Matrices and spectra satisfying the Newton inequalities.(English)Zbl 1189.15009

A sequence of real numbers $$(c_n)_{n \geq 1}$$ is called a Newton sequence if $$c_k^2 - c_{k - 1} c_{k + 1} \geq 0$$ for every $$k \geq 1$$ (considering that $$c_0 = 1$$).
If $$A$$ is a matrix of order $$n$$ with eigenvalues $$\{ \lambda_1, \lambda_2, \dots , \lambda_n \}$$ then we define
$S_k (A) = \sum_{1 \leq j_1 < \dots < j_k \leq n} \lambda_{j_1} \lambda_{j_2} \dots \lambda_{j_k}$
and
$c_k (A) = \frac{1}{{n \choose k}} S_k (A).$
These numbers $$c_k(A)$$ are called the Newton coefficients of the matrix.
A matrix $$A$$ is called a Newton matrix (and its spectrum a Newton spectrum) if the Newton coefficients are real and form a Newton sequence. It is known that if all eigenvalues of a matrix are real then the matrix is a Newton matrix. The authors discuss possible Newton matrices with complex eigenvalues.
First, in Section 2, all such matrices of order 2, 3 and 4 are completely characterized. Section 3 is concerned with some general properties: similarity invariance of the property, the fact that $$A$$ is a Newton matrix if and only if $$\alpha A$$ is for every real $$\alpha$$, the fact that an invertible matrix $$A$$ is a Newton matrix if and only if its inverse is a Newton matrix and that $$A$$ is a Newton matrix if and only if $$A \oplus 0$$ is a Newton matrix.
The last two sections discuss the behavior of the property of being a Newton matrix under translations.

MSC:

 15A15 Determinants, permanents, traces, other special matrix functions 15A45 Miscellaneous inequalities involving matrices 15A18 Eigenvalues, singular values, and eigenvectors 15A21 Canonical forms, reductions, classification 15A09 Theory of matrix inversion and generalized inverses
Full Text:

References:

 [1] James, G.; Johnson, C.R.; Pierce, S., Generalized matrix function inequalities on $$M$$-matrices, J. London. math. soc., 57, 2, 562-582, (1998) · Zbl 0930.15022 [2] Holtz, O., M-matrices satisfy newton’s inequalities, Proc. amer. math. soc., 133, 3, 711-717, (2005) · Zbl 1067.15018 [3] Maclaurin, C., A second letter to martin folkes, esq.; concerning the roots of equations, with the demonstration of other rules in algebra, Philos. trans., 36, 59-96, (1729) [4] I. Newton, Arithmetica universalis: sive de compositione et resolutione arithmetica liber, 1707. [5] C.P. Niculescu, A new look at Newton’s inequalities, J. Inequal. Pure Appl. Math. 1(2) (2000) (Article 17, 14 pp. (electronic)). · Zbl 0972.26010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.