The Eigenvalue Problem

   

• Equation (3.54) can be thought of as an equation in an $n = n_x \times n_y$ dimensional vector space with a linear operator F acting on a vector function a(t) on the right hand side, and a differential operator acting on a(t) on the left hand side. There is a standard way to solve such equations. The way is as follows:
  • If F is a normal   operator, i.e., $\boldsymbol{F} \boldsymbol{F}^\dag = \boldsymbol{F}^\dag\boldsymbol{F}$ then it is possible to find such rotation in ${\mathbb{R} }^n$, i.e., such transformation $\boldsymbol{\Lambda} \in \hbox{SO}(n)$ that when we rotate the basis of the vector space, the operator F in the new basis will be diagonal.
  • If F is not normal then a diagonalizing transformation still exists, but it may not be in $\hbox{SO}(n)$. This is not a big problem: things just get a little more messy.
  • Once F has been diagonalized equation (3.54) splits into n independent differential equations that can be solved trivially.
  • Then we have to rotate things back to the original basis.