Handbook of Linear Algebra, 2nd Edition

Chapter 20

Invariant Subspaces

G. W. Stewart

University of Maryland at College Park

An invariant subspace is a generalization of the space spanned by an eigenvector of a matrix. Specifically, an invariant subspace χ of a matrix A is a subspace that is mapped by A into itself. If the columns of X form a basis for χ, then there is a unique matrix L, called the Rayleigh quotient, such that AX = XL — which corresponds to the equation Ax = λx defining an eigenvector and its eigenvalue.

There are good reasons, both theoretical and practical, for working with invariant sub-spaces. For example, the theoretically important Jordan canonical form can be regarded as a decomposition of A into invariant subspaces in which the corresponding Rayleigh quotients ...