Let A be an $n \times n$ matrix. Prove that $dim(span{I_n,A,A^2,...}) \leq n$.

Asked Apr 23 '20 at 18:45

Active Apr 23 '20 at 18:45

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Let A be an $n \times n$ matrix. Prove that $dim(span{I_n,A,A^2,...}) \leq n$.

I am thinking about the theorem( Let T be a linear operator on a finite-dimensional vector space V, and let W denote the T-cyclic subspace of v generated by a noznero vector $v \in V$. Let k=dim(W), then ${v,T(v),T^2(v),...,T^{k-1}(v)}$ is a basis for W). I am also encouraged to think about Cayley-Hamilton theorem, but I don't see the interconnection here. Can someone point out what to do? Thanks.

asked Apr 23 '20 at 18:45

spruce

2

Cayley Hamiltion shows that $A^n \in \operatorname{sp} { I, A, ..., A^{n-1} }$. – copper.hat Apr 23 '20 at 19:02

Let A be an $n \times n$ matrix. Prove that $dim(span{I_n,A,A^2,...}) \leq n$.

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