# Finding an analytical expression for the eigen values

I would like to know if there is any way to find an analytical expression for the eigen values of the following matrix.

$$A^h = \frac{1}{h^4} \begin{pmatrix} 5&-4&1&&&&&&\\ -4&6&-4&1&&&&\bigcirc&\\ 1&-4&6&-4&1&&&&\\ &\ddots & \ddots & \ddots & \ddots&&& \\ &&\ddots & \ddots & \ddots & \ddots&& \\ &&&&1&-4&6&-4&1\\ &\bigcirc&&&&1&-4&6&-4\\ &&&&&&1&-4&5\\ \end{pmatrix}$$

I have heard that it is possible through DFT, but I am not sure how to proceed with that. The size of the matrix is $N \times N$ and $h = \frac{1}{N}$.

Thanks!

• Does this come from cubic splines? Nov 13, 2015 at 22:18
• Well, it actually came from the discretization of the bi-harmonic equation with a 5 point finite difference stencil. Nov 13, 2015 at 22:38

Let $h=1$ for simplicity (rescaling is trivial). Note that $A^h=B^2$, where $B$ is the symmetric tridiagonal Toeplitz matrix with non-zero elements $(-1,2,-1)$, i.e. $$B= \begin{bmatrix} 2 & -1 & 0 & 0 & \dots \\ -1 & 2 & -1 & 0 &\dots \\ 0 & -1 & 2 & -1 & \dots \\ & & \ddots & \ddots & \ddots \end{bmatrix}.$$ The eigenvalues of symmetric tridiagonal Toeplitz matrices are well known and can be found in many texts. For $B$, the eigenvalues are $$\lambda_k(B) = 2 \left( 1-\cos{\left( \frac{k \pi}{N+1} \right)} \right).$$ It immediately follows that the eigenvalues of $A^h$ are given by $$\lambda_k(A) = \lambda_k^2(B) = 4 \left( 1-\cos{\left( \frac{k \pi}{N+1} \right)} \right)^2.$$