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I am having a hard time understand these two concepts

Algebraic multiplicity and Geometric multiplicity of a matrix regarding its eigenvalues

for example

if I have the matrix:

| 5 0 0 |
| 1 5 0 |
| 0 1 5 |

The eigenvalues are 5,5,5, so what does this mean about its multiplicity?

Is geometric multiplicity the number of similar eigenvalue? In this case, 3

and algebraic multiplicity the number of unique eigenvalue? In this case, 1

thanks

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2 Answers 2

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The geometric multiplicity of an eigenvalue is defined to be the number of linearly independent eigenvectors associated with that eigenvalue.

The algebraic multiplicity of an eigenvalue is defined as the eigenvalue's multiplicity as a root of the characteristic polynomial.

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    $\begingroup$ what do you mean by " as a root of the characteristic equation" ? If the characteristic equation has n roots, does it mean it has algebraic multiplicity of n? $\endgroup$
    – JLL
    Apr 9, 2014 at 1:51
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    $\begingroup$ the characteristic equation is det(A-$\lambda$I)=0. And yes if it has n roots then it has algebraic multiplicity of n. $\endgroup$
    – TheBluegrassMathematician
    Apr 9, 2014 at 1:54
  • $\begingroup$ For example, consider the characteristic equation $\lambda^2-4\lambda+4$. Then it does have two roots, but note that it has only have 1 distinct root: $\lambda = 2$. Hence, $\lambda$ would have algebraic multiplicity of 2. $\endgroup$
    – Kaj Hansen
    Apr 9, 2014 at 1:55
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You have the types of multiplicity reversed. Geometric multiplicity here basically means how many different eigenvectors can you create given these eigenvalues. the eigenvalues are the same, therefore the number of distinct eigenvectors is the same. You have geometric multiplicity of 1. Algebraic multiplicity is how many solutions does the solving for eignenvalues give you. It doesn't matter if the solutions are the same. So you have algebraic multiplicity of 3.

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