The geometric multiplicity of an eigenvalue 
of an 
matrix 
is the dimension of its eigenspace,
 |
where 
denotes the null space, 
is the identity matrix,
and 
denotes matrix rank. It is the maximum number of linearly independent eigenvectors
belonging to 
.
If 
is the algebraic multiplicity of 
, then
 |
A complex matrix can be diagonalized iff these multiplicities agree for every eigenvalue.
For a real matrix to be diagonalizable over 
, all its eigenvalues must additionally
be real numbers.
