A matrix has all elements 1 except the diagonal elements. It is an $n\times n$ matrix. What are the eigenvectors and eigenvalues ?
Solving book problems in Strang book and stuck on this one and I have no idea where to begin ?
[Math] Find the eigenvalues and eigenvectors of the matrix with all diagonal elements as $d$ and rest $1$
linear algebra
Best Answer
Denote the matrix as $M$.
Let $J$ be the matrix with all entries $1$.
The matrix $J$ has rank $1$ therefore only one non zero eigenvalue which is $n$ with corresponding eigenvector $(1,1,\ldots,1)^T$ (what are the eigenvectors corresponding to the eigenvalue(s) $0$?).
Now note that $M=J+(d-1)I_n$ and that for any matrix $A\in\mathbb R^{n\times n}, \ x\in\mathbb R^n$ and $r\in\mathbb R$ if $Ax=\lambda x $ then $(A+rI_n)x=(\lambda+r)x$.