In a nuclear reactor, neutrons are born (from fission or n,2n reactions) and neutrons are lost (by diffusion or absorption). The rates of these processes are all dependent on how many neutrons there are (call it X), and by conservation of neutrons, the mechanisms can be written:
Losses * X = Gains * X
or equivalently: (Losses - Gains) * X = 0
As it turns out, when you discretize this (matrix) equation over some spatial mesh (and sometimes energy and angular meshes as well), that is a classic eigenvalue problem AX=0 or (L - lambdaG)X = 0. You can just write down the loss and gain terms in every spatial mesh point and pass the matrix into an eigenvalue solver like the power method and you'll get a value for lambda and X. X (the eigenvector) is your neutron distribution in the reactor, it tells you where the neutrons are, where they're going, and how fast they are. This determines how much fission power is being produced where and how much each material is transmuting into fission products and whatnot. The eigenvalue lambda is a number that tells you how critical or not the chain reaction is. If it's 1.0, you're critical. If it's less than 1 you're exponentially growing (going up in power or exploding), and if you're above one, you're exponentially decaying.
And that my friends is how computing eigenvalues is the key to nuclear engineering.
Losses * X = Gains * X
or equivalently: (Losses - Gains) * X = 0
As it turns out, when you discretize this (matrix) equation over some spatial mesh (and sometimes energy and angular meshes as well), that is a classic eigenvalue problem AX=0 or (L - lambdaG)X = 0. You can just write down the loss and gain terms in every spatial mesh point and pass the matrix into an eigenvalue solver like the power method and you'll get a value for lambda and X. X (the eigenvector) is your neutron distribution in the reactor, it tells you where the neutrons are, where they're going, and how fast they are. This determines how much fission power is being produced where and how much each material is transmuting into fission products and whatnot. The eigenvalue lambda is a number that tells you how critical or not the chain reaction is. If it's 1.0, you're critical. If it's less than 1 you're exponentially growing (going up in power or exploding), and if you're above one, you're exponentially decaying.
And that my friends is how computing eigenvalues is the key to nuclear engineering.