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REVIEW OF MATRICES AND VECTORS
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assume asymptotically stable characteristic equation coefficient matrix completely observable completely state controllable Consider the following Consider the system derivative determined diagonal matrix dynamic system eigenvalues eigenvectors elements equilibrium example following form following system fundamental matrix given by Eq Gram matrix Hence Hermitian form Hermitian matrix initial condition input Jordan canonical form Liapunov function linearly independent matrix differential equation method of Liapunov minimal polynomial multiple n x n matrix negative definite nonsingular matrix nonzero Notice obtain optimal control optimal control system origin orthogonal orthogonal matrix output performance index positive definite Problem proof prove quadratic form rank real symmetric matrix satisfies scalar differential equation Section semidefinite solution of Eq solving space equation space representation sufficient condition symmetric matrix system is completely Theorem time-varying transfer function transformation matrix unique variables vector matrix differential vector space vector state vector x'Ax zero