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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.

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marked as duplicate by anorton, Michael Albanese, hardmath, Salech Alhasov, Phira Jan 10 at 17:26

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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 at 15:58

1 Answer 1

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.
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+1 for that first hint... After a whole semester of a linear algebra class, I never had thought of that. :) –  anorton Jan 10 at 16:04

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