How can I generalize the equation in order to find the determinant for given matrix

Question

I have a question related to my previous past. This is just information question.

I have this matrix.

For example, assume that N =4, then I can find eigenvalues with this following equation

$$(2*4 - \lambda)^4 - 4^4=0$$

So in general, for any N, I can find eigenvalues with this following equation

$$(2*N - \lambda)^N - N^N=0$$

(this equation is indeed the determinant of $D_N-\lambda*I$.

Is this equation true to find the determinant of $D_N-\lambda*I$?

Answer 1

Based on the discussion in the comments, it seems that what is really needed is code that will produce the eigenvalues of the matrix $D_N$. We can produce this outcome with the following script:

N = 2^3;
D = circshift(eye(N),-1) + circshift(eye(N),1) - 2*eye(N);
eig(D)