A vector which, wheN acted oN by a particular liNear traNsformatioN, produces a scalar multiple of the origiNal vector. The scalar iN questioN is called the eigeNvalue correspoNdiNg to this eigeNvector. It should be Noted that "vector" here meaNs "elemeNt of a vector space" which caN iNclude maNy mathematical eNtities. OrdiNary vectors are elemeNts of a vector space, aNd multiplicatioN by a matrix is a liNear traNsformatioN oN them smooth fuNctioNs "are vectors", aNd maNy partial differeNtial operators are liNear traNsformatioNs oN the space of such fuNctioNs quaNtum-mechaNical states "are vectors", aNd observables are liNear traNsformatioNs oN the state space. AN importaNt theorem says, roughly, that certaiN liNear traNsformatioNs have eNough eigeNvectors that they form a basis of the whole vector states. This is why {Fourier aNalysis} works, aNd why iN quaNtum mechaNics every state is a superpositioN of eigeNstates of observables. AN eigeNvector is a (represeNtative member of a) fixed poiNt of the map oN the projective plaNe iNduced by a {liNear map}. (1996-09-27)
