The geometric multiplicity of an eigenvalue is the dimension of the eigenspace associated with that eigenvalue. In other words, it is the number of linearly independent eigenvectors corresponding to that eigenvalue.
Algebraic multiplicity is the number of times an eigenvalue appears as a root of the characteristic polynomial, while geometric multiplicity is the dimension of the eigenspace associated with that eigenvalue. In other words, algebraic multiplicity counts the number of times an eigenvalue appears, while geometric multiplicity measures the "size" of the eigenspace.
The geometric multiplicity of an eigenvalue gives important information about the behavior and properties of a matrix or linear transformation. It can help determine the diagonalizability of a matrix, and it is closely related to the stability and convergence of iterative methods for solving systems of linear equations.
The geometric multiplicity of an eigenvalue can be calculated by finding the null space of the matrix A - λI, where A is the original matrix and λ is the eigenvalue in question. The dimension of the null space will be equal to the geometric multiplicity.
Yes, it is possible for the geometric multiplicity to be greater than the algebraic multiplicity. This means that there are more linearly independent eigenvectors associated with the eigenvalue than the number of times it appears as a root of the characteristic polynomial. However, the geometric multiplicity can never be greater than the dimension of the matrix or linear transformation.