Let A, i = 1, 2,...,n be n distinct eigenvalues of a matrix A C***...

 

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Let A₁, i = 1, 2,...,n be n distinct eigenvalues of a matrix A € C*** and let z and y be the corresponding right- and left-unit eigenvectors such that y = 1. Show that and 12 y₁xj=dij, A=Miriyi (Cri)(s; B) 8-A₁ C(sI-A)-¹B= Furthermore, show that the mode A, is controllable iff y; B 0 and the mode A, is observable iff Ca #0. Let A₁, i = 1, 2,...,n be n distinct eigenvalues of a matrix A € C*** and let z and y be the corresponding right- and left-unit eigenvectors such that y = 1. Show that and 12 y₁xj=dij, A=Miriyi (Cri)(s; B) 8-A₁ C(sI-A)-¹B= Furthermore, show that the mode A, is controllable iff y; B 0 and the mode A, is observable iff Ca #0.

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SOLUTION IN DETAILS To prove these statements well use the properties of eigenvalues and eigenvectors along with basic matrix algebra Lets go through ... View the full answer