Sep 23, 2004 - alize and extend the GaussâLucas theorem and prove the old .... Gauss-Lucas theorem for the products of roots (see Proposition 4.3 and Corol-.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 357, Number 10, Pages 4043–4064 S 0002-9947(04)03649-9 Article electronically published on September 23, 2004
INVERSE SPECTRAL PROBLEM FOR NORMAL MATRICES AND THE GAUSS-LUCAS THEOREM S. M. MALAMUD
Abstract. We establish an analog of the Cauchy-Poincare interlacing theorem for normal matrices in terms of majorization, and we provide a solution to the corresponding inverse spectral problem. Using this solution we generalize and extend the Gauss–Lucas theorem and prove the old conjecture of de Bruijn-Springer on the location of the roots of a complex polynomial and its derivative and an analog of Rolle’s theorem, conjectured by Schoenberg.
1. Introduction ∗
Let A = A be a Hermitian n × n matrix and An−1 its principal (n − 1) × (n − 1) submatrix obtained by deleting the last row and column. Let σ(A) = (λ1 ≤ . . . ≤ λn ) and σ(An−1 ) = (µ1 ≤ . . . ≤ µn ) be the ordered lists of (real) eigenvalues of A and An−1 , respectively. By the Cauchy–Poincare interlacing theorem, the sequences interlace each other, that is, (1.1)
λ1 ≤ µ1 ≤ λ2 ≤ µ2 ≤ . . . ≤ λn−1 ≤ µn−1 ≤ λn .
It is known (see [18], [26]) that the converse is also true, that is, for any two of real numbers satisfying (1.1), there exists a (nonsequences (λj )n1 and (µj )n−1 1 unique) n×n Hermitian matrix A such that σ(A) = {λj }n1 and σ(An−1 ) = {µj }n−1 . 1 We say that such a matrix A solves the inverse spectral problem for the sequences . A theorem of Hochstadt [17] asserts that there exists a unique (λj )n1 and (µj )n−1 1 Jacobi, that is, tridiagonal Hermitian matrix A solving the inverse spectral problem for these sequences. In the present paper we generalize both the Cauchy-Poincare interlacing theorem and the Hochstadt theorem to the case of normal matrices. Since the eigenvalues of a normal matrix are not real, the statements of these generalizations must be different from the Hermitian case. Moreover, the principal submatrix An−1 of a normal matrix A is only normal in trivial cases (see [12] or Lemma 4.14 below). The main result of the first part of the paper is Theorem 3.11. It provides necesto be sary and sufficient geometric conditions for two sequences {λj }n1 and {µj }n−1 1 the spectra of a normal matrix A and of its submatrix An−1 respectively. In order to formulate these geometric conditions we introduce in Section 2 several notions of majorization for sequences of vectors from Rn , which are natural generalizations of the classical notions for the case n = 1. Received by the editors July 6, 2003 and, in revised form, November 7, 2003. 2000 Mathematics Subject Classification. Primary 15A29; Secondary 30C15, 30C10. Key words and phrases. Zeros of polynomials, normal matrices, inverse spectral problem, majorization. c 2004 American Mathematical Society
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S. M. MALAMUD
Note that the sufficient conditions for the tuples {λj }n1 and {µj }n−1 of complex 1 numbers to be the spectra of a normal matrix A and its principal submatrix An−1 respectively can be expressed analytically without the notion of majorization and read n−1 j=1 (λ − µj ) (1.2) ck := Resλk n ≥ 0, k ∈ {1, . . . , n}. j=1 (λ − λj ) The second part of our paper concerns the location of the roots of a polynomial and its derivative and in particular the Gauss-Lucas theorem [33], stating that the of the derivative p of any complex polynomial p(∈ C[z]) of degree roots {µk }n−1 1 n lie in the convex hull of the roots {λj }n1 of p. The following proposition provides a “bridge” between the first and second parts of the paper. Proposition 1.1. Let p(z) be a polynomial of degree n with zeros {λj }n1 and {µj }n−1 the zeros of its derivative. Then there exists a (nonunique) normal matrix 1 A ∈ Mn (C) such that σ(A) = {λj }n1 and σ(An−1 ) = {µj }n−1 . 1 Proposition 1.1 immediately follows from (1.2) since in this case ck = 1, k ∈ {1, . . . , n}. Combining Proposition 1.1 and Theorem 3.11 below, we immediately arrive at the following result (Proposition 4.3): {µj }n−1 ≺uds {λj }n1 . 1 This is a generalization of the Gauss-Lucas theorem. Proposition 1.1 allows us to apply linear algebra to study the location of the zeros of a polynomial and its derivative. For example, using exterior algebra we obtain a generalization of the Gauss-Lucas theorem for the products of roots (see Proposition 4.3 and Corollary 4.4). In particular we show that µk1 µk2 ∈ conv{λj1 λj2 : 1 ≤ j1 < j2 ≤ n} for any pair {k1 , k2 } with 1 ≤ k1 = k2 ≤ n − 1. Note that while the second topic has attracted a lot of attention during the last two decades (see [2], [4], [7], [9], [29], [30], [36], [39]), our approach seems to be new and promising. In particular in the framework of this approach we get simple (and short) solutions to the old problems of de Bruijn-Springer [8] and Schoenberg [38]. Let us briefly describe these problems. In 1947 de Bruijn and Springer [8] conjectured that the following inequality holds for any convex function f : C → R: n−1 n 1 1 f (µj ) ≤ f (λj ). n − 1 j=1 n j=1
They succeeded in proving this inequality for a class of convex functions. It is not difficult to see that this class actually coincides with the class CV S(C), defined in (2.10) below. We provide a complete proof of this conjecture by showing that a doubly stochastic matrix S in the above-mentioned representation µ = Sλ can be chosen in such a way that all entries in the last row equal 1/n. Moreover, we prove (see
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ZEROS OF POLYNOMIALS AND NORMAL MATRICES
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Theorem 4.10) that the following inequality is valid for any k ∈ {1, . . . , n − 1} k 1 n−1 f (µij − α) k
(1.3)
1≤ii
