Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

How would you find eigenvalues/eigenvectors of a $n\times n$ matrix where each diagonal entry is scalar $d$ and all other entries are $1$ ? I am looking for a decomposition but cannot find anything for this.
For example:

$\begin{pmatrix}2&1&1&1\\1&2&1&1\\1&1&2&1\\1&1&1&2\end{pmatrix}$

share|improve this question
    
Dupe of math.stackexchange.com/questions/1873/…. You can edit the original question to make it clearer instead of asking a new one. –  Graviton Aug 9 '10 at 7:31
    
You should have clarified your original question; finding exact eigenvectors / eigenvalues of structured matrices as opposed to finding approximate eigenvectors / eigenvalues of arbitrary matrices are very different problems. –  Qiaochu Yuan Aug 9 '10 at 7:47
    
they are different questions. one requires a general process of finding eigenvectors while this is specific to this matrix. –  user957 Aug 9 '10 at 8:02

1 Answer 1

up vote 7 down vote accepted

The matrix is $(d-1)I + J$ where $I$ is the identity matrix and $J$ is the all-ones matrix, so once you have the eigenvectors and eigenvalues of $J$ the eigenvectors of $(d-1)I + J$ are the same and the eigenvalues are each $d-1$ greater. (Convince yourself that this works.)

But $J$ has rank $1$, so it has eigenvalue $0$ with multiplicity $n-1$. The last eigenvalue is $n$, and it's quite easy to write down all the eigenvectors.

share|improve this answer
1  
It really should be hammered often enough: exploit any structure you find in your problem to the hilt. –  J. M. Aug 9 '10 at 9:28

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.