## What is/are Real Eigenvalues?

Real Eigenvalues - In particular, we pose and solve a new type of inverse spectral problems involving the interior transmission eigenvalue problem with complex transmission eigenvalues except for at most finite real eigenvalues.^{[1]}Instead, we find that the operational signature is the “sign characteristic” of real eigenvalues of Hermitian matrix pencils.

^{[2]}Unlike their Hermitian counterparts, nonconservative systems do not exhibit a priori real eigenvalues and hence unitary evolution.

^{[3]}We study the distribution of the eigenvalue condition numbers $$\kappa _i=\sqrt{ ({\mathbf{l}}_i^* {\mathbf{l}}_i)({\mathbf{r}}_i^* {\mathbf{r}}_i)}$$ κ i = ( l i ∗ l i ) ( r i ∗ r i ) associated with real eigenvalues $$\lambda _i$$ λ i of partially asymmetric $$N\times N$$ N × N random matrices from the real Elliptic Gaussian ensemble.

^{[4]}We show the existence of the spectrum, prove the stability of the system if the kinematic coefficient of viscosity and the coefficient of temperature conductivity are sufficiently large and the existence of a set of positive real eigenvalues having a point of the real axis as point of accumulation.

^{[5]}The product of a Hermitian matrix and a positive semidefinite matrix has only real eigenvalues.

^{[6]}Non-Hermitian Hamiltonians may still have real eigenvalues, provided that a combined parity-time (ƤƮ) symmetry exists.

^{[7]}The present paper examines the embedding problem for discrete-time Markov chains with three states and with real eigenvalues.

^{[8]}its leading principal submatrix, two distinct real eigenvalues, and part of the corresponding eigenvectors.

^{[9]}However, it is found that several non-Hermitian Hamiltonian still have real eigenvalues, especially the parity-time symmetric system.

^{[10]}For a continuous-time linear positive system, it has been known that if there exists a stabilizing linear time-invariant controller, the number of nonnegative real eigenvalues cannot be greater than one.

^{[11]}We identify a class of operator pencils, arising in a number of applications, which have only real eigenvalues.

^{[12]}There are mainly three types of eigenvalues: diffusion tensor fields with all positive real eigenvalues; the tensor field with negative real eigenvalues; the tensor field with imaginary eigenvalues.

^{[13]}In this paper we consider the case when the matrix Lyapunov inequality, which is part of the Lurie equation, has a matrix with real eigenvalues, some of which may be zero.

^{[14]}We show that readily computable and explicit lower bounds can be found by computing the real eigenvalues of a constant matrix, and LTI controllers, potentially of a low order, can be synthesized to achieve the bounds based on the H ∞ control theory.

^{[15]}The corresponding time independent Schr\"odinger equation yields real eigenvalues with complex eigenfunctions.

^{[16]}The question of which nonsingular commuting complex matrices with real eigenvalues have the same characteristic polynomial is formulated via determinant and trace conditions.

^{[17]}In this paper we formulate concepts of statistical thermodynamics for systems described by non-Hermitian Hamiltonians with real eigenvalues.

^{[18]}Most previous studies on platoon control have only focused on specific communication topologies, especially those with real eigenvalues.

^{[19]}Moiseyev, “Atomic and Molecular Complex Resonances from Real Eigenvalues Using Standard (Hermitian) Electronic Structure Calculations”, J.

^{[20]}The study is restricted to such asymmetric second rank tensors, for which it is still possible to keep the notion of real eigenvalues, but not to accept the mutual orthogonality of the directors of the principal trihedron.

^{[21]}Particularly, this paper shows that the peculiar Neimark-Sacker bifurcations regaining the stability of near-grazing period-one impact motion can be induced by two different ways, either through the interaction between the singular and regular real eigenvalues or via a grazing bifurcation directly.

^{[22]}The corresponding iterations for the matrix square root converge to $A^{1/2}$ for any input matrix $A$ having no nonpositive real eigenvalues.

^{[23]}Existence of infinitely many real eigenvalues will be proven as well as showing that the eigenvalues depend monotonically on the refractive index and boundary parameter.

^{[24]}Furthermore, $$\sigma (\mathcal {A})=\overline{\bigcup _{n=1}\sigma (\Lambda _n)}$$σ(A)=⋃n=1σ(Λn)¯, and there is a sequence of real eigenvalues of $$\mathcal {A}$$A that diverges to negative infinity.

^{[25]}In this paper bounds on the absolute values and the imaginary parts of the non-real eigenvalues of A are proved for uniformly locally integrable potentials q and potentials q ∈ L s ( R ) for some s ∈ [ 1 , ∞ ].

^{[26]}Furthermore, the Hermitian Laplacian guarantees some desirable properties, such as non-negative real eigenvalues and the unitarity of the Fourier transform.

^{[27]}The key contribution is that the tuning can be implemented for complex eigenvalues of the arising graph Laplacian of the network, complementing the current state of the art, which is limited to real eigenvalues.

^{[28]}The present paper deals with non-real eigenvalues of regular nonlocal indefinite Sturm–Liouville problems.

^{[29]}For the systems with system matrix having only real eigenvalues, the near-controllability problem has been solved.

^{[30]}By simply solving the eigenvalue problem of the underlying undamped vibration system, the real eigenvalues and eigenvectors can then be combined with the nonviscous damping matrix to develop an iterative procedure from which required complex eigenvalues and eigenvectors of the damped system can be computed.

^{[31]}Non-Hermitian parity-time (PT) symmetric systems that possess real eigenvalues have been intensively investigated in quantum mechanics and rapidly extended to optics and acoustics demonstrating a lot of unconventional wave phenomena.

^{[32]}The free-motion equation of this nonviscous system yields a nonlinear eigenvalue problem where it has a certain number of real eigenvalues corresponding to the non-oscillatory nature.

^{[33]}Optical systems with gain and loss that respect parity-time (PT) symmetry can have real eigenvalues despite their non-Hermitian character.

^{[34]}Using complex and real eigenvalues, five unstable and three stable rolls are found respectively.

^{[35]}This real symmetric matrix has real eigenvalues and eigenvectors.

^{[36]}We give an estimated bound for the real eigenvalues of the NESS preconditioned matrix whose left end is positive and show that the non-real eigenvalues of the NESS preconditioned matrix are located in an intersection of two rings, particularly, these non-real eigenvalues are located in an intersection of two rings and one circle if t ≥ 1 2.

^{[37]}The first method is applicable to systems with purely real eigenvalues while the other works for partially conjugate complex ones.

^{[38]}

## Positive Real Eigenvalues

We show the existence of the spectrum, prove the stability of the system if the kinematic coefficient of viscosity and the coefficient of temperature conductivity are sufficiently large and the existence of a set of positive real eigenvalues having a point of the real axis as point of accumulation.^{[1]}There are mainly three types of eigenvalues: diffusion tensor fields with all positive real eigenvalues; the tensor field with negative real eigenvalues; the tensor field with imaginary eigenvalues.

^{[2]}