The Largest Entry in the Inverse of a Vandermonde Matrix

08/03/2020 ∙ by Carlo Sanna, et al. ∙ University of Waterloo 0

We investigate the size of the largest entry (in absolute value) in the inverse of certain Vandermonde matrices. More precisely, for every real b > 1, let M_b(n) be the maximum of the absolute values of the entries of the inverse of the n × n matrix [b^i j]_0 ≤ i, j < n. We prove that lim_n → +∞ M_b(n) exists, and we provide some formulas for it.



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1 Introduction

Let be a list of real numbers. The classical Vandermonde matrix is defined as follows:

As is well-known, the Vandermonde matrix is invertible if and only if the are pairwise distinct. See, for example, [3].

In what follows, is a positive integer and is a fixed real number. Let us define the entries by

and let , the maximum of the absolute values of the entries of . The size of the entries of inverses of Vandermonde matrices have been studied for a long time (e.g., [1]). Recently, in a paper by the first two authors and Daniel Kane [2]

, we needed to estimate

, and we proved that . In fact, even more is true: the limit exists and equals . In this paper, we generalize this result, replacing with any real number greater than .

Our main results are as follows:

Theorem 1.

Let and . Then for . Hence .

Theorem 2.

Let and . Then .

Theorem 3.

For all real the limit exists.

2 Preliminaries

For every real number , and for all integers , let us define the power sum

The following lemma will be useful in later arguments.

Lemma 4.

Let be integers with , , and let be a positive real number.

  1. [(a)]

  2. If , then .

  3. If , then .


We have



Now there is a bijection given by

where is the unique integer such that . Hence, it follows easily that for , and for . ∎

Recall the following formula for the entries of the inverse of a Vandermonde matrix (see, e.g., [4, §1.2.3, Exercise 40]).

Lemma 5.

Let be pairwise distinct real numbers. If then

For define

We now obtain a relationship between the entries of and and .

Lemma 6.

Let . Then


for .


By Lemma 5, we have

which in turn, by Vieta’s formulas, gives


for . The result now follows by the definitions of and . ∎

Next, we obtain some inequalities for .

Lemma 7.

Define . Then


For , we have

A quick computation shows that the inequality

is equivalent to

Let be the minimum positive integer such that . Then . Hence, for , we have

so that


Finally, we have the easy

Lemma 8.

For we have .


is a symmetric matrix, so its inverse is also. ∎

3 Proof of Theorem 1


Suppose . Then

and so we get




by Lemma 8. Make the substitutions for and for in (4) to get


The result now follows by combining Eqs. (4), (5), and (6). ∎

4 Proof of Theorem 2


Since , it follows that . Hence in Theorem 1 we can take , and this gives . However, by explicit calculation, we have

so that



and the result follows. ∎

5 Proof of Theorem 3


We have

where the equality

arises from the one-to-one correspondence between the subsets of of cardinality and those of cardinality .

For define

Hence the limits


exist and are finite.

From Theorem 1 we see that

and the proof is complete. ∎

From this theorem we can explicitly compute for .

Corollary 9.

Let be the real zero of the polynomial .

  1. [(a)]

  2. If , then .

  3. If , then .


From Theorem 2 we know that for we have . Now an easy calculation based on (8) shows that

By solving the equation , we see that for we have , while if we have . This proves both parts of the claim. ∎

Remark 10.

The quantity converges rather slowly to its limit when is close to . The following table gives some numerical estimates for .

3 1.785312341998534190367486
2 5.194119929182595417
1.5 67.3672156
1.4 282.398
1.3 3069.44
1.2 422349.8

6 Final remarks

We close with a conjecture we have been unable to prove.

Conjecture 11.

Let and . Then, for all sufficiently large , we have for some , .


  • [1] W. Gautschi. On inverses of Vandermonde and confluent Vandermonde matrix. Numer. Mathematik 4 (1962), 117–123.
  • [2] D. M. Kane, C. Sanna, and J. Shallit. Waring’s theorem for binary powers. Combinatorica 39 (2019), 1335–1350.
  • [3] A. Klinger. The Vandermonde matrix. Amer. Math. Monthly 74 (1967), 571–574.
  • [4] D. E. Knuth. The Art of Computer Programming, Vol. 1, Fundamental Algorithms. Addison-Wesley, third edition, 1997.