Statistics and characterization of matrices by determinant and trace
type: Item
instance of: scholarly article
title: Statistics and characterization of matrices by determinant and trace
zbMATH Open document ID: 1482.15030
author: Emre Alkan
author: Ekin Sıla Yörük
published in: The Ramanujan Journal
publication date: 2019-03-21T00:00:00Z
full work available at URL: http://cdm21054.contentdm.oclc.org/cdm/ref/collection/IR/id/7767
review text: This paper presents results on different aspects concerning square matrices with integer and complex entries. The first part of the paper is about proving that the probability of a random \(n\times n\) matrix, whose entries are (positive) integer prime numbers less than or equal to \(x\), being singular goes to zero as \(x\) goes to infinity (Theorem 1). The second part of the work concerns unitary, Hermitian and normal matrices over the complex numbers. Theorems 2 and 3 state that a (real) complex \(n\times n\) square matrix \(A\) is unitary with respect to a positive definite (bilinear form or scalar product over \(\mathbb{R}\)) Hermitian form if and only if its determinant is \(\pm 1\) and the trace of \(AA^*\) equals \(n\), where \(A^*\) denotes the conjugate transpose of \(A\). Moreover, the authors give a characterization of non-singular commuting complex square matrices having the same characteristic polynomial in terms of determinant and trace. They also show that a complex square matrix being Hermitian is equivalent to it having the same eigenvectors as its conjugate adjoint. Finally, it is proved that a complex matrix \(A\) is normal if and only if \(AA^*\) and \(A^*A\) have the same eigenvectors corresponding to each of their (common) eigenvalues.
reviewed by: Roswitha Rissner
Mathematics Subject Classification ID: 15B36
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15B52
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 11C20
Mathematics Subject Classification ID: 11N13
Mathematics Subject Classification ID: 11B25
zbMATH DE Number: 7040082
zbMATH Keywords: matrices with prime number entries
zbMATH Keywords: prime numbers in progressions
zbMATH Keywords: determinant
zbMATH Keywords: trace
zbMATH Keywords: unitary matrix
zbMATH Keywords: Hermitian matrix
zbMATH Keywords: normal matrix
MaRDI profile type: Publication
OpenAlex ID: W2748215695
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DOI: 10.1007/S11139-017-9930-5
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description: scientific article; zbMATH DE number 7040082