China
Africa
Explore chapters and articles related to this topic
Canonical Forms
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
A Jordan matrix (or a matrix in Jordan canonical form) is a block diagonal matrix having Jordan blocks as the diagonal blocks, i.e., a matrix of the form Jk1(λ1) ⊕ ⋯ ⊕ Jkt (λt) for some positive integers t, k1, …, kt and some λ1, …, λt ∈ F. (Note: the λi need not be distinct.)
State-Space Representation of Control Systems
Published in Anatasia Veloni, Alex Palamides, Control System Problems, 2012
Anatasia Veloni, Alex Palamides
Remarks: The elements of the main diagonal of the Jordan matrix are the eigenvalues of the system.The order of the Jordan submatrices is equal to the multiplicity of the corresponding eigenvalue, and their number is equal to the number of the linear independent eigenvectors.
Linear Algebra Problems
Published in Dingyü Xue, YangQuan Chen, Scientific Computing with MATLAB®, 2018
When the generalized eigenvector matrix V is found, the Jordan canonical form can be obtained from J = V−1AV. It should be noted that the main diagonal elements in the Jordan matrix are the eigenvalues, with the main sub-diagonal elements taking 1’s.
Hybrid control design for limit cycle stabilisation of planar switched systems
Published in International Journal of Control, 2018
Mohammed Benmiloud, Atallah Benalia, Mohamed Djemai, Michael Defoort
The matrix given by fi(x*i, i + 1)CTi, i + 1 has a rank 1. Hence, it has only one nonzero eigenvalue λ1 = CTi, i + 1fi(x*i, i + 1) associated with the eigenvector fi(x*i, i + 1). The other eigenvector, corresponding to the eigenvalue λ2 = 0, is the orthogonal complement to vector Ci, i + 1 (denoted C⊥i, i + 1). Furthermore, let us consider the following first order matrix polynomial: From (Horn and Johnson, 2012), the matrix with X = fi(x*i, i + 1)CTi, i + 1 has the same eigenvectors of X, which are the column vectors of Vi, i + 1 as proved above. Besides, the eigenvalues associated to these eigenvectors can be obtained by evaluating the characteristic polynomial for the eigenvalues of fi(x*i, i + 1)CTi, i + 1. One can get the same eigenvalues of the Jordan matrix J. Using Jordan decomposition, one can get the results of Proposition 3.1.