Eigenvectors

When studying linear transformations, it is extremely useful to find nonzero vectors whose direction is left unchanged by the transformation. These are called eigenvectors (also known as characteristic vectors). If v is an eigenvector for the linear transformation T, then T(v) = λv for some scalar λ. This scalar is called an eigenvalue. The eigenvalue of greatest absolute value, along with its associated eigenvector, have special significance for many physical applications. This is because whatever process is represented by the linear transformation often acts repeatedly—feeding output from the last transformation back into another transformation—which results in every arbitrary (nonzero) vector converging on the eigenvector associated with the largest eigenvalue, though rescaled by a power of the eigenvalue. In other words, the long-term behaviour of the system is determined by its eigenvectors.

Finding the eigenvectors and eigenvalues for a linear transformation is often done using matrix algebra, first developed in the mid-19th century by the English mathematician Arthur Cayley. His work formed the foundation for modern linear algebra.