Advanced Linear Algebra - Advanced Linear Algebra, Lecture 4.4: Invariant subspaces

Advanced Linear Algebra, Lecture 4.4: Invariant subspaces An invariant subspace of a linear map A:X→X is any subspace Y such that A(Y)⊆Y. If X is a direct sum of A-invariant subspaces, then the matrix of A can be written in block-diagonal form, with the blocks corresponding to the subspaces. The generalized eigenvectors of A span an A-invariant subspace, and the matrix of A with respect to this basis is a Jordan matrix. We weave an example throughout this lecture, of an 11x11 matrix with only one eigenvalue and 4 eigenvectors. By drawing the generalized eigenvectors in rows, we can read off features of the minimal and characteristic polynomials right from this diagram. This leads us to definitions of algebraic multiplicity, geometric multiplicity, and the index of an eigenvalue. We characterize these three ways: algebraically in terms of polynomials, geometrically in terms of generalized eigenvectors, and in terms of the Jordan canonical form. We conclude with a key technical lemma for why we can always construct such a "row diagram" of generalized eigenvectors. This will be needed in the following lecture, when we prove that X always has a basis of generalized eigenvectors. Course webpage: http://www.math.clemson.edu/~macaule/math8530-online.html