Analysis II Lecture 18 Part 1 the matrix exponential
Channel: Arthur Parzygnat
Date: 19/05/2018

The matrix exponential is defined for a linear transformation from a vector space to itself (a square matrix) by using the power series definition of the exponential of a number. The matrix exponential is used to solve autonomous linear ODE's. There are three convenient ways to compute the matrix exponential depending on what sort of matrix you have: diagonal, diagonalizable, arbitrary. The last one can always be put into Jordan form, which is the sum of a diagonal matrix plus a certain nilpotent matrix (whose exponential is relatively easy to calculate). A shear is a simple example of the latter. Although I do not mention it here, one way to put a matrix into Jordan form is to compute a set of generalized eigenvectors for that matrix.

This is part of a series of lectures on Mathematical Analysis II. Topics covered include continuous and differentiable multi-variable functions on Euclidean space, the chain rule, the implicit function theorem, manifolds, tangent spaces, vector fields, the degree and index of a smooth map, the Euler characteristic, metric spaces, the contraction mapping theorem, existence and uniqueness of solutions to ordinary differential equations, and integral equations.

I speak rather slowly, so you may wish to increase the speed of this video.

These videos were created during the 2017 Spring semester at the UConn CETL Lightboard Room.