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Eigenvectors and eigenvalues
Corresponding to the general, abstract definition of a linear space (i.e. abstract vector space) and the general, abstract definition of a linear transformation is a general, abstract definition for eigenvectors and eigenvalues. It is the following:
Def. Eigenvector. An eigenvector is a nonzero vector X which is imaged by a linear transformation A into a vector λX , a scalar multiple of itself. That is, it is a vector X such that AX = λX where λ is a scalar called an eigenvalue. An eigenvector of a linear transformation A corresponds to any of those vectors in the domain which are imaged by A into scalar multiples of themselves.
Eigenvectors are also called invariant vectors, characteristic vectors, or latent vectors. Eigenvalues are also called characteristic roots or latent roots.
Eigenvalues and eigenvectors are highly importance in applications. They arise in many areas of mathematics, physics, chemistry and engineering. They arise in analytic geometry in connection with finding that particular coordinate system in which a conic in the plane or a quadric surface in three-dimensional space finds its simplest canonical expression. In physics and engineering they arise in connection with finding, for example, the critical frequencies of a vibrating string, suspension bridge or rotating shaft, the critical load of a supporting column or the energy levels of a system in quantum mechanics.
For the case when the vector space V is n-space and the operator A is an n-square matrix, the eigenvalues are given as the roots of the characteristic equation of matrix A and the eigenvectors are vectors in n-space that A images into scalar multiples of themselves. For the case when A is an integral operator and V is Hilbert space, eigenvectors correspond to those vectors (functions) of Hilbert space that A images into scalar multiples of themselves. In this case the eigenvectors are called eigenfunctions. For example, the homogeneous integral equation
can be written as Af = λf and the solution f(x) of the equation corresponds to those functions that A carries into scalar multiples of themselves.