I obtained 6 eigenpairs of a matrix using eigs
of Matlab.
How can I demonstrate that these eigenvectors are orthogonal to each other? I am almost sure that I normalized in the right way modulus and phase but they do not seem to be orthogonal. The matrix should be normal.
The matrix comes from the discretization of the Euler-Bernoulli beam problem for a beam of length 1 with hinged free boundary conditions:
$$ \frac{\partial u }{\partial t} + \gamma \frac{\partial^4 y}{\partial x^4} = 0,\\
\frac{\partial y}{\partial t} - u = 0.\\$$
Then, the eigenproblem can be written as:
$$ \lambda \left[ \begin{matrix} I & 0 \\ 0 & I \end{matrix} \right] \left\{ \begin{matrix} y \\ u \end{matrix} \right\} = \left[ \begin{matrix} 0 & I \\ -\gamma B & 0 \end{matrix} \right] \left\{ \begin{matrix} y \\ u \end{matrix} \right\},$$ where $I$ is the identity matrix and $B$ is the bilaplacian operator discretized using finite difference.