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Let $A$ be an n *n matrix all of whose entries are 1. Find all the eigenvalues and eigenvectors of $A$. I have checked for 2*2 ,3*3 matrices and guessing the answer but in general how to show.

EuReka
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  • They are all rank 1, and you can explicitly write the eigenvectors, right? Tell us more on what you think the answer should be and why. – Memming Jan 10 '14 at 15:58

1 Answers1

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Hint

  • The dimension of the kernel of $A$ is $n-1$ (why?) so $0$ is an eigenvalue with multiplicity $n-1$;
  • The last eigenvalue is determined by the trace (why?)
  • For the eigenvectors solve the equation $$Ax=\lambda x$$ for $\lambda$ the two founded eigenvalues.
  • +1 for that first hint... After a whole semester of a linear algebra class, I never had thought of that. :) – apnorton Jan 10 '14 at 16:04