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Background
Published in Quan Zheng, David Skillicorn, Social Networks with Rich Edge Semantics, 2017
From another perspective, an eigendecomposition represents a transformation of an initial space, where the matrix entries are coordinates with respect to the standard basis, to a new space, spanned by the eigenvectors, and coordinates in this space that are better behaved. For example, if the space is not of full rank, such a transformation can reveal that the data lie on a lower-dimensional manifold.
Solving systems of algebraic equations
Published in Victor A. Bloomfield, Using R for Numerical Analysis in Science and Engineering, 2018
The familiar process of finding the eigenvalues and eigenvectors of a square matrix can be viewed as eigendecomposition. It factors the matrix into VDV−1, where D is a diagonal matrix formed from the eigenvalues, and the columns of V are the corresponding eigenvectors.
Data-driven feedback stabilisation of nonlinear systems: Koopman-based model predictive control
Published in International Journal of Control, 2023
Abhinav Narasingam, Sang Hwan Son, Joseph Sang-Il Kwon
Calculating Λ: The time-series data of snapshot pairs satisfying the dynamical system of (1) are generated and organised in the following matrices: where , and is the discretisation time. Note is used here instead of because the data need not necessarily be temporally ordered as long as the corresponding pairs are obtained as shown above.A library of nonlinear observable functions is selected to define the vector-valued function where is used to lift the system from a state space to a function space of observables.A least-squares problem is solved over all the data samples to obtain which is the transpose of the finite-dimensional approximation to the Koopman operator, : The value of that minimises (18) can be determined analytically as: where denotes the pseudoinverse, and the data matrices are given by where It has been previously shown that the matrix asymptotically approaches the Koopman operator as we increase (Korda & Mezić, 2018b), and hence approximates the evolution of observables.An eigendecomposition of is performed to determine the eigenvalues and eigenvectors for .The eigenvalues are converted to continuous time as , and the eigenfunctions, , are computed, using , according to the procedure described in (9).The system matrix Λ is constructed using the block-diagonalisation described in (11).