We review several sufficient conditions for the positive definiteness of a tridi-
agonal matrix and propose a different approach to the problem, recalling and ...
SUFFICIENT CONDITIONS FOR POSITIVE DEFINITENESS OF TRIDIAGONAL MATRICES REVISITED ´ AND C.M. DA FONSECA MILICA AN¯DELIC Abstract. We review several sufficient conditions for the positive definiteness of a tridiagonal matrix and propose a different approach to the problem, recalling and comprising little-known results on chain sequences.
1. Introduction The positive definiteness of symmetric (more generally, Hermitian) matrices is a key property in many areas of pure and applied mathematics, and it can be seen as a generalization of the notion of positive real number. Additionally, though structurally simple, tridiagonal matrices play an important role in different modern fields of research, from computational programming, PDE’s to statistics or engineering, with many problems involving them still remaining open. We will focus our attention on (irreducible) real symmetric tridiagonal matrices a1 b 1 b 1 a2 b 2 ... ... b2 (1.1) An = , . . .. . . bn−1 bn−1 an but all the results can be easily extended to Hermitian matrices. One of the most standard way to characterize the positive positive definiteness of a tridiagonal matrix is based on the positiveness of the successive principal minors of the matrix, known as the Sylvester’s Criterium (cf., e.g, [5, 9]): Theorem 1.1. The real symmetric tridiagonal matrix An , defined in (1.1), is positive definite if and only if its principal minors det Ak , for k = 1, . . . , n, are positive. We remark that the general characterization in terms of the eigenvalues, i.e., An is positive definite if and only if all its eigenvalues are positive, seems inadequate as far as applied or numerical matrix theory is concerned, due to the high computational effort required. A more practical but still standard criterion can be derived from Gerˇsgorin Circle Theorem (cf., e.g., [10]): Date: January, 2010. 2000 Mathematics Subject Classification. 15A15, 15A57, 15A45, 15A60, 15A48. Key words and phrases. Chain sequences, Wall-Wetzel Theorem, tridiagonal matrix, positive definite matrix. The first author is supported by FCT - Funda¸c˜ ao para a Ciˆencia e a Tecnologia, grant no. SFRH/BD/44606/2008, and by CIDMA - Center for Research and Development in Mathematics and Applications. The second author is supported by CMUC - Centro de Matem´ atica da Universidade de Coimbra.
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´ AND C.M. DA FONSECA MILICA AN¯DELIC
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Theorem 1.2. Suppose that the real symmetric tridiagonal matrix An , defined in (1.1), with diagonal entries all positive, is strictly diagonally dominant, i.e., |ak | > |bk−1 | + |bk |, for k = 1, . . . , n. Then An is positive definite. In this brief and unpretentious note, the recent refined characterizations on the positive definiteness of tridiagonal matrices are reviewed, and proved using some earlier works on chain sequences. 2. Tridiagonal positive definite matrices From the definition, it is well-known that if An is positive definite, then all diagonal entries are positive and b2i < ai ai+1 , for i = 1, . . . , n − 1. In 1977, Barrow et al. [1] in an attempt to prove a particular unicity property of the best spline approximations from a certain nonlinear manifold for functions having a positive second derivative, proposed the following sufficient condition for the positiveness of the determinant of a tridiagonal matrix. Proposition 2.1. Let An be the real symmetric tridiagonal matrix defined in (1.1), with diagonal entries positive. If 1 π2 2 (2.1) bi ≤ ai ai+1 1 + , f or i = 1, . . . , n − 1 . 4 1 + 4n2 then det An > 0. Under the conditions of the previous proposition, one can easily deduce that An is positive definite. The original proof of Proposition 2.1 involves second order differential equations and Green’s functions. In 1997 Johnson et al. [6] refined the inequality (2.1) using a matrix and a graph theoretic approach. Proposition 2.2. Let An be the real symmetric tridiagonal matrix defined in (1.1), with diagonal entries positive. If b2i
0 such that αk = gk (1 − gk−1 ) ,
with 0 ≤ g0 < 1 and 0 < gk < 1 , for k > 0 .
Chain sequences are used, for example, to formulate procedures to find bounds for zeros of orthogonal polynomials (cf. [7]). We will assume here that chain sequences are finite, and recall that a chain sequence does not, in general, uniquely determine its parameter sequence (cf. [2]). n−1 The simplest example of a chain sequence is the constant sequence 14 k=1 . In [7], Ismail and Li characterized the largest constant finite chain sequence. n−1 Theorem 3.1 ([7]). A constant sequence {α}k=1 is a (finite) chain sequence if and only if 1 . 0
