Geometric Multiplicity of an Eigenvalue

Geometric multiplicity of an eigenvalue is equal to the dimension of the eigenspace associated with the eigenvalue.

• It gives us an idea of the set of vectors that satisfy $(A-\lambda I)\vec{x}=0$.

  • Do they form a plane, are they all on a line, stuff like that.

• So it gives us an idea of the sort of vectors that are unchanged during the vector transformations.

  • Maybe a particular line through the origin doesn’t get knocked off it’s span after the transformation.

• This too is defined for each eigenvalue.

• Denoted by $g_{\lambda }$.