I have to find the eigenvalues of a matrix and find out whether the matrix is ill posed or well-posed, however having done this before on regular matrices, this matrix is a little different.
It is a 2x2 matrix with two elements being a real and an imaginary number, and the other two elements being two operators, the D3 and D4 operator (third differential and fourth diff). The reason for doing this is that I am evaluating the properties of a Hamiltonian in a Hilbert space, and analyse it (without the wavefunction) as a matrix for eigenvalues and eigenvectors.
After solving the secular equation of it, I end up with two eigenvalues which have the differential operators acting on the vector coordinates, x_1 and x_2, among other combinations of the elements.
How do I solve and interpret these two eigenvalues for stability, ill-posedness and well-posedness as well as condition number? Do I have to include the wavefunction the original Hamiltonian acts on in the matrix representations, or is it fully acceptable to analyse the operator as it is in the given matrix for eigenvalues :
\begin{bmatrix} D_3 & D_4 \\ -i3 & 6 \\ \end{bmatrix}
\begin{bmatrix} (D_3 -\lambda)x_1 & (D_4)x_2 \\ (-i3)x_1 & (6-\lambda)x_2 \\ \end{bmatrix}
This last matrix gives the eigenvalues of the secular equation, by forming the quadratic equation of lambda as variable. How do I solve these eigenvalues with the operators acting on vector coordinates, such as x_1D_4 x_2 ?
Thanks!