Quantcast

Got Homework?

Connect with other students for help. It's a free community.

  • across
    MIT Grad Student
    Online now
  • laura*
    Helped 1,000 students
    Online now
  • Hero
    College Math Guru
    Online now

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

windsylph Group Title

: Linear Algebra: (Diagonalization) Let A = {{1,1},{0,1}}. Use the definition of the matrix exponential to compute e^A.

  • 2 years ago
  • 2 years ago

  • This Question is Closed
  1. windsylph Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    A = \[\left[\begin{matrix}1 & 1 \\ 0 & 1\end{matrix}\right]\]

    • 2 years ago
  2. joemath314159 Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    Write A as S + N, where S is the identity, and N is the nilpotent matrix with only a 1 in the 1,2 position.

    • 2 years ago
  3. windsylph Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    Okay, here's what I got before I posted this question: {{e, 1+ 2/2! + 3/3! + ... + n/n!},{0,e}}. The book says {{e,e},{0,e}}, but I don't get how 1+ 2/2! + 3/3! + ... + n/n! = e, lest I'm wrong on the row 1 col 2 entry.

    • 2 years ago
  4. joemath314159 Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    more scanning, one sec.

    • 2 years ago
  5. joemath314159 Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    the idea is if you matrix isnt diagonalizable, you want to write it as the sum of a matrix that is, and a matrix that is nilpotent, which means that after some finite power, the matrix will be 0. Also, these matrices must commute.

    • 2 years ago
  6. joemath314159 Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    • 2 years ago
    1 Attachment
  7. joemath314159 Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    • 2 years ago
    1 Attachment
  8. windsylph Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    hm, I really can't use the fact that A = S+N since our haven't gone over nilpotent matrices yet (or we won't at all), so here's what I did, haha I don't know how to go from here to {{e,e},{0,e}}.

    • 2 years ago
    1 Attachment
  9. joemath314159 Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    oh cool, so you came up with an idea of what A^n might be by observation, and used that, very clever :) To make that method a little more solid, prove your guess for A^n by Induction. Then nobody will have any problem with that solution :)

    • 2 years ago
  10. joemath314159 Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    Something like this, its short, but it gives your argument more foundation :) very nice observation.

    • 2 years ago
    1 Attachment
  11. joemath314159 Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    oops forgot some 0's in there <.<

    • 2 years ago
  12. windsylph Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    wow, thanks!

    • 2 years ago
  13. joemath314159 Group Title
    Best Response
    You've already chosen the best response.
    Medals 1

    i hope they teach you about nilpotent matrices =/ thats really important when calculating matrix exponentials. In general, you might not be able to guess a formula for A^n if it isnt diagonalizable. thats why you need to be able to write it as S + N, where S is diagonalizable, and N is nilpotent, and S and N commute. If you are interested in that sorta stuff, look up the Jordan Canonical Form.

    • 2 years ago
    • Attachments:

See more questions >>>

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.