How Do You Determine the Algebraic Multiplicity of an Eigenvalue?

[dlevanchuk]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

I did find the basis, and geometric multiplicity (the dimention of eigenspace).. but I can't 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.