Nilpotent matrices

  • Thread starter angelz429
  • Start date
  • #1
24
0
[SOLVED] Nilpotent matrices

Homework Statement




Use the McLaurin series for 1/(1+x) to show that I + N is invertible where N is a nilpotent matrix.



Homework Equations


n/a



The Attempt at a Solution


It has something to do with the inverse of 1+x is 1/(1+x), but i'm lost. I'm not sure where to start nor what exactly I need to do.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,260
619
You might want to start with the hint and find the McLaurin series for 1/(1+x).
 
  • #3
24
0
Ok, I understand that I need to start with the maclaurin series, and I've done that, but I'm not sure what that shows me.
 
  • #4
Dick
Science Advisor
Homework Helper
26,260
619
You might want to show us what you found for the MacLaurin series so we can make sure you are finding the correct thing. Then substitute N for x. What else? What does the nilpotency of N tell you about the infinite MacLaurin series?
 
  • #5
24
0
ok well we know:
P(n) = 1 + x + ... + x^n
xP(n) = x + x^2 + ... + x^n + x^(n+1)

(x-1)P(n)=x^(n+1)-1

P(n) = x^(n+1)-1/(x-1) goes to -1/(x-1) = 1/(1-x) <== close to 1/(1+x)

if you use -x, P(n)= 1/(1+x)


because the MacLaurin series for 1/(1+x)
f(x) = f(0) + f'(0)x + f''(0)x^2 + ...
f(x) = 1 + x + x^2 + ...
 
  • #6
Dick
Science Advisor
Homework Helper
26,260
619
The MacLaurin series for 1/(1-x)=1+x+x^2+x^3+x^4+..., yes. Doesn't that make the series for 1/(1+x)=1-x+x^2-x^3+x^4-...?? Taking x->-x, just as you said?
 
  • #7
24
0
yes...
Let P(n) = 1-x+x^2-x^3+x^4-...+x^n
xP(n) = x-x^2+.................-x^n+x^(n+1)

add them
(x+1)P(n)=1+x^(n+1)

therefore P(n)=1+x^(n+1)/(x+1) which goes to 1/(1+x)
 
  • #8
Dick
Science Advisor
Homework Helper
26,260
619
yes...
Let P(n) = 1-x+x^2-x^3+x^4-...+x^n
xP(n) = x-x^2+.................-x^n+x^(n+1)

add them
(x+1)P(n)=1+x^(n+1)

therefore P(n)=1+x^(n+1)/(x+1) which goes to 1/(1+x)
Ok, so what happens when x is the nilpotent matrix N??
 
  • #9
24
0
P(n) approaches 1/(1+0) = 1
 
  • #10
24
0
so how does this show that I + N is invertible?
 
  • #11
Dick
Science Advisor
Homework Helper
26,260
619
so how does this show that I + N is invertible?
The inverse of I+N is 1/(I+N). Does the MacLaurin expansion converge??
 
  • #12
24
0
Thanks!!!!
 

Related Threads on Nilpotent matrices

  • Last Post
Replies
1
Views
968
  • Last Post
Replies
2
Views
1K
Replies
10
Views
3K
  • Last Post
Replies
8
Views
2K
  • Last Post
Replies
1
Views
7K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
2
Views
4K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
6
Views
734
Top