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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 Spring '11
 Prof.Ramachandran
 Linear Algebra, Eigenvalue, eigenvector and eigenspace, Singular value decomposition, geometric multiplicity

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