Given is the following nxn matrix for $n >1$:
$$ \begin{pmatrix} b & a & .. & a \\ a & b & ... & : \\ : & ... & ... & a \\ a & .. & a & b \end{pmatrix} $$
a) Show that $b-a$ is an eigenvalue.
b) Determine the dimension of the eigenspace for $b-a$.
c) Find an eigenvector for an eigenvalue different from $b-a$
I have already shown a). For b), I think that the dimension is $n-1$, given that there are $n-1$ "free variables". Is that correct ? And for c), I don't really see how to proceed. Thanks for your help.
b) The dimension of the eigenspace associated with $b-a$ is $n-1$ because it contains $n-1$ linearly independente vectors:$$(1,-1,0,0,\ldots,0),(1,0,-1,0,\ldots,0),\ldots,(1,0,0,0,\ldots,-1).$$The dimension cant-t be $n$ because otherwise the matrix would be a diagonal one.
c) If $A$ is your matrix, then $A.(1,1,1,\ldots,1)=\bigl((n-1)a+b\bigr)(1,1,1,\ldots,1)$.