1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Show exp(A) is invertible for a matrix A

  1. May 4, 2013 #1
    1. The problem statement, all variables and given/known data

    Define [itex]e^A=I_n+A+\frac{1}{2!}A^2+\frac{1}{3!}A^3+...+\frac{1}{2012!}A2012[/itex]
    where A is an nxn matrix such that [itex]A^{2013}=0[/itex]. Show that [itex]e^A[/itex] is invertible and find an expression for [itex](e^A)^{-1}[/itex] in terms of A.

    3. The attempt at a solution

    To first show that it's invertible, can I just show the function [itex]e^x[/itex] has an inverse?
     
  2. jcsd
  3. May 4, 2013 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Where x is a real number? That would work fine if A were diagonal. I don't think assuming ##A^{2013}=0## makes it any easier. I would just try to show ##e^{sA}e^{tA}=e^{(s+t)A}## first. It's really just the binomial theorem. Then put s=1 and t=(-1).
     
  4. May 5, 2013 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Do you know that [itex]x^n- 1= (x- 1)(1+ x+ x^2+ \cdot\cdot\cdot+ x^{n-1})[/itex]?

    If not try proving it by induction on n. Notice that proof works for matrices as well as numbers.
     
  5. May 5, 2013 #4

    statdad

    User Avatar
    Homework Helper

    Try it first for a matrix B with B^4 = 0 (to make the problem smaller): if you can do it with one where you can see all the terms in the sum and see what happens, writing down the work for your case will just require a little careful writing.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Show exp(A) is invertible for a matrix A
  1. Invertible matrix (Replies: 11)

Loading...