algebraic multiplicity

dlevanchuk

1. The problem statement, all variables and given/known data
I have a matrix A [1 -1 -1 -1; -1 1 -1 -1; -1 -1 1 -1; -1 -1 -1 1], its characteristic polynomial p(t) = (t + 2)(t-2)³, and given value of lambda = 2. I need to find basis for eigenspace, and determine algebraic and geometric multiplicities of labmda.


2. Relevant equations



3. The attempt at a solution

I did find the basis, and geometric multiplicity (the dimention of eigenspace).. but I cant figure out how to figure out algebraic multiplicity! I know the correct answer is 3, but why? i was trying to find simple explanation of alg. mult. on google, but the answer come up waay too tangled up for me to understand :-S

EDIT: Is it because the characteristic polynomial is p(t) = (t + 2)(t-2)³ and since my lambda = 2, i need to take (t-2)³ (which is t-2=0 => t=2), and the power is the value of algebraic mult...? am I on the right track?

phyzguy

You're on the right track. The multiplicity is basically how many "copies" of each eigenvalue exist. If each eigenvalue is unique, the multiplicity is 1. Since you have (t-2)^3, the eigenvalue of 2 is repeated three times, so the multiplicity is three, as you said.