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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.

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.

Reviewer: Gabriel Prajitura (Brockport)

### 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 |

### Keywords:

determinantal inequalities; Newton inequalities; Newton matrix; translation; Newton sequence; Newton coefficients; Newton spectrum; complex eigenvalues; similarity invariance; invertible matrix
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\textit{C. R. Johnson} et al., Linear Algebra Appl. 430, No. 11--12, 3030--3046 (2009; Zbl 1189.15009)

### 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 |

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