My Math Forum  

Go Back   My Math Forum > College Math Forum > Linear Algebra

Linear Algebra Linear Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
September 16th, 2013, 09:29 AM   #1
Newbie
 
Joined: Sep 2013

Posts: 2
Thanks: 0

Eigenvalue Problem

If A is the 4 by 4 matrix of ones, what are the eigenvalues and the determinant of A - I? How to solve this without expanding?
subhajit028 is offline  
 
September 16th, 2013, 10:15 AM   #2
Senior Member
 
Joined: Jun 2013
From: London, England

Posts: 1,316
Thanks: 116

Re: Eigenvalue Problem

The determinant will be 0, as you will have 1 -1 + 1 - 1 times the determinant of the same 3x3 matrix with all 1's.

Any vector (a, b, c, d) will become (a+b+c+d, a+b+c+d, a+b+c+d, a+b+c+d) under the action of this matrix. If this is an eigenvector with eigenvalue k, then a+b+c+d = ka = kb =kc = kd. Hence a = b = c = d and k = 4.
Pero is offline  
September 16th, 2013, 12:16 PM   #3
Newbie
 
Joined: Sep 2013

Posts: 2
Thanks: 0

Re: Eigenvalue Problem

I think you wrote about the eigenvalues and determinant of the matrix A. But I need those of the matrix A-I. I understand A has eigenvalues 4,0,0,0 and Determinant is 0. But how do I get the results for A-I from this?
subhajit028 is offline  
September 16th, 2013, 01:00 PM   #4
Senior Member
 
Joined: Jun 2013
From: London, England

Posts: 1,316
Thanks: 116

Re: Eigenvalue Problem

The matrix A - I acting on the vector (a, b, c, d) will produce the vector (b+c+d, a+c+d, a+b+d, a+b+c). If this is an eigenvector then either:

a = b = c =d and the eigenvalue k = 3 or

k = -1 and a = -(b+c+d)

This can be seen by solving the system of simultaneous equations.

The determinant of A is -9.
Pero is offline  
Reply

  My Math Forum > College Math Forum > Linear Algebra

Tags
eigenvalue, problem



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Not finding one of the eigenvalue colt Linear Algebra 3 April 24th, 2013 11:46 AM
Gerschgorins theorem in eigenvalue problem raymondp44 Applied Math 0 May 8th, 2010 05:57 AM
Eigenvalue wannabe1 Linear Algebra 2 April 17th, 2010 03:33 PM
eigenvalue tinynerdi Linear Algebra 3 April 5th, 2010 01:56 AM
prove it's an eigenvalue 450081592 Linear Algebra 2 February 7th, 2010 12:57 PM





Copyright © 2019 My Math Forum. All rights reserved.