China
Africa
Explore chapters and articles related to this topic
Fundamentals of Systems of Differential Equations
Published in Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski, A Course in Differential Equations with Boundary-Value Problems, 2017
Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski
We stated in (5.84) that we combine Jordan blocks together to give us the complete Jordan Canonical Form of a matrix. The size of each Jordan block Jvi in (5.84) is the dimension of each respective set of chain vectors and the total number of blocks in J is the total number of these sets. If we allow for complex eigenvalues, then every matrix can be decomposed into Jordan Canonical Form. However, for our study of differential equations, we will keep our focus on Jordan blocks that are of size 3 × 3 or lower, arising from real repeated eigenvalues, and will refer the interested reader to an advanced linear algebra text such as [23],[52] for further study. As such, we finish with a classification system that will allow us to apply Theorem 5.6.6.
Linear algebra
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
Such a matrix is called a Jordan block; it cannot be diagonalized. Jordan blocks arise whenever a matrix has (over the complex numbers) too few eigenvectors to span the space. In particular, they arise in the basic theory of linear differential equations. We illustrate with the differential equation (D–λI)2y=y″–2λy′+λ2y=0.
Matrix Decomposition with Applications
Published in Crista Arangala, Exploring Linear Algebra, 2019
A Jordan Block is of the form (λ100000λ100000λ100⋮⋮⋮⋱⋱⋮0000λ100000λ).
Actuator fault detection for uncertain systems based on the combination of the interval observer and asymptotical reduced-order observer
Published in International Journal of Control, 2020
Xiangming Zhang, Fanglai Zhu, Shenghui Guo
Consider a Jordan matrix whose all eigenvalues have negative real parts. where each diagonal block matrix is a Jordan block associating with a single eigenvalue. Without loss of generality, we assume that the first r Jordan block matrices associate with a single real eigenvalue with multiplicity of . That is we assume that The last Jordan blocks associate with single conjugated complex eigenvalue with of multiplicity of . That is we assume that where , , , . Denote , then . Moreover, we denote so .