1 Introduction 2 Preliminaries and notations - Any2Any ...

On the Inversion of Vandermonde Matrices
Shui-Hung Hou
Department of Applied Mathematics
Hong Kong Polytechnic University
mahoush@polyu.edu.hk
Abstract
Anovel and simple recursive algorithm for inverting Vandermonde
matrix and its generalized form is presented. The algorithm is suitable
for classroom use in both numerical as well as symbolic computation.
1 Introduction
The importance of the Vandermonde matrix is well known. Inversion of
this matrix is necessary in many areas of applications such as polynomial
interpolation [4, 10], digital signal processing [2], and control theory [6], to
mention a few. See also for example Klinger [8], Kalman [7]. However, an
explicit recursive formula for the inversion of Vandermonde matrices seems
unavailable in most linear algebra textbooks.
The purpose of this paper is to present anovel and simple recursive algorithm
for inverting Vandermonde matrix, as well as its generalized (or conuent)
form, in a way more readily accessible for use in classroom and suitable for
both numerical as well as symbolic computation.
2 Preliminaries and notations
Let m be a nonnegative integer. For the sequence 1; (s , );:::;(s , ) m,1
of polynomials we write s(; m) =[1; (s , );:::;(s , ) m,1 ] T . In particular,
s(0;m)=[1;s;:::;s m,1 ] T :
Let 1 ; 2 ;:::; r be given distinct zeros of the polynomial
p(s) =(s , 1 ) n1 (s , r ) nr

with n 1 + :::+ n r = n. The generalized (or conuent) Vandermonde matrix
related to the zeros of p(s) is known to be
V =[V 1 V 2 V r ] ; (1)
where the block matrix V k = V ( k ;n k ) is of order n n k , having elements
i,1
V ( k ;n k ) ij =
j,1
i,j
k
for i j and zero otherwise (k = 1; 2;:::;r; i =
1; 2;:::;n; j = 1; 2;:::;n k ). More specically, V k is the n n k matrix of
coecients that appears in the truncated Taylor expansion at k , modulo
(s , k ) n k,ofs(0;n). That is,
s(0;n)=V ( k ;n k )s( k ;n k ) mod (s , k ) n k
:
In the case the zeros 1 ;:::; r of p(s) are simple, we have the usual Vandermonde
matrix, namely,
V =
2
6
4
1 1 1
1 2 r
7
. . . 5 :
n 1,1
1
n 2,1
2
nr,1
r
It will be shown that the inverse of the generalized Vandermonde matrix V
in (1) has a form
2 3
W 1
W
V ,1 =
2
6
4
7
. 5 ;
W r
where each block matrix W k is of order n k n, and may be computed by
means of a recursive procedure.
Generally, the inverse of the usual Vandermonde matrix [3], as well as the
inverse of the generalized Vandermonde matrix [9] are based on using interpolation
polynomials.
Our approach is based on using the Leverrier-Faddeev algorithm [1, 5, 10],
which states that the resolvent ofagiven n n matrix A is given by
(sI , A) ,1 = B 1s n,1 + B 2 s n,2 + + B n
s n + a 1 s n,1 + + a n
; (2)
where det(sI , A) =s n + a 1 s n,1 + + a n is the characteristic polynomial
3

of the matrix A, andall the B j matrices are of order n n, satisfying
B 1 = I; a 1 = , 1 1 tr(AB 1);
B 2 = AB 1 + a 1 I; a 2 = , 1 2 tr(AB 2);
.
.
B n = AB n,1 + a n,1I; a n = , 1 tr(AB n n)
with 0=AB n + a n I terminating as a check of computation. Here tr stands
for the trace of a matrix.
3 Main result
Let J = diag(J 1 ;:::;J r ) be the block diagonal matrix, where
2
3
k 1 0 0
0 k 1 .
J k = J( k ;n k )=
0
...
... 0
6
4
7
. k 1 5
0 0 0 k
is the n k n k Jordan block with eigenvalue k . Then J has characteristic
polynomial det(sI , J) =(s , 1 ) n1 (s , r ) nr
= p(s).
Substituting A = J in equations (2) and (3) of the Leverrier-Faddeev algorithm,
we see immediately that
where
p(s)(sI , J) ,1 = B 1 s n,1 + B 2 s n,2 + + B n ; (4)
B 1 = I;
B 2 = JB 1 + a 1 I;
B n = JB n,1 + a n,1I;
0 = JB n + a n I:
J = diag(J 1 ;:::;J r ) being block diagonal, so are all the B j matrices. In fact,
B j =diag(B j;1 ;B j;2 ;:::;B j;r );
j =1; 2;:::;n;
and each block matrix B j;k is of order n k n k , satisfying
B 1;k = I k ;
B 2;k = J k B 1;k + a 1 I k ;
B n;k = J k B n,1;k + a n,1I k ;
0 = J k B n;k + a n I k ;
(3)
(5)

where I k is the n k n k identity matrix.
Let us now put
p k (s) =
p(s)
; k =1;:::;r:
(s , k ) n k
Dene also the n k -dimensional column vector k = [0; ; 0; 1] T , and write
=[ T 1
; ;r T ] T .
If we postmultiply both sides of equation (4) by the column vector , we
easily get 2
3 2 3
p 1 (s)s( 1 ;n 1 ) H
6
4 .
7
5 = 1
6
4 .
7
5 s(0;n): (6)
p r (s)s( r ;n r ) H r
Each H k is of the form
H k = h B n;k k B 1;k k
i
and has order n k n.
Comparing in turn for k =1; 2;:::;r the truncated Taylor expansions at k ,
modulo (s , k ) n k, of both sides in (6) and putting these results together, we
get
2 3
H 1
diag(P 1 ;:::;P r )= 6
h i
4 .
7
5 V1 V r ;
H r
where each block P k is a n k n k upper triangular matrix given by
P k = p k (J k )=
n k ,1
X
j=0
p (j)
k
( k )
(N k ) j :
j!
It is noted here that N k = J(0;n k )=J k , k I k is nilpotent of order n k .
If we can show that each P k is invertible, then
2 3
P ,1
1 H 1
V ,1 = 6
4 .
7
5 : (8)
P
r ,1 H r
To this end we require the following lemma which is an easy consequence of
the partial fraction expansion of 1=p(s) andthe fact that N k is nilpotent.
(7)

Lemma 1 Let there be given the partial fraction expansion
1
rX
p(s) =
k=1
Then for k =1; 2;:::;r
K k;nk
(s , k ) n k
X
n k ,1
P ,1
k
=
j=0
where the polynomial K k (s) is given by
+ K k;n k ,1
(s , k ) + + K !
k;1
:
n k,1
s , k
K k;j (N k ) j = K k (J k );
K k (s) =K k;nk + K k;nk ,1(s , k )++ K k;1 (s , k ) n k,1
:
Putting the above results together with equations (7) and (8), we are now
ready to state our main result:
Theorem 1 The inverse of V = [V 1 V 2 :::V r ] related to the distinct zeros
1 ;:::; r of p(s) is given by
2 3
W 1
W
V ,1 =
2
6
4
7
. 5 ;
W r
where each block matrix
W k = W ( k ;n k )= h K k (J k )B n;k k K k (J k )B n,1;k k K k (J k )B 1;k k
i
is of order n k n.
Taking into account of (5), we nd that K k (J k )B j;k = B j;k K k (J k );j =
1; 2;:::;n, so that B 1;k K k (J k )=K k (J k ), and for j =2;:::;n
B j;k K k (J k ) = J k B j,1;kK k (J k )+a j,1K k (J k )
= ( k I k + N k )B j,1;kK k (J k )+a j,1K k (J k ):
Moreover, ( k I k + N k )B n;k K k (J k )+a n K k (J k )=0.

4 The Algorithm
Based on the results obtained in the last section, we are now ready to give a
recursive algorithm for inverting generalized Vandermonde matrix.
The Algorithm:
Let 1 ; 2 ;:::; r be distinct zeros of the polynomial
p(s) = (s , 1 ) n1 (s , r ) nr
= s n + a 1 s n,1 + + a n
given together with the partial fraction expansion of
1
rX
p(s) = K k;nk
+ K k;n k ,1
(s , k ) n k
(s , k ) + + K !
k;1
:
n k,1
s , k
k=1
For each k 2f1; 2;:::;rg, compute recursively polynomials h 1 ;h 2 ;:::;h n of
degree at most n k , 1by means of the following scheme:
terminating at
h 1 (s) = K k;nk + sK k;nk ,1 + + s n k,1
K k;1 ;
h 2 (s) = ( k + s)h 1 (s)+a 1 h 1 (s) mod s n k
;
h 3 (s) = ( k + s)h 2 (s)+a 2 h 1 (s) mod s n k
;
.
h n (s) = ( k + s)h n,1(s)+a n,1h 1 (s) mod s n k
;
0=( k + s)h n (s)+a n h 1 (s) mod s n k
:
Obtain a block matrix W k = W ( k ;n k ) of order n k n via the equality
h
s
n k ,1
s n k,2
1 i W ( k ;n k )= h h n h n,1 h 1
i
:
The inverse of the generalized Vandermonde matrix V related to the distinct
zeros 1 ; 2 ;:::; r of p(s) may then be given by
V ,1 = [V ( 1 ;n 1 )V ( 2 ;n 2 ) V ( r ;n r )] ,1
=
2
6
4
W ( 1 ;n 1 )
W ( 2 ;n 2 )
.
W ( r ;n r )
3
7
5 :

Let us now give some supplementary remarks on the above algorithm.
(i) A check on the accuracy of the computation of polynomials h 1 ;:::;h n is
provided by the last polynomial ( k + s)h n (s)+a n h 1 (s), which should
result identically in the zero polynomial 0 when modulo s n k
is performed.
(ii) The coecients a 1 ;a 2 ;:::;a n of the polynomial p(s) may be recursively
computed using (3) with A = diag( 1 ;:::; 1 ;:::; r ;:::; r ).
| {z }
n 1
| {z }
(iii) The partial fraction coecients K k;nk ;K k;nk ,1;:::;K k;1 used in the construction
of the starting polynomial h 1 (s) may be obtained by expanding
n
1
p k (s) = k ,1
X
K k;nk ,j(s , k ) j +
j=0
in powers of (s , k ). They may also be recursively computed using
the following scheme:
K k;nk = 1=p k;0 ;
K k;nk ,j =
P j
i=1
p k;i K k;nk ,j+i
,
p k;0
(j =1;:::;n k , 1);
where p k (s) = n,n P k
j=0
p k;j (s , k ) j .
5 Illustrative example
The following example will serve to illustrate the recursive algorithm presented
above. Let the generalized Vandermonde matrix V in (1) be given
by
2
3 2
3
1 0 0 1 1 0 0 1
V =
1 1 0 2
6
4 2 1
2 1 1 2 7
2 5 = ,2 1 0 3
6
4 4 ,4 1 9 7
5 ;
3 1
3 2 1
3 1 3 2
,8 12 ,6 27
for which 1 = ,2;n 1 = 3, and 2 = 3;n 2 = 1. The coecients of the
polynomial p(s) = (s +2) 3 (s , 3) are given by a 1 = 3;a 2 = ,6;a 3 = ,28,
nr

and a 4 = ,24. It is easy to determine the partial fraction expansion of 1=p(s)
to be
1
(s +2) 3 (s , 3) = , 1 5
(s +2) + , 1
25
3 (s +2) + , 1
1
125
2 s +2 + 125
s , 3 :
Let us consider rst the case 1 = ,2. Clearly
Then
h 1 (s) =, 1 5 ,
s
25 , s2
125 :
h 2 (s) = (,2+s)h 1 (s)+3h 1 (s) mod s 3
= , 1 5 , 6s
25 , 6s2
125 ;
h 3 (s) = (,2+s)h 2 (s) , 6h 1 (s) mod s 3
= 8 5 + 13s
25 , 12s2
125 ;
h 4 (s) = (,2+s)h 3 (s) , 28h 1 (s) mod s 3
= 12 5 + 42s
25 + 117s2
125 :
As a check of computation, we verify that
(,2+s)h 4 (s) , 24h 1 (s) mod s 3 = 117s3
125 mod s3 =0:
h i h i
Thus it follows from s 2 s 1 W 1 = h 4 h 3 h 2 h 1 that
2
3
117
, 12 , 6 , 1
125 125 125 125
42 13
W 1 = 6
,
4
6 , 1 7
25 25 25 25 5 :
Similarly, for 2 =3we nd that
h 1 (s) = 1
125 ;
12
5
8
5
, 1 5
, 1 5
h 2 (s) = (3 + s)h 1 (s)+3h 1 (s) mod s = 6
125 ;
h 3 (s) = (3 + s)h 2 (s) , 6h 1 (s) mod s = 12
125 ;
h 4 (s) = (3 + s)h 3 (s) , 28h 1 (s) mod s = 8
125 :

and
(3 + s)h 4 (s) , 24h 1 (s) mod s =3
Then h 1 i W 2 = h h 4 h 3 h 2 h 1
i
gives
8 24
125 , 125 =0:
W 2 = h 8
125
12
125
6
125
1
125
i
:
Finally, we have
V ,1 =
"
W1
#
=
W 2
2
6
4
117
125
42
25
12
5
8
125
, 12
125
13
25
8
5
12
125
, 6
125
, 6
25
, 1 5
6
125
, 1
125
, 1
25
, 1 5
1
125
3
7
5
:
Acknowledgment
This work was supported by the Research Committee of The Hong Kong
Polytechnic University (Grant No.G-S744).
References
[1] D. K. Faddeev and V. N. Faddeeva, Computational Methods of
Linear Algebra, Freeman, San Francisco, 1963.
[2] H. K. Garg, Digital Signal Processing Algorithms, CRC Press, 1998.
[3] F. A. Graybill, Matrices with Applications to Statistics, second ed.,
Wadsworth, Belmont, Calif., 1983.
[4] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University
Press, 1985.
[5] S. H. Hou, On Leverrier-Faddeev Algorithm, Proceedings of the
Third Asian Technology Conference in Mathematics, Yang et al (Eds.),
Springer Verlag, 399-403, 1998.
[6] T. Kailath, Linear Systems, Prentice Hall, Inc., Englewood Clis,
New Jersey, 1980.
[7] D. Kalman, The Generalized Vandermonde Matrix, Math. Mag., 57:15-
21, 1984.

[8] A. Klinger, The Vandermonde Matrix, Amer. Math. Monthly, 74:571-
574, 1967.
[9] N. Macon and A. Spitzbart, Inverses of Vandermonde Matrices,
Amer. Math. Monthly, 65:95-100, 1958.
[10] C. Pozrikidis, Numerical Computation in Science and Engineering,
Oxford University Press, 1998.