| > GIVEN A an n x n matrix
> p a postive integer
> x any vector
>
> and p
> A x does NOT equal 0
>
> p+1
> A x = 0
>
> 2 p
> Question : are the elements of the set (x Ax A x......A x)
>
> dependent or independent
These p+1 vectors must perforce (a word I haven't used in a while) be
independent. Say not, then there are a bunch of coefficients, t_i, such
that S = sum(i=0...p) t_i * A(^i) * x = 0. And not all the t_i are 0. Then
there will be some smallest I such that t_I is non zero.
Now what is A(^(p-I)) * S ? It is evidently just t_I * A(^p) * x,
since all higher terms vanish. But S = 0. Contradiction. So the vectors
are independent.
Regards,
Andrew.
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