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Question is, what does "index" mean?

For systems of order greater than the number of characteristic roots of $C$ of index one

Also, can anyone explain why is $u_1 + u_2 + n -1 =0$ and what "equivalence theorem" it is referring to?

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The index here appears to refer to the index of a root of the characteristic equation (otherwise called the eigenvalue of the corresponding matrix). Say the characteristic equation is $(\lambda-a_1)^2(\lambda-a_2)=0$. Then $a_1$ has index 2, while $a_2$ has index 1. As for the equivalence theorem, I'm not sure, but you could refer to the index of the book for help.

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