Advanced Linear Algebra - Advanced Linear Algebra, Lecture 4.7: Jordan canonical form

Advanced Linear Algebra, Lecture 4.7: Jordan canonical form The spectral theorems says that if A:X→X is a linear map on a finite-dimensional vector space over an algebraically closed field, then X has a basis of generalized eigenvectors, and we gave an explicit construction of this in the previous lecture. The matrix form of such a basis is the Jordan canonical form, which is a block-diagonal matrix of "Jordan blocks". After introducing this, we consider two commuting maps A and B. We show that X always has a basis of common generalized eigenvectors. In the case when A and B are diagonalizable, then A and B are simultaneously diagonalizable. In matrix form, this means that for some matrix P, whose columns are common eigenvectors to A and B, both P^{-1}AP and P^{-1}BP are diagonal. Course webpage: http://www.math.clemson.edu/~macaule/math8530-online.html