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Algebraic multiplicity

by dlevanchuk
Tags: algebraic, multiplicity
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Mar16-10, 03:04 AM
P: 29
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?
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Mar16-10, 05:48 AM
P: 2,179
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.

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