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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
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References:

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