线性代数关于特征向量和特征值的定理证明

一个关于特征值和特征向量的定理，最好用英文详细证明。谢谢！

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从我们的书里找的，希望对你有用

it is an interesting problem.  the result as stated is not quite correct (the reason it that not all real eigenvalues can be >=n,  it is only the largest eigenvalue that is guaranteed to be >=n.  this can be seen from the simplest case in which a_{ij} all equal to 1, in this case A is only rank-1, and it can have only one positive real eigenvalue, all the rest are zero eigenvalues which clearly cannot be >=n)

I have provided a detailed proof in the attached file (you may notice that I reword the problem to make it simpler, also, I assign it as an excise with a given answer).  please check.  thanks.