The algebraic multiplicity of an eigenvalue 
of a square matrix 
is its multiplicity
as a root of the characteristic
polynomial. Thus, if
 |
where 
and 
is the identity matrix, then 
has algebraic multiplicity 
. For an 
complex matrix, the
algebraic multiplicities of its distinct eigenvalues
sum to 
.
Algebraic multiplicity need not equal geometric multiplicity, which counts the number of linearly independent eigenvectors for the eigenvalue. For example,
![A=[2 1; 0 2].](/images/equations/AlgebraicMultiplicity/NumberedEquation2.svg) |
The eigenvalue 2 has algebraic multiplicity 2 but geometric multiplicity 1.
