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44 Multiplicity of Eigenvalues

# 44 Multiplicity of Eigenvalues - Multiplicity of...

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Multiplicity of Eigenvalues Learning Goals: to see the difference between algebraic and geometric multiplicity. We have seen an example of a matrix that does not have a basis’ worth of eigenvectors. For example. 1 1 0 1 ! " # \$ % (note: this is not the Fibonacci matrix!). The characteristic polynomial of this matrix is (1 - λ ) 2 , so 1 is a double root of this polynomial. Finding the nullspace of A - λ I leads us to consider 0 1 0 0 ! " # \$ % , whose nullspace only contains (1, 0) (and multiples of it). There is no place else to look for eigenvectors, so this matrix does not have an adequate supply! This leads us to two definitions: Definition: the algebraic multiplicity of an eigenvalue e is the power to which ( λ e ) divides the characteristic polynomial. Definition: the geometric multiplicity of an eigenvalue is the number of linearly independent eigenvectors associated with it. That is, it is the dimension of the nullspace of A eI . In the example above, 1 has algebraic multiplicity two and geometric multiplicity 1.

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44 Multiplicity of Eigenvalues - Multiplicity of...

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