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Systems of First Order Linear Differential Equations
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
Example 8.1.7: Consider the vector differential equation with a defective matrix: tx˙(t)=Ax(t),whereA=-32-21 $$ t{{\dot{x}}}(t) = {\text{A}}\,{\text{x}}(t),{\text{ where}}\,\,{\text{A}} = \left[ {\begin{array}{*{20}l} { - 3} \hfill & 2 \hfill \\ { - 2} \hfill & 1 \hfill \\ \end{array} } \right] $$
Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2018
Example 8.1.7: Consider the vector differential equation with a defective matrix: tx˙(t)=Ax(t),whereA=-32-21
First order coupled differential equations with constant coefficients
Published in P. Mohana Shankar, Differential Equations, 2018
Because all the eigenvalues are equal and the matrix is defective, this case can be treated as an example of a defective matrix with algebraic multiplicity of 4. The four generalized eigenvectors in this case constitute an identity matrix of size 4 and these eigenvectors are 1000,0100,0010,0001. $$ \left[ {\begin{array}{*{20}l} 1 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{array} } \right],\left[ {\begin{array}{*{20}l} 0 \hfill \\ 1 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{array} } \right],\left[ {\begin{array}{*{20}l} 0 \hfill \\ 0 \hfill \\ 1 \hfill \\ 0 \hfill \\ \end{array} } \right],\left[ {\begin{array}{*{20}l} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 1 \hfill \\ \end{array} } \right]. $$
Complex ground-state and excitation energies in coupled-cluster theory
Published in Molecular Physics, 2021
Simon Thomas, Florian Hampe, Stella Stopkowicz, Jürgen Gauss
At this point, the question arises in which cases the matrix becomes defective. For the case that the eigenvectors describe crossing states of different symmetries, they cannot become linearly dependent. Assuming that none of the other eigenvalues coincide at this point the matrix is non-defective. In the absence of a constraint that ensures linear independence of the eigenvectors, the defectiveness of the matrix seems to be rather the rule than the exception for the following reasons: First, in general, to ensure a non-defective matrix with multiple eigenvalues more conditions have to be fulfilled than for a defective one [43]. Second, in numerical examples providing crossing states of the same symmetry defective matrices occur in all cases [7,8].
Stability of bimodal planar switched linear systems with both stable and unstable subsystems
Published in International Journal of Systems Science, 2022
Swapnil Tripathi, Nikita Agarwal
Consider a Hurwitz stable (defective) matrix with Jordan form , and an unstable (defective) matrix , with . Both and are defective matrices. The matrix is already in normal form, hence is the identity matrix. Consider Jordan basis matrix , then of with Jordan form . We will now use the notations and results from Section 7 to discuss stability of the system (1) with subsystem matrices given here. Using Theorem 7.1, for fixed and , the dwell-flee relation is given by the unique positive solution τ of Observe that the relation does not depend on . Refer to Figure 11 for the graphs of as functions of η, for . With this procedure, the ‘best’ dwell-flee relation is obtained when . Let us take , then .
Heteroepitaxy of diamond semiconductor on iridium: a review
Published in Functional Diamond, 2022
Weihua Wang, Benjian Liu, Leining Zhang, Jiecai Han, Kang Liu, Bing Dai, Jiaqi Zhu
For question (2), the key is related with the role of the ion bombardment and highlighted based on diamond epitaxial nucleation on Ir substrates. When the electric field is applied in the plasma, the charged species (ions and electrons) move directionally. The positively charged ions bombard the substrate while the negatively species move toward the anode. The process of positive ions moving toward the negative substrate is called the ion bombardment [29]. The sub-plantation, preferential etching, and secondary-electron emission are three main consequences under the ion bombardment. The widely used sub-plantation model is proposed by Lifshitz et al. [139, 140]. They carefully studied the mechanism of the film growth from hyper-thermal species. The model involves a shallow sub-plantation process, energy loss, preferential displacement of atoms with low displacement energy, leaving the atoms with the high displacement energy intact, sputtering of substrate material, and inclusion of a new phase due to incorporation of a high density of interstitials in a host matrix. More specifically [141–143], a dense amorphous hydrogenated carbon (a-C:H) layer is firstly formed, and then pure sp3 carbon clusters containing dozens of atoms are spontaneously precipitated in the a-C:H layer, caused by the “thermal spike” of the impinging energetic species [144, 145]. By converting amorphous carbon to diamond at the amorphous matrix-diamond interface, diamond clusters grow to a few nanometers. This transition is caused by a “preferential displacement” mechanism mainly caused by the influence of high-energy hydrogen atoms. On this basis, Schreck et al. [29] proposed the ion bombardment induced buried lateral growth mechanism on Ir substrates in Figure 7. In their model, there are at least five stages in the nucleation process, and Figure 7(a-1–a-5) represents the exposed Ir surface before BEN, a-C:H layer formation, primary nucleation from the a-C:H layer at the interface or the domain formation, a great many secondary nuclei formation from highly defective crystalline matrix or the domain spreading, nuclei growth after BEN. By comparing the height difference across the domain (Figure 7(b-1,b-2)), a mechanism of the ion bombardment induced buried lateral growth is carefully described in Figure 7(c). The a-C:H layer is firstly formed (region III), and then the highly defective matrix including a large amount of sp3 C–C bonds is formed (region II). Diamond nuclei with the crystallinity structure are further formed from this defective matrix (region I). The preferential etching [146, 147] was once thought to be the key mechanism for epitaxial nucleation. However, it just refers to the difference of the ability of epitaxial or non-epitaxial nuclei resisting the ion bombardment. The secondary emission [113, 115] refers to that diamond has the higher electron emission ability than the substrate. Thus, the bias current representing the number of moving species can increase once diamond is deposited. This has been shown in Section 3.2, but not thought to be so important in diamond epitaxial nucleation. Overall, the sub-plantation is more acceptable to researchers with the further research.