wu :: forums
« wu :: forums - Linear Algebra ---- eigenvalues/eigenvectors »

Welcome, Guest. Please Login or Register.
Dec 4th, 2024, 10:17am

RIDDLES SITE WRITE MATH! Home Home Help Help Search Search Members Members Login Login Register Register
   wu :: forums
   general
   wanted
(Moderators: SMQ, Eigenray, william wu, Grimbal, towr, ThudnBlunder, Icarus)
   Linear Algebra ---- eigenvalues/eigenvectors
« Previous topic | Next topic »
Pages: 1  Reply Reply Notify of replies Notify of replies Send Topic Send Topic Print Print
   Author  Topic: Linear Algebra ---- eigenvalues/eigenvectors  (Read 3292 times)
MonicaMath
Newbie
*





   


Gender: female
Posts: 43
Linear Algebra ---- eigenvalues/eigenvectors  
« on: Mar 3rd, 2009, 3:00pm »
Quote Quote Modify Modify

hi,,
 
could anyone help me !!
 
What is the relation between the eigenvalues of the matrix A and thos for the matrix B, where
 
A= [ b    a+c   0 0      
   a     b    a+c    0  
   0     a     b     a+c    
   0     0     a b    ]      
 
and  
B =[ d    e    0    0    
   e   d    e     0      
   0    e    d    e    
   0    0    e     d ]  
 
 
with  d ,e  are related to a, b , and c.
 
These are both tridiagonal matrices.
So Im trying to find the eigenvalues and eigenvectors of a nonsymmetric tridiagonal matrix A by finding the eigenvalues and eigenvectors of a symmetric tridiagonal matrix B, with some relation between d,e and b,a,and c .
 
 
 
so, what u suggest ??
 
IP Logged
Eigenray
wu::riddles Moderator
Uberpuzzler
*****






   


Gender: male
Posts: 1948
Re: Linear Algebra ---- eigenvalues/eigenvectors  
« Reply #1 on: Mar 4th, 2009, 1:08am »
Quote Quote Modify Modify

Please don't post the same problem to multiple forums.
 
The idea is probably that for any a,b,c, you can find d,e such that B is similar to A.
 
If this is the case, you need tr A = tr B and det A = det B.  This gives you two equations to relate a,b,c with d,e.  Under these conditions, it turns out A and B actually are similar (assuming e 0).  In fact, show that there is a diagonal matrix D such that AD=DB.
« Last Edit: Mar 4th, 2009, 1:11am by Eigenray » IP Logged
MonicaMath
Newbie
*





   


Gender: female
Posts: 43
Re: Linear Algebra ---- eigenvalues/eigenvectors  
« Reply #2 on: Mar 4th, 2009, 12:54pm »
Quote Quote Modify Modify

HI,
 
Thank you for your replay,
 
I solved the problem, and both A and B have the same eigenvalues, but the eigevectore differ !
 
my solution is as follows,
 
d will be: d =b - 2a,
and: e=sqrt(a^2 + ac ),
 
A and B have the same trace and same determinant, but there still some wrong ??
 
( take as an example a=2, b=3, c=-7  
  so d=3, and e=sqrt(-5) )  
 
so what is my mistake ??
IP Logged
towr
wu::riddles Moderator
Uberpuzzler
*****



Some people are average, some are just mean.

   


Gender: male
Posts: 13730
Re: Linear Algebra ---- eigenvalues/eigenvectors  
« Reply #3 on: Mar 4th, 2009, 1:47pm »
Quote Quote Modify Modify

on Mar 4th, 2009, 12:54pm, MonicaMath wrote:
d will be: d =b - 2a?
Correct me if I'm wrong, but isn't the trace the sum of the elements on the diagonal? And therefore since we have 4 times b on the diagonal in A, and 4 times d on the diagonal in B, we must have d=b.
IP Logged

Wikipedia, Google, Mathworld, Integer sequence DB
Eigenray
wu::riddles Moderator
Uberpuzzler
*****






   


Gender: male
Posts: 1948
Re: Linear Algebra ---- eigenvalues/eigenvectors  
« Reply #4 on: Mar 4th, 2009, 1:56pm »
Quote Quote Modify Modify

If A and B are similar there will be an isomorphism that takes the eigenvectors of one to the eigenvectors of the other.  That is, suppose B = P-1AP.  Then v is an eigenvector of B if and only if Pv is an eigenvector of A (with the same eigenvalue).
IP Logged
MonicaMath
Newbie
*





   


Gender: female
Posts: 43
Re: Linear Algebra ---- eigenvalues/eigenvectors  
« Reply #5 on: Mar 4th, 2009, 2:06pm »
Quote Quote Modify Modify

Sorry for my misstype ..
 
d= - c - 2a  , (not b - 2a)
 
 
and e is still the same.
IP Logged
MonicaMath
Newbie
*





   


Gender: female
Posts: 43
Re: Linear Algebra ---- eigenvalues/eigenvectors  
« Reply #6 on: Mar 4th, 2009, 2:09pm »
Quote Quote Modify Modify


Thanks Mr. Eigenray ,,,  
you are right .... but how I can fine a formula for that matrix P ?
 
if you can help me then my problem will be solved ...
 
thanks
IP Logged
Eigenray
wu::riddles Moderator
Uberpuzzler
*****






   


Gender: male
Posts: 1948
Re: Linear Algebra ---- eigenvalues/eigenvectors  
« Reply #7 on: Mar 4th, 2009, 2:15pm »
Quote Quote Modify Modify

The relation should be d = b,  e2 = a(a+c).  As a hint, it turns out you can take P to be a diagonal matrix.  Since any scalar multiple of P will also work, you can assume the upper-left entry is a 1.  Then there are only three variables you should be able to solve for to get AP = PB.
IP Logged
Pages: 1  Reply Reply Notify of replies Notify of replies Send Topic Send Topic Print Print

« Previous topic | Next topic »

Powered by YaBB 1 Gold - SP 1.4!
Forum software copyright © 2000-2004 Yet another Bulletin Board