Linear Algebra / Geomtric Multiplicity

[Wildcat]

Homework Statement

Let A = a 2x2 matrix row 1 [3/2 ½ ] row 2 [-½ ½]
a) Find the eigenvalue(s) of A
b) Find the algebraic multiplicity of each eigenvalue.
c)Find the geometric multiplicity of each eigenvalue.

Homework Equations

The Attempt at a Solution

I found the eigenvalue to be 1 with an algebraic multiplicity of 2.
I'm not sure about the geometric multiplicity. I think it is 1.
When I look at the dim(ker(A-1I)) I get ½x1 + ½x2=0 which I think means you have 1 degree of freedom therefore the geometric multiplicity is 1.
Can someone tell me if I'm doing this right?

 

[Dick]

That's right. 1 is a double eigenvalue but it only corresponds to a single eigenvector.

 

[Wildcat]

That's right. 1 is a double eigenvalue but it only corresponds to a single eigenvector.

Thanks! I also have to find P and J such that A=PJP^-1 where J is the Jordan canonical form of A, but I'm not finished with that.

 

[Wildcat]

Thanks! I also have to find P and J such that A=PJP^-1 where J is the Jordan canonical form of A, but I'm not finished with that.

I know I need to find the eigenvector corresponding to the eigenvalue 1, I found it to be
[1 -1] but I need another one. Do I choose one where (A-1I)v‡0, can I choose [1 0] for the other??

 

[Wildcat]

I know I need to find the eigenvector corresponding to the eigenvalue 1, I found it to be
[1 -1] but I need another one. Do I choose one where (A-1I)v‡0, can I choose [1 0] for the other??

if I can do that, I get P=[1 1] [-1 0] J=[1 ½] [0 1] P^-1=[0 -1] [1 1] (these are 2x2 matrices)


which does = A [3/2 ½] [-½ ½] ?