Eigenvalues and eigenvectors of this matrix

[shouvikdatta8]

Consider the nXn matrix A whose elements are given by,

$$A_{ij} = 1 if i=j+1 or i=j-1 or i=1,j=n or i=n,j=1<br /> = 0 otherwise$$
What are the eigenvalues and normalized eigenvectors of A??

 

[CompuChip]

You could start by calculating the characteristic polynomial. Put $-\lambda$ on the diagonal entries and calculate the determinant. Expanding along the first row or column should be easy, because most entries are zero anyway. You might get a recursive equation ;)