Lets say i have a linear system of equations arising from the discretization of a space-time dependent PDE. Now, if i could find out all the eigen-values,vectors of the linear system, how or where would i be able to effectively use it ? I am quite sure i can use the eigenmodes to solve the linear problem at hand but since the process of finding all the eigenvalues is expensive, this seems like a very small value addition. Then the primary question becomes, "Is there any physical system where you would be interested in how all the eigenmodes behaved or do most systems primary care only about the first few dominant modes ?". To give you a little background, I'm doing my PhD in reactor physics and work mostly on computational methods. Please feel free to ask me relevant questions to make this clearer or if you see a use for such a numerical scheme while dealing with a physical system.