Some Properties of the Q-Adic Vandermonde Matrix

The Electronic Journal of Linear Algebra. A publication of the International Linear Algebra Society. Volume 1, pp. 18-33, June 1996. ELA

ISSN 1081-3810. http://gauss.technion.ac.il/iic/ela

SOME PROPERTIES OF THE Q-ADIC

VANDERMONDE MATRIX

y y

VAIDYANATH MANI AND ROBERTE.HARTWIG

Abstract. The Vandermonde and con uentVandermonde matrices are of fundamental

signi cance in matrix theory. A further generalization of the Vandermonde matrix called the

q -adic co ecient matrix was intro duced in [V. Mani and R. E. Hartwig, Lin. Algebra Appl.,

to app ear]. It was demonstrated there that the q -adic co ecient matrix reduces the Bezout

matrix of two p olynomial s by congruence. This extended the work of Chen, Fuhrman, and

Sansigre among others. In this pap er, some imp ortant prop erties of the q -adic co ecient

matrix are studied. It is shown that the determinant of this matrix is a pro duct of resultants

like the Vandermonde matrix. The Wronskian-like blo ck structure of the q -adic co ecient

matrix is also explored using a mo di ed de nition of the partial derivative op erator.

Key words. Vandermonde, Hermite Interp olation, q -adic expansion.

AMSMOS sub ject classi cation. Primary 15A15, 15A21, 41A10, 12Y05

1. Intro duction. Undoubtedly, the Vandermonde matrix is one of the

most imp ortant matrices in applied matrix theory. Its generalization to the

con uentVandermonde matrix has numerous applications in engineering. This

concept was further generalized to the q -adic Vandermonde matrix in [9], ex-

tending the work of Chen [4], Fuhrman [7] and Sansigre [3] among others. In

this pap er, weinvestigate some of the prop erties of the q -adic Vandermonde

matrix. In particular, we shall evaluate its determinant and analyze its blo ck

structure. The former extends the Vandermonde determinant, while the lat-

ter shows a remarkable analogy to the layered Wronskian structure of the

con uentVandermonde matrix.

It is well known that the con uentVandermonde matrix for the p olynomial

x= x ;

i=1

deg = m = n, can b e constructed by forming the Taylor expansion of

Received by the editors on 5 Decemb er 1995. Final manuscript accepted on 29 May

1996. Handling editor: Daniel Hershkowitz.

North Carolina State University, Raleigh, North Carolina 27695-8205, USA

[email protected], [email protected]. 18

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Prop erties of q -adic Vandermonde Matrix 19

x in terms of the p owers of x , i.e.,

1 = 1

x = +1x

i i

2 2 2

x = +2 x +1x

i i i

t t

x + ::: x = + t

n 1

n1 j

x = x :

j =1

j 1

If we set x = q and denote the column containing the rst m co ecients

i i i

t t

in the ab ove expansion of x by[x ] , then

q ;m

i i

3 2

7 6

2 : [x ] =

7 6

q ;m

i i

7 6

5 4

tm +1

m 1

In case t

of by zero. Collecting these co ordinate columns, wemay construct the nm

i i

matrix as

i h

n T

[1] [x] ::: [x ]

: =

q ;m q ;m q ;m

i i i i i i

In other words, a is made up of rst m columns of Caratheo dory's n n

n1;n1

k j

. Lastly, the n n con uent binomial matrix a= a

k;j =0

Vandermonde is de ned by matrix

n n n

3 : ; ::; ; =

s 2 1

m m m

2 1

To extend this concept further, we recall the q -adic expansion of a given

p olynomial f x relative to a monic p olynomial q x. Indeed, the following

lemma from [2] is elementary.

Lemma 1.1. Let q x be a monic polynomial of degree l over F.Iff x

is any polynomial over F, then there exist unique polynomials r x over F,of

degree less than l , such that

f x= r x[q x]

j =0

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20 Vaidyanath Mani and Rob ert E. Hartwig

for some nite non-negative integer k .

Let us now use the ab ove lemma to de ne the q -adic co ordinate column of

a p olynomial. For an arbitrary p olynomial f x= a +a x+:::+a x , its stan-

0 1 k

dard co ordinate column relative to the standard basis B = f1;x;:::;x g

is de ned and denoted by

3 2

7 6

: f =[f ] =

7 6

5 4

Now for a given pair q; m made up of a p olynomial q x of degree l and a

p ositiveinteger m, the q-adic expansion of f xmay b e written as

m1 m

f x= r x+r xq + :::+ r xq + q ?;

0 1 m1

where ? denotes the p olynomial quotient after division by q . The q-adic

co ordinate column asso ciated with f x for the pair q; misnow de ned as

2 3

6 7

[f ] = ;

6 7

q;m

4 5

lm1

where r is the standard co ordinate column for r x. It should b e noted that

i i

i if the degree of f x is larger than q x ,we simply ignore the quotient

after division by q .

ii the statements q jf and [f ] = 0 are equivalent, i.e.,

q;m

4 [f ] =0 , q jf:

q;m

iii if deg f = k , then [f ] is a k +1 1 column vector.

Throughout this pap er, we shall denote the i unit column vector by e .

1.1. The q -adic Vandermonde matrix. We are now ready for the

construction of the q -adic Vandermonde matrix. In order to rep eat the con-

struction given in 1 over elds which are not algebraically closed, let x

b e a foundation p olynomial with the factorization

x= q

i=1

with deg q =l 1 and deg = m l = n. Again, using the q-adic

i i i i

i=1

expansion of the standard basis elements x ,wehave

m 1

t j m

x = r xq x + q x ?:

i i

j =0

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Prop erties of q -adic Vandermonde Matrix 21

The q -adic co ordinate column [x ] maynow b e formed as

q ;m

i i

2 3

6 7

[x ] = 5 ;

6 7

. q ;m

i i

. 6 7

4 5

i;m 1

t t

where r is the standard co ordinate column for r x. Lastly, collecting

ij ij

co ordinate columns, we set

i h

[1] [x] ::: [x ]

6 W =

q ;m q ;m q ;m

i i i i i i

l m n

i i

and construct the n nq-adic Vandermonde matrix W as

2 3

6 7

2 T

6 7

W =

4 5

:::

It is clear that when the q 's are linear p olynomials then, 2 reduces to 6 and

T T

W to .We shall as such refer to the matrix W as the q -adic Vandermonde

matrix. If W is nonsingular, it solves the q -adic interp olation problem for

p olynomials as discussed in [9]. Moreover, for any p olynomial f x with degree

less than n and standard co ordinate column [f ] , it is clear from 5 that

7 W [f ] =[f ] ;

st q ;m

i i

In other words, W acts as a change of basis matrix from the standard to the

q -adic co ordinates. This is a crucial observation which will b e used in the

next section.

To analyze W further, we need the relation b etween the q-adic co ordinate

columns of [f ] and [ f ] . This relation is b est handled via the concept

q;m q;m

of a p olynomial in a hyp ercompanion matrix whichwe shall now address.

1.2. Some basic results hyp ercompanion matrices. If q x= a +

a x + ::: + x is a monic p olynomial of degree l , then the hyp ercompanion

matrix for q x is the m m blo ck matrix

2 3

L 0 : :: 0

6 7

N L 0 :: 0

6 7

: : : : 0

H = 8 ;

6 7

4 5

0 0 N L 0

0 0 0 N L q

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22 Vaidyanath Mani and Rob ert E. Hartwig

and L is the companion matrix of q x de ned by where N = E = e e

q 1;l 1

3 2

0 0 0 ::: a

7 6

1 0 0 ::: a

7 6

0 1 0 ::: ::

7 6

: L =

7 6

: : : ::: ::

7 6

5 4

: : : ::: a

k 2

0 0 0 ::1 a

k 1

The key result whichwe need is the following lemma dealing with p olynomials

of hyp ercompanion matrices.

Lemma 1.2. Let px be a monic polynomial of degree l and let H =

H [px ] be the ml ml hypercompanion matrix inducedby px.Iff x is

any polynomial over F with p-adic expansion

m1 m

f x= x+ xpx+ :::+ xpx 9 + px ?;

0 1 m1

and coordinate columns , then

3 2

A 0 ::: 0

7 6

A A 0

1 0

7 6

10 = U; JU; :::; J U ; f H =

7 6

. . .

. . . .

5 4

. . .

A A ::: A

m1 m2 0

where A = L , A = L + L + N L , i =1;:::;m 1,

0 0 p i i p i1 p i1 p

N = E and U =[a;Ha;:::;H a] with J = J 0 I = pH and

1;l m l m

3 2

7 6

. a =

7 6

5 4

Proof.To calculate a p olynomial in the hyp ercompanion matrix H =

H [px ], we pro ceed in two stages. First supp ose that f x has degree

deg f < deg p= l . Now split H as H = I L + J 0 N , and

m m m m

since

11 NL N =0;k=0; 1;:::;l 2;

k k

it follows by induction that L + N = L + Y ;k=0; 1;:::;l, and that

k k

12 H = I L + J 0 Y ;k=0; 1;:::;l

m k

k 1 k 2 k 1

where Y = L N + L NL + ::: + NL and Y =0.

k 0

Next, we de ne the di erence op erator for xed px and f xby

f = f L + N f L:

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Prop erties of q -adic Vandermonde Matrix 23

Summing up 12 shows that if deg f

13 f H = I f L+J 0 f:

m m m

This formula breaks down when deg f l , since 11 do es not hold for values

of k l 2. But wemay pro ceed in the following way. Supp ose f x admits

the p-adic expansion

m1 m

f x= x+ xpx+ :::+ xpx + px ?:

0 1 m1

Now recall from [6] that J = pH =J 0 I . Clearly, pH = 0 for

m m l m

k l . Hence by 13, H pH has the form

k m m

k k k +1

H pH = J 0 L+ J 0 ;

k m m m k m k

for k =0; 1;:::;l 1. Summing this for k =0; 1;:::;l 1, we see that f H

is blo ckTo eplitz of the form given in 10.

For the remaining part it suces to show that U agrees with the matrix

2 3

6 7

A = .Itiswell known that for a p olynomial

4 5

:::

n1 n1

hx= h + h x + ::: + h x , hL= [h Lh ::: L h], where L is

0 1 n1

any n n companion matrix. Moreover, L=L + N; N = E is also a

1;n

companion matrix. From the ab ovetwo observations, it follows that

A e = Le + Le Le = + = :

0 1 k 1 k 1 1 k 1 1 k k 1 k 1 k

Needless to say, this ensures that Ae = a = f H e . Consequently, H a =

1 m 1 m

a = f H e for k =0; 1;:::;l 1 f H H e = f H e . Similarly, H

m k +1 m m 1 m 2

a], completing the pro of. and hence U =[a;H a ::: H

An immediate corollary to Lemma 1.2 gives us the required result relating

the q -adic co ordinate column of [f ] and [ f ] .

q;m q;m

Corollary 1.3. Let [f ] = a for polynomials f x and q x as in

q;m

k k k k

Lemma 1.2. Then [ f ] =H a and [q f ] = J a, where H =

m m

q;m q;m

H [q ] and J = J 0 I .

m l

Proof.From Lemma 1.2, we see that for any p olynomial g x, [g ] =

q;m

g H e : Consequently,ifg x= f x, then

m 1

k k

[g ] = g H e =H f H e =H a:

m 1 m m 1 m

q;m

The remaining result is clear since q H = J .

Given the sp ecial construction of W , it comes as no surprise that W is

directly related to the hyp ercompanion matrix. Indeed, since [1] = e ,

q ;m

i i

k k

it is clear from Corollary 1.3 that [ ] = H e ; where H = H [q ].

1 i

q ;mi

Consequently,

h i

14 W = :

e H e ::: H e

1 i 1 1 i

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24 Vaidyanath Mani and Rob ert E. Hartwig

1.3. The E matrix. In order to calculate the determinantofW , our

strategy shall b e to blo ck diagonalize W rst, and then compute the deter-

minant of the resulting blo ck diagonal matrix. In other words, we wish to

nd a suitable non singular matrix E =[E ;E ;:::;E ] such that W E =

1 2 s

diag D ;D ;:::;D , i.e., wewant

1 2 s

0 i 6= j

W E =

i j

D i = j;

where W is l m n and E is n l m .

i i i j j j

Based on 4 and 7, it suces to take as the columns of E , the standard

co ordinate columns of suitable p olynomials all divisible by q . In addition, we

want E to b e such that the blo cks D = W E are manageable and detE 6=0.

i i i

A convenientchoice is the chain of p olynomials

m 1

k r

] =[x q x x] = x[ ;q ;:::;q

j j j j j j j j

l 1

, =[1;x;:::;x ], j =1;:::;s, r =0;:::;m 1 and where = =q

j j j

divides every p olynomial in for i 6= j .As k =0; 1;:::;l 1. Clearly, q

j j

mentioned earlier, E is de ned by

15 = B E :

j ST j

From 4, wehave

k r k r

16 W [x q x x] =[x q x x]

i j j j j

st q ;m

i i

and hence for j 6= i, W E = 0. On the other hand when j = i,we recall

i j

Corollary 1.3, so that 16 reduces to

k r k r r k

W [x q x x] =[ q ] = J H a;

i i i i

st q ;m

i i i

i i

where H = H [q ], J = J I and a =[ ] . Nowif[U ]=

i i m l i

q ;m

i i

l 1

[a;H a;:::;H a] then an application of Lemma 1.2 yields

h i

m 1

W E = U JU ::: J U = H :

i i i i

Wehavethus shown the following result.

Lemma 1.4. For the matrices W and E describedabove

T s

17 W E = diag [ H ] ;

i i

i=1

m m

i i

is the hypercompanion matrix associated and H = H q where = =q

i i

. with q

We close this section with the following remarks.

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Prop erties of q -adic Vandermonde Matrix 25

Remark 1.5. The blo cks E in E =[E ;E ;:::;E ]may b e further

r 1 2 s

describ ed as

I I I

l l l

m 1

i i i

18 E = L ;qL ; :::::; q L ;

r r i i

0 0 0

see 5.4 of [5]. To obtain the ab ove, simply note that

h i

k k l 1

Lq L = q L ;L ;:::;L ;

r r r r r r

h i

k k l 1 k

= [q ] ; [xq ] ;:::;[x q ] ;

r r r

st st st

r r r

in which L = L and [ ] = are resp ectively the companion matrix and

r r

the standard co ordinate column of x. Since n deg + l ,we see that

r r

for k =0; 1;:::;l 1, L merely shifts the nonzero entries in without

r r r

losing any of them.

Remark 1.6. In the next section we shall explicitly calculate the deter-

minant of the E matrix. It will b e shown that detE is nonzero, and hence

E is nonsingular, if and only if the p olynomials q are pairwise coprime, i.e,

gcdq ;q =1 for i 6= j .

i j

Remark 1.7. It was shown in [9] that the p olynomials form a Jacobson

chain basis for the space F [x] of p olynomials of degree less than n with

resp ect to a shift op erator S . Hence, when the q 's are pairwise coprime, E is

achange of basis matrix from the standard to a chain basis denoted by B .

The reader is referred to [9] for more details.

Let us now capitalize on 17 and simultaneously calculate the determinant

of E as well as that of W .

2. Determinants. It is well known that if is the con uentVander-

monde matrix for the p olynomial x= x , then

i=1

m m

i j

det = ;

i j

where the pro duct is taken over all the ro ots .We shall now show that the

determinant of the q -adic Vandermonde matrix W induced by x= q

i=1

is given by

m m

i j

Rq ;q detW = ;

i j

where Rq ;q is the resultantofq and q . This shows that detW is non-zero

i j i j

if and only if the q 's are pairwise coprime.

From 17, we see at once that

19 det[W ]det[E ]= det[W ]det[E ]= det[ H ];

i i

i=1

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26 Vaidyanath Mani and Rob ert E. Hartwig

and thus in order to calculate detW , we shall need to evaluate detE and

det[ H ], i =1;:::;s.

i i

Our strategy shall b e to factor E as E = MT , in which detT = 1 and

M is a generalized Sylvester matrix. The determinantofM can b e calculated

with the aid of some results from [5] and this will enable us to compute detE .

We then evaluate det[ H ] and combine the results to express the detW

i i

in terms of the factors of the foundation p olynomial x= q .

i=1

In the next subsection, we undertake this factorization of the E matrix.

2.1. Resultants and the E matrix. Before we can factor E ,we shall

need to intro duce the Sylvester matrix of two p olynomials. We rst de ne the

r + k r strip ed matrix S f for a p olynomial f x= a + a x + :::+ a x

r 0 1 k

to b e

2 3

a ::: ::: 0

6 7

a a :::

1 0

6 7

6 . 7

. .

6 7

. a

6 7

S f = 20 ;

6 7

a a

k 0

6 7

a a

k 1

6 7

. .

4 5

0 ::: a

The strip ed matrices of two p olynomials f x and f x of degrees n and

1 2 1

n resp ectively can b e combined to yield the square Sylvester matrix S f ;f

2 1 2

given by

S f ;f = [S f jS f ]

1 2 n 1 n 2

2 1

whose determinant Rf ;f is the resultantoff and f . The Sylvester

1 2 1 2

matrix can b e constructed only for two p olynomials. It is p ossible to extend

this construction to a set of more than two p olynomials as follows. A more

thorough discussion can b e found in [5].

Let ff x;f x;:::;f xg b e a set of p olynomials with deg f = n .

1 2 s i i

Moreover, let f = f f :::f , deg f = n = n and construct the set

1 2 s i

i=1

of p olynomials fF ; F ;:::;F g by F = f=f .Now de ne the square n n

1 2 s i i

generalized Sylvester matrix M f jf :::jf tobe

1 2 s

M = M f jf :::jf = [S F ;S F ;:::;S 21 F ] :

1 2 s n 1 n 2 n s

1 2

Clearly

22 M f jf = S f ;f :

1 2 2 1

The following lemma corollary 5, page 24, [5] gives a factorization of the

matrix M .

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Prop erties of q -adic Vandermonde Matrix 27

Lemma 2.1. For the generalized Sylvester matrix M = M f jf :::jf

1 2 s

describedabove,

M f jf :::f 0

s1 1 s2

M = M f jf :::f

s 1 s1

0 I

M f jf f 0 M f jf 0

3 1 2 2 1

::: :

0 I 0 I

where I denotes the identity matrix of appropriate size.

Using the ab ove factorization, it is easy to calculate detM as follows.

Corollary 2.2. The determinant of the matrix M = M f jf :::jf

1 2 s

de nedabove is

detM = Rf ;f ;

i j

where Rf ;f is the resultant of f and f .

i j i j

Proof. The pro of easily follows from 22, Lemma 2.1 and the pro duct

rule for resultants i.e.

Rf ;f f =Rf ;f Rf ;f :

1 2 3 1 2 1 3

We are now ready to give the desired factorization. Let us rst factor the

chain E given in 18 as follows

I I I

l l l

m 1

r r

i r

23 = LT ; E = L ;q L ; :::; q L

r r r r r r

0 0 0

where

I I I

l l l

m 1

r r

i r

; :::; q L ;qL T =

r r r

0 0 0

is upp er triangular with a diagonal of ones and L contains the rst m l

r r r

columns of L . Since the deg x = m l 1 and L is n n,wemay

r r r r

write

L = S ;

r m l r

r r

where the strip ed matrix S f was de ned as in 20. Using 18 and 23,

the induced factorization of E b ecomes

h i

^ ^ ^

E = diag [T ] = MT:

L L ::: L

1 2 s

i=1

Wenow observe that

i h

^ ^ ^

M =

L ; L ; :::; L

1 2 s

S ; :::; S ;

m l 1 m l s

s s

1 1

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28 Vaidyanath Mani and Rob ert E. Hartwig

m m

1 2

is the generalized Sylvester matrix of the p olynomials fq ;q ;:::;q g de-

1 2

ned in 21. Since T is upp er triangular and blo ck diagonal, with a diagonal

of ones, detT = 1 and hence from Corollary 2.1

Y Y

m m

i j

detE = detM = Rf ;f = Rq ;q 24 :

i j i j

Wehavethus shown the following result.

Lemma 2.3. For the matrix E given in 15,

m m

i j

detE = Rq ;q

i j

where Rq ;q is the resultant of q x and q x.

i j i j

In the next subsection, weevaluate det[ H ] and then calculate detW

i i

according to 19.

2.2. The determinantof W . Recall from 8 and Lemma 1.2 that

H isam m blo ck upp er triangular matrix with diagonal blo cks L

i i i i i i

where L = L is a companion matrix. Note further that

i q

L = 25 q L :

i i j i

j =1

j 6= i

It is well known see [1] for example that

26 det[q L ] = Rq ;q ;

j q i j

where Rq ;q is the resultantofq and q . Using 25, 26, and the pro duct

i j i j

rule for determinants, the quantity det[ L ] can b e readily calculated to b e

i i

det[ L ] = Rq ;q :

i i i j

j =1

k 6= i

Since H consists of m diagonal blo cks of the form L , wehave that

i i i i i

m m

i j

det[ H ] = Rq ;q ;

i i i j

j =1

k 6= i

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Prop erties of q -adic Vandermonde Matrix 29

and rep eating this for i =1;:::;s,we arrive at the expression

s s

Y Y

s m m

i j

det diag [ H ] = Rq ;q 27 :

i i i j

i=1

j =1

j 6= i

We are now ready to put the pieces together. Let us recall from 17 that

T s

W E = diag [ H ] :

i i

i=1

and hence taking determinants on b oth sides of this equation yields

detW detE = detW detE = 28 det[ H ]:

i i

i=1

Substituting from 24 and 27, equation 28 can b e rewritten as

s s

Y Y Y

m m m m

i j i j

29 detW Rq ;q : Rq ;q =

i j i j

i=1

j =1

j 6= i

Wenow need to consider two cases. First supp ose that gcdq ;q = 1for

i j

i 6= j . In this case, detE and the right hand side of 29 are b oth non zero

since Rq ;q 6= 0. Hence wemay cancel detE from b oth sides of 29 to

i j

obtain

m m

i j

30 : detW = Rq ;q

i j

Next, assume that gcdq ;q = hx, deg h 1 for some distinct r;t,1

r t

e e

To show this, consider the p olynomial = =h where deg n 1.

j and thus from 7 and 4, wehave Now note that for 1 i s, q

e e

=[ x] =0; W

q ;m

i i

i.e, W = 0 and consequently detW = 0. Needless to say Rq ;q = 0 and

r t

hence 30 also holds in this case.

We reiterate the fact that in the ab ove discussion, no assumption has b een

made ab out the p olynomials q b eing prime or irreducible. Hence 30 holds

of the p olynomial .Wehavethus for every factorization of x= q

i=1

proven the following theorem.

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30 Vaidyanath Mani and Rob ert E. Hartwig

Theorem 2.4. Let x= q be any not necessarily prime fac-

i=1

torization of a polynomial x of degree n.Let W be the q -adic matrix cor-

responding to this factorization. Then

m m

i j

detW = Rq ;q :

i j

In particular W is nonsingular if and only if gcdq ;q =1 for i 6= j .

i j

So far wehave demonstrated one way in which the q -adic Vandermonde

matrix generalizes the Vandermonde matrix. In the next section, we consider

the Wronskian structure of the con uentVandermonde matrix and show that

the q -adic Vandermonde matrix has a similar blo ck structure.

3. The Structure of W . We conclude this pap er by examining the blo ck

structure of the q -adic Vandermonde matrix W . Recall from 3 that the

con uentVandermonde matrix , for = x has the following

i=1

structure

3 2

7 6

;

5 4

:::

where

3 2

2 n1

1 :::

r r

7 6

0 1 2 ::: n 1

7 6

0 0 1

7 6

31 : =

7 6

r .

. .

7 6

. 0 0

7 6

5 4

n 1

0 0 :::

m 1

m n

Clearly, for 1 j n and 1 i m ,

j 1

j i

i 1

[ ] = 32

r ij

0 other w ise

Wemay express the entries in via partial di erentiation. Indeed if D is

the k derivative with resp ect to ,wemay denote the quantity on the right

hand side of 32 by1=i!D , j =0;:::;n 1.

Let us now extend the idea of partial di erentiation to matrices thereby

, showing that the q -adic Vandermonde matrix induced by x= q

i=1

has a blo ck structure that is very similar to that of the Wronskian structure exhibited in 31.

ELA

Prop erties of q -adic Vandermonde Matrix 31

l 1

of the p olynomial q x= + x+ :::+ x .We Supp ose L = L

r 0 1 r q

de ne the asso ciated partial derivative op erator D acting on an l l matrix

r r r

@ a

D A= ;

i.e., the derivativeistaken with resp ect to the negative of the constant term

in q x. For example, D L =N = E and hence

0 r r r 1;l

D L x= D L x + L D x= N x + L D x:

r r r r r r r r

It follows swiftly by induction that

i i1 i

33 D L x= iN D x + L D x:

r r r

Our essential step is to relate the q -adic co ordinate columns to the op erator

D . In fact, wehave the following lemma.

3 2

7 6

i 7 6

j j

e for L , then a =1=i!D Lemma 3.1. If [x ] =

7 6

r .

1 i

q ;m

r r

5 4

m 1

i =0;:::;m 1 and j =0;:::;n 1.

Proof. The pro of is by induction on j . The lemma clearly holds for j =0,

3 2

7 6

.Now assume that since [x ] =

q ;m

r r

5 4

:::

m l 1

r r

3 3 2 2

b a

0 0

7 7 6 6

b a

1 1

7 7 6 6

j +1 j

; ; and [x ] = [x ] =

7 7 6 6

. .

q ;m q ;m

r r r r

. .

5 5 4 4

. .

b a

m 1 m 1

r r

j j

where a =1=i!D L e . From Corollary 1.3 with f = x we see that

i 1

j +1 j

[x ] = H [x ] which in matrix form b ecomes

q ;m q ;m

r r r r

b = L a ;:::; b = N a + L a

0 r 0 i i1 r i

for i =1;:::;n 1. Using the induction hyp othesis, this gives

j j +1

b = L a = L L e = L e

0 r 0 r 1 1

r r

as well as

1 1

i1 j i j

b = N D L e + L D L e :

i 1 r 1

r r r r

i 1! i!

ELA

32 Vaidyanath Mani and Rob ert E. Hartwig

Using 33 with x = L e ,we see that 3 reduces to

1 1

i i1 j +1 j i j

D b = iN D L L e ; e +L D L e =

1 1 r 1

r r r r r r

i! i!

completing the induction.

Our nal result on the structure of the matrix W is given by the following

theorem.

Theorem 3.2. The q -adic Vandermonde matrix W inducedby q has

3 2

7 6

, where the block form W =

7 6

5 4

2 3

2 n1

e L e L e ::: L e

1 r 1 1 1

r r

6 7 2 n1

0 e D L e D L e

1 r 1 r 1

6 7

r r

6 7

. .

6 7

. .

W = :

. . 0 e

6 7

:::

4 5

m 1

e L 0 0 ::: 1=m 1!D

1 r

m n

In other words, for i =0; ::; m 1 and j =0; ::; n1, [W ] =1=i!D L e :

r r ij 1

Proof.We recall from 6 that the columns of W are of the form [x ] .

q ;m

r r

The structure of these columns is precisely given by Lemma 3.1 as

3 2

e L

7 6

D L e

r 1

7 6

. [x ] =

q ;m

7 r r 6

5 4

m 1

e 1=m 1!D L

1 r

and hence the pro of of the theorem follows.

Remark 3.3. It has b een p ointed out to the authors by an anonymous

referee that eachoftheW is a Kalman reachability matrix. Indeed, from

equation 14, wehave that

h i

W = e H e :::H e ;

i 1 i 1 1

]isahyp ercompanion matrix. This where e =[1; 0;:::;0] and H = H [q

1 i

structure of this matrix is similar to the one given in Theorem 3.18, page 41

of [8]. The control theoretic asp ects of this matrix are still to b e explored.

ELA

Prop erties of q -adic Vandermonde Matrix 33

REFERENCES

[1] S. Barnett, Polynomials and Linear Control Systems, Marcel Dekker, New York, 1983.

[2] S. Lang, Algebra, 3rd edition, Prentice Hall, New York, 1993.

[3] G. Sansigre and M. Alvarez, On Bezoutian Reduction with Vandermonde Matrices,

Linear Algebra and its Applications, 121:401{408, 1989.

[4] G. Chen and Z. Yang, Bezoutian Representation Via Vandermonde Matrices, Linear

Algebra and its Applications, 186:37{44, 1993.

[5] R. E. Hartwig, Resultants, Companion Matrices, and Jacobson Chains, Linear Algebra

and its Applications, 94:1{34, 1987,

[6] R. E. Hartwig, Some Prop erties of Hyp ercompanion Matrices, Industrial Mathematics,

Volume 24, Part 2, 1974.

[7] U. Helmke and P.A.Fuhrman, Bezoutians, Linear Algebra and its Applications, 122{

124:1039{1097, 1989.

[8] R. E. Kalman, P.L.Falb and M. A. Arbib, Topics in Mathematical System Theory,

McGraw-Hill, New York, 1969.

[9] V. Mani and R. E. Hartwig, Bezoutian Reduction over Arbitrary Fields, Linear Algebra

and its Applications, to app ear.