Register to reply

Algebraic Multiplicity and Eigenspace

by Hashmeer
Tags: algebraic, eigenspace, multiplicity
Share this thread:
Hashmeer
#1
Apr9-10, 10:14 AM
P: 17
1. The problem statement, all variables and given/known data
Find h in the matrix A such that the eigenspace for lambda=5 is two-dimensional.

A= [5,-2,6,-1] [0,3,h,0] [0,0,5,4] [0,0,0,1]
A-lambda*I(n) = [0,-2,6,-1] [0,-2,h,0] [0,0,0,4] [0,0,0,-4]


2. Relevant equations



3. The attempt at a solution
I'm not really sure how to do this. Is there some relation between algebraic multiplicity and the eigenspace of a matrix that would help?

I tried solving this by row operations to solve for the eigenvectors in the hope that I would be able to eliminate values of h that would have resulted in more or less than a 2 dimensional eigenspace, but it didn't work out. This is the matrix I got (keep in mind this is after applying the lambda value to the diagonals) A= [0,-2,6,-1] [0,0, h-6,1] [0,0,0,4] [0,0,0,0].

I'm pretty confused about this any help would be greatly appreciated. Thanks.
Phys.Org News Partner Science news on Phys.org
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100
icystrike
#2
Apr9-10, 11:33 AM
P: 436
Quote Quote by Hashmeer View Post
1. The problem statement, all variables and given/known data
Find h in the matrix A such that the eigenspace for lambda=5 is two-dimensional.

A= [5,-2,6,-1] [0,3,h,0] [0,0,5,4] [0,0,0,1]
A-lambda*I(n) = [0,-2,6,-1] [0,-2,h,0] [0,0,0,4] [0,0,0,-4]


2. Relevant equations



3. The attempt at a solution
I'm not really sure how to do this. Is there some relation between algebraic multiplicity and the eigenspace of a matrix that would help?

I tried solving this by row operations to solve for the eigenvectors in the hope that I would be able to eliminate values of h that would have resulted in more or less than a 2 dimensional eigenspace, but it didn't work out. This is the matrix I got (keep in mind this is after applying the lambda value to the diagonals) A= [0,-2,6,-1] [0,0, h-6,1] [0,0,0,4] [0,0,0,0].

I'm pretty confused about this any help would be greatly appreciated. Thanks.

Hey there! the eigenspace of having lambda=5 is exactly the nullspace of A-lambda I . and since it is 2 dimensional , it suggest that the rank of the matrix is ?
Hashmeer
#3
Apr9-10, 04:51 PM
P: 17
So the rank = 2 since rank = # columns (4 in this case) - dimNul A (in this case 2). So if the rank is to equal 2 then I will need another free variable, or I need to remove a pivot position. So since h-6 is in a pivot position I can easily make it a nonpivot column by setting h=6. This would ensure that the dimension of the null space is 2.

Am I going the right way with this? It makes sense to me if this is right. Thanks for the help!

HallsofIvy
#4
Apr10-10, 06:27 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,565
Algebraic Multiplicity and Eigenspace

Quote Quote by Hashmeer View Post
1. The problem statement, all variables and given/known data
Find h in the matrix A such that the eigenspace for lambda=5 is two-dimensional.

A= [5,-2,6,-1] [0,3,h,0] [0,0,5,4] [0,0,0,1]
A-lambda*I(n) = [0,-2,6,-1] [0,-2,h,0] [0,0,0,4] [0,0,0,-4]
So any eigenvector of A corresponding to eigenvalue 5 must satisfy
[tex]\begin{bmatrix}0 & -2 & 6 & 1 \\ 0 & -2 & h & 0 \\ 0 & 0 & 0 & 4 \\ 0 & 0 & 0 & -4\end{bmatrix}\begin{bmatrix}w \\ x \\ y \\ z\end{bmatrix}= \begin{bmatrix}0 \\ 0 \\ 0 \\ 0\end{bmatrix}[/tex].

That gives the equations -2x+ 6y+ z= 0, -2x+ hy= 0, 4z= 0, and -4z= 0. The last two obviously give z= 0 so the first two equations become -2x+ 6y= 0 and -2x+ hy= 0. One obvious eigenvector is (u, 0, 0, 0). There will be another, independent, eigenvector, and so the eigenspace will be two dimensional if and only if there exist non-zero x and y satifying both -2x+ 6y= 0 and -2x+ hy= 0.


2. Relevant equations



3. The attempt at a solution
I'm not really sure how to do this. Is there some relation between algebraic multiplicity and the eigenspace of a matrix that would help?

I tried solving this by row operations to solve for the eigenvectors in the hope that I would be able to eliminate values of h that would have resulted in more or less than a 2 dimensional eigenspace, but it didn't work out. This is the matrix I got (keep in mind this is after applying the lambda value to the diagonals) A= [0,-2,6,-1] [0,0, h-6,1] [0,0,0,4] [0,0,0,0].

I'm pretty confused about this any help would be greatly appreciated. Thanks.


Register to reply

Related Discussions
Eigenvalues + Algebraic/Geometric Multiplicity Calculus & Beyond Homework 2
Algebraic multiplicity Calculus & Beyond Homework 1
Question of algebraic flavor in algebraic topolgy Differential Geometry 2
Dimension of eigenspace, multiplicity of zero of char.pol. Linear & Abstract Algebra 5
Eigenspace of A? Calculus & Beyond Homework 1