1. Not finding help here? Sign up for a free 30min 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!

The Matrix Exponent of the Identity Matrix, I

  1. May 23, 2015 #1
    So, essentially, all I wonder is: What is the The Matrix Exponent of the Identity Matrix, [itex]I[/itex]?

    Silly question perhaps, but here follows my problem. Per definition, the Matrix Exponent of the matrix [itex]A[/itex] is,

    [itex]
    e^{A} = I + A + \frac{A^2}{2} + \ldots = I + \sum_{k=1}^{\infty} \frac{A^k}{k!} = \sum_{k=0}^{\infty} \frac{A^k}{k!}
    [/itex]

    as [itex]e^0 = I[/itex]. I suspected that, since [itex]I^k = I[/itex] for any integer [itex]k[/itex], we would get

    [itex]
    e^{I} = I + I + \frac{I}{2} + \ldots = I \cdot \left( \sum_{k=0}^{\infty} \frac{1}{k!} \right) = I \cdot e,\quad e\approx 2.72
    [/itex]

    such that for an arbitrary constant [itex]a[/itex] we could write

    [itex]
    e^{aI} = I \left( \sum_{k=0}^{\infty} \frac{a^k}{k!} \right) = I e^{a}
    [/itex]

    However, apparently this is not the case as a (suggested) solution to some (homework) problem I've been working on claims that

    [itex]
    e^{aI} = e^{-a} I
    [/itex]

    With a sign change of a!! I think I'm just missing something trivial and fundamental, but I'd really appreciate some help to sort this one out. Might it also be a misprint in the solution?
     
  2. jcsd
  3. May 23, 2015 #2

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    This is correct.

    This is wrong.
     
  4. May 23, 2015 #3

    Mark44

    Staff: Mentor

    @mhsd91, when you post a question, please do not delete the three parts of the homework template. The template is required.
     
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: The Matrix Exponent of the Identity Matrix, I
  1. Identity matrix (Replies: 2)

  2. Identity matrix (Replies: 1)

  3. Matrix exponent rules (Replies: 28)

Loading...