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

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.