Geometric multiplicity

Geometric multiplicity refers to the number of independent directions (or vectors) associated with an eigenvalue of a matrix. Think of it as how many different ways you can "stretch" or "compress" along distinct, non-overlapping directions without changing the overall structure. In mathematical terms, it’s the dimension of the eigenspace corresponding to that eigenvalue, indicating how many independent eigenvectors exist for it. This concept helps understand the matrix’s behavior and whether it can be fully "diagonalized," meaning simplified into a form that makes calculations easier.