia t ing wi th n the two n u m b e r s

n ' . = 2 k _ n , n ' : = n _ 2 k-1 .

L e m m a 5.1. The followin9 relations hold for n > 5 :

(5.2) '(") - t) ") j = 1, n" - 1 - 2 k - I + j - - , - - . , ,

( 5 . 3 ) -2k'(")- I - - to t ~ - 2 k ,

(5.4) '~") - tJ ") .., 2 k-z ~ , j = l , . - 1 ,

(5.5) ,(,) _ ,(,) ,2 k- i "j - ' 2 k - ' - / , j = l . . . . - 1 ,

(5.6) t~(")-- tJ "') , j = 1 . . . . . n " - 1 ,

(5.7) t j(")-- -- tJ "') , j = 1 . . . . . n ' - - 1 .

Proof. (5.2): F o r j odd , this is clear . N o w , f o r j even, we have

l < j = 2 U ( 2 m - 1 ) _ - n - l - 2 k-1 < 2 k - l - 1 ,

so tha t 1 < # < k - 2 . Hence ,

2 k - l + j = 2 U ( 2 m - - l + 2 k - l - " ) = 2 U ( 2 m * - - l ) , m * ~ N ,

and the resul t fo l lows f r o m the de f in i t i on o f t! "). (5.3): W e have n = (n m o d 2 k) = 2 k- 1 + (n m o d 2 k- 1). Hence ,

t~'~)- 1 = 2(n m o d 2 k- 1) _ (n m o d 2 k)

= _ 2 k + 2(2 k- 1 + (n m o d 2 k- 1)) _ n

= n - 2 k

= t~o") _ 2 k"

(5.4) : As (5.2), bu t n o w us ing 1 < # < k - 3, i f k > 3. I f k = 3, we on ly h a v e j = 1, an ' o d d ' case.

(5.5): T h e case j o d d is aga in obv ious . Le t j be even wi th 1 < j = 2 ~ ( 2 m - l ) < 2 k- 1 _ 2, so tha t 1 < # < k - 2. Hence ,

2k-1 _ j = 2 u ( 2 ( 2 k - 2 - U _ m + 1 ) - 1)

= 2 " ( 2 m * - 1).

Since the n u m b e r on the left is pos i t ive , we d e d u c e m* ~ N , a n d the resul t fol lows. (5.6): W e h a v e

n + l n " + l + 2 k-1 n " + l _ _ _ = + 2 k-2 ,

2 2 2


which gives the result f o r j odd. Next, f o r j even, write 1 < j = 2~(2m - 1) < n" < 2 k- x, so that 1 < #_-< k - 2. Hence,

(n rood 2 u) = (n - 2 k- l) rood 2 u ,

(n mod 2u+l) = ( n - 2k- x) mod 2 ~+l '

which proves this case. (5.7): We have

n ' + l 2 k - - n + I n + l - - = = 2 k - l + 1 _ _ _

2 2 2 which shows

' ,

and hence the assertion f o r j odd. F o r j even, we use the definition o f tJ ") and (2.10), (5.1) to obtain ( j = 2 U ( 2 m - 1),/~> 0):

tJ") = o-~")

( 2 . t O ) . . _ k = 1(2 - n) rood 2 u + 1) _ 2 ((2 k - n) rood 2")

= - t ) . ' ) . []

To simplify the p roo f for the next lemma, we introduce the following periodic doubly infinite sequences:

a(") = t) ") , j = 1 .... ,2 k- 1 2 k- 1 j + m 2 k - i , meT/, < n < 2 k,

which have the following properties (n > 5):

(5.8) a~"~-,r j ~ Z , - - j - - - _ j ,

(5.9) at."")=a(. ") 1 < j < 2 k-1 - 1 n">n',

(5.10) a~"'~ = - a ~ "~ , l < j < 2 k - 1 -- 1, n"<n'.

To prove (5.8), we assume without loss o f generality that l < j < 2 k - 1 - 1 (periodicity). But then we have by (5.5):

a ( n ) _ r , (n) _ ~ ( n ) _ j - - - - _ j + 2 k - 1 - - t ~ j .

Proper ty (5.9) has to be proved only for j = n " , . . . . 2 k - i - 1 because o f (5.6). The assumption n" > n' implies that 2 k - 2 < n't < 2 k - 1, and hence that a~. "') has period 2 k- 2. Then, using 1 __<j-2k-2< 2 k - 2 in the following application o f (5.4), we get

at,") _ ,~(,") (s.6) (,) (5_.4) a~.) j - - ~ j - - 2 k - 2 = a j _ 2 k - 2 �9 .

A similar a rgument proves (5.10). We note that by (5.3),

a ( n ) __ t ( n ) __ ")k 0 - - ' 0 ~ '

and hence, using (5.8),

�9 _ _ (n ) _ _ k A n . - ( a j _ i ) o ~ _ i , j < = n _ 1 - T . - 2 I , .

It is convenient to associate with A, the matrix

B . :

O < j ~ n ' - i


The fol lowing in fo rma t ion abou t the s t ructure o f A, is basic for the induct ive a rgument in the next section.

Lemma 5.2. L e t n >_ 5. T h e n we have

T . , , - - 2 R - 1 I . , , M T , , , - - 2 k - l I . , ,

where

Proo f . We write

M = { Bn'' n ' < n " ,

- B t . , , n' > n " .

M l l M12 M13~

\M~I M~ M~/

where Mix , Ml3 , Max, M33 and n" x n" matr ices, Mi2,/1432, M~l , Mi3 are n" x n ' matr ices, and M22 is a n ' x n ' matr ix . Since n' + n " = 2 k- 1, it follows immedia te ly f rom the per iodic i ty o f a) ") and (5.8) tha t

M l l = MI.3 = M31 = M 3 3 ,

M 1 2 = M 3 2 = M ~ l = M ~ 3 .

Hence, we can confine ourselves to the ident i f icat ion o f M l l , M z z , M l z .

As for M l l , we have

m l 1 = r ~ ~, j-i/O<=i,j<=n"-I

=(alT)-il)o<=i.j<=n,,-1

= T o e p ( t (o") -- 2k, t ~ ") , . . . , t (") ," - l J ~

= T o e p (t~o"")-2 k-1 , t~ "') , �9 �9 �9 ,."("")- i J ~ (by (5.6))

= T . , , - - 2 k - l I , , .

F o r M22 we have

_ (n) M22 - (a j_ i)o ~ i,~ ~ . ' - 1

= Toep (tot,)_ 2k, ,( . ) ,(")

= Toep ( - - to ("'), -- t~"'), �9 �9 -, -- '("')-.,- iJ ~ (by (5.7))

-- Tn,.

n' < n", which 2 k - z < n . < 2 k-1. We also N o w let implies n ' = 2 k - i _ n", which means n' = (n") ' ; likewise, ,, ,, 1 ,, ( n ) = i - (n - n ' ) . We then have

M - t a (~) ~ B.,,=['a(",'~ ) , ~o_<i-<."-1- 12 - - \ n " + j - i / O < i < n " - i ' O~=j~n, 1 ~ n 2 n +j i ) o < j < n - 1

If we set s = n" + j - i, then 1 < s < n ' + n" - 1 = 2*- 1 _ 1, and we have to show

a(") = a ("') 1 < s < 2 *-1 - 1 s s n'+n" ~

2

observe tha t


But a} "') has per iod 2 k-2 n'+n" - and we are left with 2 '

at.) _ a("") 1 < s < 2 k- 1 _ 1 s - - - - s , - - __

which is (5.9). Hence, M12 = B.,, in this case. A similar a rgument works for n' > n", where (n ' )"=�89 and we have to use (5.10) instead of (5.9). []

6 Proof of Corollary 2

As we saw above, only par t iii) o f Corol lary 2 for n odd remains to be proven. We shall prove the following, more precise statement.

L e m m a 6.1. Let n e N be odd. Then the following hoM: i) 2 = 1 is the smallest eigenvalue of T,.

ii) Every eigenvalue 2 o fT . , except for 2 k, is also eigenvalue ofT. , , with the same multiplicity.

iii) To every eigenvalue 2 o f T. there exists a polynomial e.(2, z) o f degree < n - m . ( 2 ) , where m.(2) is the multiplicity o f 2, such that

zSe,(2, z), 0_<s_<m, (2 ) - 1,

represents an eigenvector e~")(2) of T . for 2. iv) These polynomials obey the followin9 recursion:

e l ( 1 , z ) = l ,

e3(1,z) = 1 + z + z z ,

zk-, ,- ,, , ( l + z )e. , , (2,z)+z e . , (2 ,z ) , n > n , n > 5 , 2 < 2 k, e. (2, z) = ) / 2 k- 1 \

L ~ I - - ~ - - - ) ( 1 +z2k-')e. , ,(2,z)+z""e.,(2,z), n '>n" ,

k - 1 ( n + l 2 j ' ~ k - I e.(2k, Z) = 1--I l+d j ( ( - -1) - -~- - -z ) / ' n ' = E d~ 2j"

j=O ~ .it j = O

In this formula, i f2 does not happen to be an eioenvalue ofT.,,, we set e.,,(~,,z)--0. v) The vectors e~ ") satisfy the following relations:

B,- e~"')(2) = - 2e~")(2), m . , ( 2 ) . 0 ,

B. t" e~ ")(2) = ( 2 - 2k)e~ "')(2), m.(2) 4: 0,

for s = 0 , 1 ... . , m . ( 2 ) - - 1.

Proof The t ruth o f this l emma is readily checked for n = 1,3, which permits us to start an induction. Note that i) is contained in ii). Assume we are done up to n - 2, and let 2 k-1 < n < 2 k, k > 3 . We now have to distinguish between two cases.

a) n">n'. In this case we have 2 k - 2 < n " < 2 k - 1 , and hence

(n")' = n' = 2 k-1 - -n" .

Thus, by assumpt ion ii), applied to n", every eigenvalue 2 o f T., is also eigenvalue o f T.,,, with the same multiplicities (the largest one, 2 k- 1, o f T.,, is not a m o n g them !).


Hence, for all eigenvalues 2 of T,, we find from v) (applied to n"),

B.,,- el"')(2) = - 2e(~"")(2),

B.L,. el"")(2) = (2 - 2 k- ' ) e~"')(2),

0 _ < s < m . , ( 2 ) - 1 . Our recurrence iv) says that

e~")(2) =~ el")(2)} , s = 0 .... , m , , ( 2 ) - 1,

\ el.")(2)/

should be eigenvectors for T. and 2. By Lemma 5.2 and the assumptions we find that

/ ( T.,, -- 2 k - l I.,,) . e~"")(2)\

(T . - 2kI . ) �9 e~ ") (2) = 2 [ B.',.- e(~ "') (2) ]

\ ( T . , , - - 2 k - l I , , , ) - e("")(2)/

= ( 2 - 2k)e(~")(2),

[8..,- ~I"')(2) \ + 1-

\B., ," el"')(2) /

which proves this assertion. Since we do know already that T. has the eigenvalue 2 k with multiplicity n - n' and the mentioned polynomial en(2 k, z) (see (4.2) and recall the sign changes made in the definition o f t)")), the p roo f o f iv) is completed for the present case a). We need to prove v). We have by Lemma 5.2

/ 8: e;""(2)\ 8 . - e :" (2 )= | - I = - 2e:)(2)

\ B..-e~ ")(2) / for all eigenvalues of T,. by the assumptions, and similarly,

B~-e]">(2) - ' - e ~ " " ) ( 2 ) - T . , . (.') - B.'., e~ (2)+B.,. �9 e~"")(2)

= [(2 - 2 t - x) _ 2 + (2 - 2 k- l ) ] - e(. ") (2)

= ( 2 - 2 a) - e("')(2)

for all eigenvalues of T,,. Fo r the remaining case /],----2 k we note that e~")(2 k) is eigenvector to the eigenvalue 0 of the symmetric matrix A . = 7 " . - 2 k I . . This implies by Lemma 5.2 that

B~ �9 e(s")(2k) = 0, O < _ s < _ m , ( Z k ) - - l ,

which completes the p roo f o f case a). b) n " < n ' . In this case we have 2k-Z<n'<2 k-l, and hence

(n ') ' = n" = 2 k - 1 _ n ' .

Thus, by assumption ii), applied to n', every eigenvalue 2 of T.,, is also eigenvalue o f T.,, with the same multiplicities. However, T., has in addit ion the eigenvalue 2 k- 1 of multiplicity n ' - n " . Here we find from the assumption that

B.,-e~"")(2) = - 2e~"')(2), 2 < 2 k-1 ,

B.L. el""(2) = (2--2k- 1)e(f)(2), 2 < 2 k-x ,

590 A. C 6 r d o v a et al.

for 0_< s < m,, ( 2 ) - 1. The claimed eigenvectors for T, and 2 now have the form

l ( 2 k- l \ \

e~")(2) = e~"')(2) ] ' s = 0 ..... m,, ().) -- 1, !

2 k-1 \ 1 1 ~-/e~"")(2) /

z / /

where e~"")(2 k-l) =0. Using Lemma 5.2 and the assumptions, we now find

( -2t-1I""). el"")(2)\ 2t_ 1 (T.,, --B., "e~"")(2) } (T. - 2kI.) "e~")(2) = 2 ( \1 -

(T.. 2 k - 1

= (2-- 2/)e]")(2)

//-- B. t, �9 el"')(2)\

+ { - T,, "e!]')(2)} \ - - S~," e] "(2) /

(2 eigenvalue of T.,), which proves iv). v) is verified in a similar fashion as in ease a). []

It is clear that Lemma 6.1 completes the proof of Corollary 2. By a backward induction n~n'~(n')'--,... (check the alternating binary representations of n and n') we deduce easily that Corollary 2 implies Theorem 1, and we are done.

7 Proofs of Theorems 4 and 5

The polynomials e.(z) as described in the introduction are now immediately identified with the polynomials e,(1, z), which represent the eigenvector of the simple eigenvalue 1 of T,, n odd. This eigenvalue is the smallest one of T,, and thus T, - 1, is positive semidefinite (with eigenvalue zero), and therefore we must have

. . - 2 2j "-1~ E

- j=0 1--ejz where

2 j > 0 , les[=l, j = 0 ..... n - 2 , and

n - 2

~_, )~j=n--1. j = 0

From Lemma 4.1 we can now deduce that we must have

n - 2

1-I (~j -- Z) = C e . (z) , j=O

since the product on the left represents, up to a factor, the only eigenvector to T , - I . , exactly as does the polynomial on the right. This proves Theorem 4. Theorem 5 is also a consequence of the fact that Sm+l -Im+1 is positive semidefinite, using the representation (1.1) of the xtm+l) We omit the simple details.


Acknowledoements. The second author is indebted to Professor G. Opfer for bringing a preprint of [4] to his attention, for initial computations of the singular values of V,, and for discussions of the problem at hand. The authors also like to thank Ruth Ruscheweyh, who did some of the computational work necessary to get better insight into the eigenspace structure, and also derived the recursion (1.4) from the numerical data obtained.

References

1. Gautschi, W.: Norm estimates for inverses of Vandermonde matrices. Numer. Math. 23, 337-347 (1975)

2. Gautschi, W., Inglese, G. : Lower bounds for the condition number of Vandermonde matrices. Numer. Math. 52, 241-250 (1988)

3. Kuipers, L., Niederreiter, H.: Uniform distribution of sequences. New York: Wiley 1974 4. Reichel, L., Opfer, G. : Chebyshev-Vandermonde systems. Math. Comput. (to appear)

Documents

Vandermonde matrices on the circle: Spectral properties ...Vandermonde Matrices on the Circle: Spectral Properties and Conditioning Antonio C6rdova 1' *, Walter Gautschi 2'**, and