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Review of Linear Algebra
Published in Mohammad Monir Uddin, Computational Methods for Approximation of Large-Scale Dynamical Systems, 2019
Invariant Subspaces: In general, a subspace S ⊂ ℂn is called invariant for the transformation A, or invariant subspaceA-invariant, if Ax ∈ S for every x ∈ S. In other words, S is invariant for A means that the image of S under A is contained in S; AS ⊂ S. For example,kerimg 0, ℂn, ker(A), img(A) are all A-invariant subspaces.
Numerical and Computational Issues in Linear Control and System Theory
Published in William S. Levine, Control System Fundamentals, 2019
A.J. Laub, R.V. Patel, P.M. Van Dooren
A large number of algebraic and dynamic characterizations of controllability have been given; see [7] for a sample. But every one of these has difficulties when implemented in finite arithmetic. For a survey of this topic and numerous examples, see [10] (p. 186). Part of the difficulty in dealing with controllability numerically lies in the intimate relationship with the invariant subspace problem [10](p. 589). The controllable subspace associated with Equation 21.1 is the smallest A-invariant subspace (subspace spanned by eigenvectors or principal vectors) containing the range of B. Since A-invariant subspaces can be extremely sensitive to perturbation, it follows that, so too, is the controllable subspace. Similar remarks apply to the computation of the so-called controllability indices. The example discussed in the third paragraph of Section 21.2 dramatically illustrates these remarks. The matrix A has but one eigenvector (associated with 0) whereas the slightly perturbed A has n eigenvectors associated with the n distinct eigenvalues.
Unsymmetric Matrix Eigenvalue Techniques
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
If the matrix A is large and sparse and just a few eigenvalues are needed, sparse matrix techniques are appropriate. Some examples of common tasks are: (1) find the few eigenvalues of largest modulus, (2) find the few eigenvalues with largest real part, and (3) find the few eigenvalues nearest some target value τ. The corresponding eigenvectors might also be wanted. These tasks are normally accomplished by computing the low-dimensional invariant subspace associated with the desired eigenvalues. Then the information about the eigenvalues and eigenvectors is extracted from the invariant subspace.
New results on algorithms for the computation of output-nulling and input-containing subspaces
Published in International Journal of Control, 2023
Lorenzo Ntogramatzidis, Fabrizio Padula, Augusto Ferrante
A subspace is said to be an -controlled invariant subspace if, for any initial state , there exists a control function such that the state trajectory generated by the system remains indefinitely on ; equivalently, is -controlled invariant if the subspace inclusion holds. The control function that maintains the trajectory on can always be expressed as a static state feedback . The -controlled invariance condition can be equivalently expressed by saying that there exists a feedback matrix F such that . In this case, we say that F is a controlled invariant friend of .