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Matrix Eigenvalue Methods
Double Bracket Isospectral Flows
Singular Value Decomposition
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action algorithm analysis Appendix applications approach approximation arbitrary balanced realizations Brockett called Chapter characterized closed compact complex computing connected consider constraint continuous Control controllable and observable converges critical points decomposition defined denote derivative determined developed differential equation dimension double bracket dynamical systems eigenvalues element equilibrium point equivalent exists exponentially fact factorizations Figure follows function geometry given GL(n global gradient flow Gramians induced initial condition invariant isospectral least squares Lemma Lie group linear systems manifold Math matrix methods minimal norm normal Note numerical obtain optimal orbit orthogonal orthogonal matrix positive definite problem projection Proof properties prove quadratic rank recursive refer respect result Riccati equation Riemannian metric satisfies sensitivity shows similar singular value smooth solution stable standard studied submanifold subset tangent space termed Theorem theory tion transfer function transformation unique vector field vector space