A community for students.
Here's the question you clicked on:
 0 viewing
anonymous
 5 years ago
: Linear Algebra: (Diagonalization) Let A = {{1,1},{0,1}}. Use the definition of the matrix exponential to compute e^A.
anonymous
 5 years ago
: Linear Algebra: (Diagonalization) Let A = {{1,1},{0,1}}. Use the definition of the matrix exponential to compute e^A.

This Question is Closed

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0A = \[\left[\begin{matrix}1 & 1 \\ 0 & 1\end{matrix}\right]\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Write A as S + N, where S is the identity, and N is the nilpotent matrix with only a 1 in the 1,2 position.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Okay, 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.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0more scanning, one sec.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0the 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.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0hm, 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}}.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0oh 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 :)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Something like this, its short, but it gives your argument more foundation :) very nice observation.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0oops forgot some 0's in there <.<

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0i 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.
Ask your own question
Sign UpFind more explanations on OpenStudy
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
 Engagement 19 Mad Hatter
 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.