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Suppose that $a = aBc$ where $a$ and $c$ are vectors and $B$ is some matrix that changes as time "continuously" goes on - making this system dynamical system. But suppose that at any time, if $B$ is linearized into square matrix, it is proven that it is impossible to satisfy $a = aBc$. Does this mean that in non-linear case, there would not be any case that satisfies $a = aBc$? Also, if nonlinear case allows satisfying the equation, what would be the condition? If the latter question is somehow ambiguous, answer to the first question is fine.

Edit: $a$ is transposed form of a vector. About $a$ and $c$ - what happens if we separate the case into two - one that has $a$ and $c$ change as time goes on(which means $B$ changes) and one that has these two fixed?

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You need to elaborate slightly. – copper.hat Dec 29 '12 at 7:27
The meaning of $a=aBc$ is unclear to me. Normally multiplication of a matrix by vectors on both sides results in a scalar. Here it somehow gives a vector... What are the dimensions of these objects? Matrix $B$ is changing; are $a$ and $c$ changing too? – user53153 Dec 29 '12 at 8:06
Can you have a look over my edit? Thanks. – Mark Hyatt Dec 29 '12 at 8:49
Any comment would be appreciated. – Mark Hyatt Dec 29 '12 at 23:28

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