Eigenvalues of Complex Matrices and their Real Counterpart with Enhanced Lower Bounds for Interval Hermitian Matrices

Hertz, David (2024) Eigenvalues of Complex Matrices and their Real Counterpart with Enhanced Lower Bounds for Interval Hermitian Matrices. In: Research Updates in Mathematics and Computer Science Vol. 2. B P International, pp. 59-69. ISBN 978-81-971889-7-8

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Abstract

The purpose of this article is threefold:
(i) To prove that each real eigenvalue of \(\mathbf{A}=\mathbf{A}_R+i \mathbf{A}_I \in \mathbb{C}^{m \times m}\) is doubled in its real counterpart \(\mathbb{A}:=\left[\begin{array}{rr}\mathbf{A}_R & -\mathbf{A}_I \\ \mathbf{A}_I & \mathbf{A}_R\end{array}\right] \in \mathbb{R}^{2 m \times 2 m}\), whereas each complex eigenvalue of \(\mathbf{A}\) appears in \(\mathbb{A}\) together with its complex conjugate. Hence, if \(f(\lambda ; \mathbf{A}):=\operatorname{det}\left(\lambda \mathbf{I}_m-\mathbf{A}\right)=\lambda^m+\sum_{k=0}^{m-1} a_k \lambda^{m-k}\) is the characteristic equation of \(\mathbf{A}\) then \(f(\lambda ; \mathbb{A}):=\operatorname{det}\left(\lambda \mathbf{I}_{2 m}-\mathbb{A}\right)=\left(\lambda^m+\sum_{k=0}^{m-1} a_k \lambda^{m-k}\right)\left(\lambda^m+\sum_{k=0}^{m-1} \operatorname{conj}\left(a_k\right) \lambda^{m-k}\right)\) is the characteristic equation of \(\mathbb{A}\), where \(\mathbf{I}_m\) denotes the \(m\)-dimensional unit matrix and \(\operatorname{conj}(a)\) denotes \(a\) 's conjugate. Notice that the proof of this result was not trivial.
(ii) To give based on (i) another simple proof of the author's result that \(\operatorname{rank}(\mathbf{A})=\) \(r\) if and only if \(\operatorname{rank}(\mathbb{A})=2 r\), where \(\mathbf{A} \in \mathbb{C}^{m \times n}\) and \(\mathbb{A} \in \mathbb{R}^{2 m \times 2 n}\).
(iii) To improve Rump's lower bound on the minimal eigenvalue of an interval Hermitian matrix that also reduces its complexity by a factor of two and decreases the dimensionality of the chosen vertex matrices.

Item Type: Book Section
Subjects: STM Academic > Mathematical Science
Depositing User: Unnamed user with email support@stmacademic.com
Date Deposited: 03 Apr 2024 09:57
Last Modified: 03 Apr 2024 09:57
URI: http://article.researchpromo.com/id/eprint/2261

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