as a part of a computer project, I need to solve
$u''=\lambda u\Longleftrightarrow (T-\lambda) u, $
with $T$ the discretized central difference second derivative (stencil/kernel: 1,-2,1).
Details are also covered in Solve the eigenvalue problem $y''=\lambda y$ numerically.
To solve this numerically, we first use the SciPy eigenvalue functions to obtain the eigenvalues and then use the first eigenvalue to solve the linear system of equations with a kernel of (1,-2-$\lambda$,1). Therefore, this equation is equal to $Au=0$.
My problem is that this matrix $A$ has a determinate $\neq 0 $, therefore it is trivially solved by $u=\vec{0}$. How can I obtain the eigenfunctions numerically if the solutions fall back to the trivial solution.
Note: There are also boundary conditions which also do not produce a determinate $\neq 0$ and/or change b from being $\vec{0}$.
I am thankful for every bit of enlightenment and I would also appreciate feedback about my posting style as I am a first time poster.
Best jw