Saturday, 30 August 2014

linear algebra - Find the eigenvalues and eigenvectors of the matrix with all diagonal elements as d and rest 1



A matrix has all elements 1 except the diagonal elements. It is an n×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 ?


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,,1)T (what are the eigenvectors corresponding to the eigenvalue(s) 0?).



Now note that M=J+(d1)In and that for any matrix ARn×n, xRn and rR if Ax=λx then (A+rIn)x=(λ+r)x.


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