If p < q the pair (A, B) is not controllable If m < q the pair (A, K2) is not observable


Proof:   The proofs to properties (1) and (2) are similar so we only prove (1). This will be accomplished by showing the existence of a row vector v' such that


v'EXI - A , B] = 0


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for some x which is an eigenvalue of

     From the structure of A, it is C and hence there is a row vector w'


A.

apparent that x is an eigenvalue of such that


                              w'[XI - C] = 0


Now define a matrix Q . Rqxqr to be the following


Q= 0


* o .   Wm]


It can be readily seen that


Q [xI - A] = 0


Now let D . RqxP be the matrix product of Q and B. That is


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