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Algebraic multiplicity 
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#1
Mar1610, 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)(t2)^{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)(t2)^{3} and since my lambda = 2, i need to take (t2)^{3} (which is t2=0 => t=2), and the power is the value of algebraic mult...? am I on the right track? 


#2
Mar1610, 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 (t2)^3, the eigenvalue of 2 is repeated three times, so the multiplicity is three, as you said.



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