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Apr 1, 1999 - To cite this Article: Gau, Hwa-long and Wu, Pei yuan (1999) 'Lucas' theorem refined ', Linear and Multilinear Algebra, 45:4, 359 — 373.
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Lucas' theorem refined

Hwa-long Gau a; Pei yuan Wu a a Department of Applied Mathematics, National Chiao Tung University, Taiwan, Republic of China Online Publication Date: 01 April 1999 To cite this Article: Gau, Hwa-long and Wu, Pei yuan (1999) 'Lucas' theorem refined ', Linear and Multilinear Algebra, 45:4, 359 — 373 To link to this article: DOI: 10.1080/03081089908818600 URL: http://dx.doi.org/10.1080/03081089908818600

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Linear and Multilinear Algebra. Vol 45. pp 359-373 Keprlnts available directly from the publisher Photocopymg perm~ttedby license only

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Lucas' Theorem Refined* HWA-LONG GAU and PEI YUAN WU+ Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan, Republic of China Communicated by T. Ando (Received 24 April 1998)

We prove a refined version of the classical Lucas' theorem: i f p is a polynomial with zeros a l , . . . ,a,+ all having modulus one and q is the Blaschke product whose zeros are those of the derivative p', then the compression of the shift S ( d ) has its numerical range circumscribed about by the (n + 1)-gon al . . . a,, with tangent points the midpoints of the n + 1 sides of the polygon. This is proved via a special matrix representation of S(4) and is a generalization of the known case for n = 2.

Keywords: Compression of the shift; numerical range; dilation A M S Subject Classification: 15A60, 47A 12

A classical result in complex analysis, variously attributed to Gauss, Lucas, Grace and others and usually called Lucas' theorem, says that zeros of the derivative of a polynomial are all contained in the convex hull of the set of zeros of the polynomial. A more refined assertion for polynomials of degree three, due to Siebeck [5] ((3also [4, p. 9]), is that if the degree-three polynomial p has zeros a l , a z and a3 and the derivative p' has zeros bl and b2, then there is an ellipse with foci at hl and bZ which is circumscribed about by the triangle Aulu2u3 with

*Dedicated to John B. Conway, the thesis advisor of the second author and the mathematical grandfather of the first, on his 60th birthday. tCorresponding author. e-mail: [email protected]

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HWA-LONG GAU AND PEI YUAN W U

tangent points the midpoints of the three sides of Aalaza3. The purpose of this paper is to prove a generalization of this latter result to degree-rt polynomials. Recall that an n-dimensional operator T is said to be in the class S,, if T IS a completely nonunitary contraction, that is, T has norm at most 1 and has eigenvalues all in D ( = {z E C : / z / < I}), with rank (1 - T * T ) = 1. Operators in S,, are exactly the so-called compre,vsions of the shijt S(+), where 4 is a finite Blaschke product with n zeros (counting multiplicity). Here S($) denotes the operator on H2C3 $H2 defined by S(4)f = P(zf ( z ) ) , where P is the (orthogonal) projection from H~onto H2@ $H2.In [I], we initiated the study of the numerical range of operators in S,,. This was continued in [2] and [3]. Among the properties we obtained is a one-to-one correspondence between (n + 1)-dimensional unitary dilations of S(q5) and (n + 1)-gons which are inscribed in the unit circle 3D and circumscribed about W(S(4)), the numcrical range of S(4) (4.[ l , Theorem 2.11). The generalization alluded in the preceding paragraph is that if p is a polynomial with zeros ul, . . . ,a,,+ all having modulus one and 4 is the Rlaschke product whose zeros are those of the derivative p', then S($) dilates to the unitary operator diag ( a l , . . . ,a, + ,) and, in particular, the numerical range of S(4) is circun~scribedabout by the (n+ 1)-gon u1 . . . u,,, with tangent points the midpoints of the n + 1 sides of the polygon (Theorem 2.1 below). In fact, this is obtained as a consequence of a more general result on a characterization of operators T in s,, with numerical range W ( T ) circumscribed about by a given (n + 1)-gon inscribed in aD (Theorem 2.5 below). These will be proved via a special matrix representation of operators in S,,, which we present in Section 1 below. Section 2 then contains the above asserted results.

1. MATRIX REPRESENTATION

We start with a matrix representation for operators in SZ,whose easy proof we omit.

LEMMA1 . I The matrix [: f]is in S2 if and only if 101, icl < I and h sutisfirs lh12 = (I-1u2)(1 - 1 ~ 1 ~ ) .

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361

Let T be an n x n matrix and CY a subset of (1, . . . ,n). The principal submatrix T[a] of T is the matrix obtained from T by deleting its rows and columns numbered by ( 1, . . . ,n) \ a . We now proceed with the main theorem of this section.

THEOREM 1.2 The following statements are equivalent for an upper triangular matrix T = [tii]yj=l(n 2 3): (a) T is in S,; (b) T[l, . . . ,n-1] and T [2, . . . ,n] are in and tl, = -t& (1 - it11 i2)/~12; 2 (c) lt;;l < I ,for i = 1 , . . . ,n, ltl,i+112= (1- 1tiil2)(1- I t i + l , i + l l )for i = 1, . . . , n - 1 and

Proof (a)+(b). Since T[1,. . . ,n - l] and T[2,. . . ,n]* are restrictions of T and T*, respectively, it is easily seen that both are in Sn-l.Then T[2,. . . ,n] is also in S , , , . To prove the expression for tl,, note that T[1,2] is in S2 and so t12 # 0 by Lemma 1.1. Since the matrix

has rank one, we deduce that (1 - itll j2)(-t12tln- iZ2t2,,)= (-tlltln) (-i12t11). A simple computation shows that 11, = -t2,G2(1 - Itl, l2)/tI2 as asserted. (b)+(c). Assume that (b) holds. As in the proof of (a)+(b), we deduce easily that T[i, . . . ,j ] is in SjPi+I for any 1 5 j - i n - 2. For j = i + l ( i = 1, . . . , n - I), thisimplies by Lemma 1.1 that ltiil < 1 and lti,i+1 / 2 = (1 - I tii12)(1 - 1 ti+ 12). On the other hand, if 1 < i 5 j - 2 < n - 2, then applying the implication (a)+(b) to T[i, . . . ,j ] for


3) from Theorem 1.2(c). Indeed, in this case, - lt,+l,l+ll 2 ) l I 2 (we may assume that since t, ,+I = ( 1 - 1 t,,12)112(1 t,.,+ I > 01, we have

as desired. On the other hand, reversing the above argument yields, by Theorem 1.2 (c), the proof for the sufficiency.

2. LUCAS' THEOREM REFINED

The main result of this section is the following

THEOREM 2.1 Let p be a polynomial with distinct zeros a,, . . . ,a,+ all having modulus one. If 4 is the Blaschke product whose zeros are those of the derivative p ' , then the numerical range of S(4) is circumscribed about by the ( n + 1)-gon al . . . a,,, with tangent points the midpoints of' the n + 1 sides of the polygon. Note that in [4, p. 1 1 , Theorem 4.21 it is shown that for a polynomial p of degree n + 1 the zeros of p' are the foci of a curve of class n inscribed in the (n + 1)-gon formed by the zeros of p. The above theorem shows that the curve can be taken to be the boundary of the numerical range of some S(4) when the zeros of p are all distinct and

HWA-LONG GAU AND PEI YUAN WU

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having modulus one since the eigenvalues of S($) are easily shown to . thank A. L. Duarte be the foci of the boundary curve of W ( S ( @ ) )We for showing ub the above reference. For the case n = 2, we obtain after scaling and translation the following

Co R O L L A R Y 2.2 Let p he a degree-three polynomial with d i s t i ~ ~ c t zeros a l , a2 and a3. lj b l and b2 are zeros of p ' , then there is a (unique) ellipse with fbci at 6 , and b2 which is circumscribed about by Aala2a3 tvith tangent points the midpoints of the three sides of Aulu2a3. To prove Theorem 2.1, we start with the following lemma, whose easy proof we omit.

L E M M A2.3 ZfS und T a r e (n + 1) x (n + 1) m ~ t r i r r swhich difer only at the

(PI

+ 1, IZ + 1)-entry:

\$,hereA, B and C are n x n , n x 1 and 1 x n matrices, respectively, then detS-detT = (s- t)drtA.

-

LEMMA2.4 Let T h e an operator in S,, represented UJ in Corollary 1.3. Then any ( n + 1)-dimensionul unitary dilation of' T is ztnitarily equiva~l, lent to a matrix of the form U [ t , ] ~ ~w7here t,,+l,j= (

1 -,,2)12 ( - 7 )

for 1 5 j 5 n.

k= l I

,

(

1

i

2

)

( i ) for 1 5 i < n k=~+l

and

6, being real, with charucteristic polynomial

e

t

-

u

=z

-t

k= 1

- e e l k- 1

-

ikkz).

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365

Proof If T is represented as in Corollary 1.3 and U = [t&Ll is any (n + 1) x (n + 1) unitary dilation of T, then since the first column of U has unit length we have t,+l,l = ei6(1- Itl1l2)'I2 for some real 0. The orthonormality of the column vectors of U then dictates that t,+ 2 5 j 5 n, be as asserted. Similar considerations of row vectors of U yield that ti,,+ 1, 1 5 i 5 n, equals the asserted expression multiplied by some ei" for real a. Since the last column of U has unit length and is 1 is orthogonal to the other columns, we infer that el" = 1 and t,, equal to the asserted expression. To prove for the characteristic polynomial, we proceed with induction on n. If n = 1, then

and thus det (zI - C T ) = (Z- t11)(z

+ e1'il1)

-

eLs(l- 1tlll2)

= Z ( Z - t l l ) - eid(l - t,,,).

Assume that the expression for the characteristic polynomial is true for n - 1. Expanding det (zI- U ) by minors on the nth row yields det (zI - U )

=

(z - t,,) . det Vl (z)

+ (1

-

/ t,,

l2)lI2. det V2 (z),

where Vl(z) (resp. V2(z)) is the n xn matrix obtained from z I - U by deleting its nth row and nth column (resp. nth row and (n + 1)th column). We then apply Lemma 2.3 to det Vl(z) and det V2(z) to obtain det (zI- U ) = ( z - I,,)(-i,,,)

+ (1

-

I

det (zI- U')

/tnn12)[det (zI- U')

+ (-z)det

(zI - TI)],

where T' is the (n - 1) x (n - 1) upper triangular matrix in S,-I and U' is its n x n unitary dilation. with eigenvalues t , . . . , tn-1

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HWA-LONG GAU AND PEI YUAN WU

Hence det ( z I - U ) = (1 - i,,,z) det ( z I - U') z [ ( z- t,,) - ( 1 - i,,z)] det ( z I - T')

+

where the second equality follows by the induction hypothesis. This completes the proof. The next theorem gives a characterization of operators T i n S , with W ( T )circumscribed about by a given ( n + 1)-gon inscribed in aD.

THEOREM 2.5 Let al, . . . ,a,+ 1 be n + 1 distinct points in 3D. Then an operator T in S, has its numerical range W ( T ) circumscribed about by the (n + 1)-gon-a l . . . a,+ 1 if and only if it,^ eigenvalues b , , . . . ,b , satisfy CY, = PI C+,+~B,+~-~,J' = 1,. . . , n , where o l ~(resp. P,) denotes the jth elementary symmetric function of the ax's (resp, bk3s) given by n+l ( 2 - ak) = (- 1) /n,z "+IF/(resp. ( z - b k ) = Cy=o(1) 4 z "-1 and /j'n+l = 0 ) .

+

cJ"'~

nip,

Proof If T is in S,, then by Lemma 2.4 the characteristic polynomial of its ( n + 1)-dimensional unitary dilation can be written as

for some real 0. Note that W ( T )is circumscribed about by the ( n + 1)gon u l . . . a,, , if and only if the matrix diag ( a l ,. . . ,an+ is a unitary

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367

dilation of T (cf. [ l , Theorem 2.11). However, the latter is equivalent to the condition that ( z - ak) coincides with (*) or, equivalently,

for j = 1,. . . , n + 1. When j = n + 1, we have an+l= & + I + (-l)"ei0po = (-l)"eie and thus the above equalities are the same as a, = Dj an+lPn+l-j,j = 1,. . . ,n. This completes the proof.

+

Now part of Theorem 2.1 follows as an easy consequence.

Proof of Theorem 2.1 Let b l , . . . ,b, be the zeros ofp', which are also the eigenvalues of S(Q). Since

we derive that p, = ((n + 1 -j)/(n + l))aj, j = 1,. . . ,n, where 9 (resp. 8;) is the jth elementary symmetric function of the ak's (resp. bk1s)We obtain

1 ((n + 1 - j)u, n+l

-

+jcyj) = a;

for j = 1,. . . , n . Thus Theorem 2.5 implies our assertion on the numerical range of S(4) circumscribed by the (n + 1)-gon a , . . . a,, I . The remaining part of the proof will be devoted to showing that the tangent points are the midpoints of the n + 1 sides of the polygon. For this, we arrange the a,'s counterclockwise around the unit circle and let T be the (unique) operator in S,, whose numerical range is circumscribed by the (n + 1)-gon a l . . . a n + with tangent points (1/2)(a, +a,+l), j = 1, . . . , n + 1 ( a n + z - a l ) (cJ[I, Theorems 3.1 and 3.21). We will prove that T = S(4).

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Note that from the proof of [I, Theorem 3.11, the operator T i n S,, whose numerical range is circumscribed by an ( n + I)-gon a1 . . . a, + 1 with tangent points v, = tlal + ( 1 - tI)al+ (0 < t, < I), j = 1 , . . . , n , is given by T = PUIK, where U = diag ( a l , . . . ,u,,+ ,), K is the subspace Of + 1 spanned by the n vectors CI1

and P is the orthogonal projection from c"+' onto K. In the present case, t, = (112) for all j and K is spanned by

'

Consider P as an operator on Cr'+ and let T' the (n + 1) x (n + 1) matrix representation

=

PUP. Note that P has

since the latter satisfies p2 = P = P* and r a n k P = t r P = n, and [0,,,0.,,11 O..,O] ' . , J = 1 , . . . , n , being equal to the sum of the jth and j th ( j + 1)th columns of P, is in the range of P. A simple computation

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LUCAS' THEOREM REFINED

yields that

where tl,

=

+

n2ai ai (- 1)"'" (,(ai

+

,j)

-

ai,,)

ifi = j, otherwise.

Here a,(resp. a,.,) denotes the sum of those a i s with 1# i (resp. 1 # i,j). Let e l , . . . , c, be the eigenvalues of T, and let a k (resp. yk), k = 1, . . . ,n, be the kth elementary symmetric function of a l , . . . ,a n + (resp. c l , . . . ,cn). To complete the proof, we need show that yk = ((n + 1 - k)/(n + l))cxk for all k. By a classical result of Cauchy, n/k equals the sum of all k x k principal minors of a matrix representation of T. Since the eigenvalues of T and T' differ by a zero, yk also equals the corresponding sum of TI. We now proceed to compute one of such principal minors of the latter:

M = det

(m 1

1 2k

det [tijI:j=I

To do this, add the first row multiplied by (- 1)' to the ith row, i = 2 , . . . ,k, in [ t c j ] f j =to , obtain

where

M=

1

(n

+ 1lkf1 det [s:.]:,=,,

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HWA-LONG GAU AND PEI YUAN WU

where Sij

(

n

+ I)

i f i = I, otherwise.

'

' (resp. Next, we add the second column multiplied by (the jth (resp. first) column, j = 3,. . . ,k, in [s b] to obtain

-

n) to

where

and -aj)

Sji(n

+ I)ai

if i = I and 3 < j < k , ifi=2and3