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I have a question: Suppose I have a $n\times n$ matrix: $$ \begin{bmatrix} 1 & 1 &...& 1 \\ 1 & 1 &...&1 \\ \vdots&\vdots &\ddots & \vdots&\\ 1 & 1 & ...&1 \\ \end{bmatrix} $$ ,then is there a easy way to compute the eigenvalues of the matrix?

How can I compute this matrix eigenvalue?

JimmyK4542
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python3
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1 Answers1

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Hint: Since all the columns are the same, the matrix has rank $1$. Thus, all but one of the eigenvalues are $0$. Can you guess an eigenvector which corresponds to the non-zero eigenvalue, and determine this non-zero eigenvalue?

JimmyK4542
  • 55,969
  • I think it should be n, but how to determine whether how many zero eigenvalues(the multiples of the eigenvalue 0) it has? – python3 Aug 20 '15 at 02:32
  • In general, if an $n \times n$ matrix has rank $r$, then exactly $n-r$ eigenvalues are $0$. – JimmyK4542 Aug 20 '15 at 02:33