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 .

Proof.

We have

where

and

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

(1)

for .

Proof.

By Lemma 5, we have

which in turn, by Vieta’s formulas, gives

(2)

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

Next, we obtain some inequalities for .

Lemma 7.

Define . Then

Proof.

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

(3)

Finally, we have the easy

Lemma 8.

For we have .

Proof.

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

3 Proof of Theorem 1

Proof.

Suppose . Then

and so we get

(4)

But

(5)

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

(6)

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

4 Proof of Theorem 2

Proof.

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

so that

(7)

Hence

and the result follows. ∎

5 Proof of Theorem 3

Proof.

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

(8)

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 .

Proof.

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
2.4862447382651613433
2 5.194119929182595417
26.788216012030303413
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 , .

References

  • [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.