How to find geometric multiplicity?
Understand the Problem
The question is asking how to determine the geometric multiplicity of a matrix. Geometric multiplicity refers to the number of linearly independent eigenvectors associated with a particular eigenvalue of a matrix, which can be found by calculating the nullity of the eigenvalue's associated eigenvector matrix.
Answer
The dimension of the eigenspace (or null space of A - λI).
The geometric multiplicity of an eigenvalue is the dimension of its eigenspace, which can be found by determining the dimension of the null space (kernel) of the matrix (A - λI).
Answer for screen readers
The geometric multiplicity of an eigenvalue is the dimension of its eigenspace, which can be found by determining the dimension of the null space (kernel) of the matrix (A - λI).
More Information
Geometric multiplicity provides insight into the nature of eigenvalues and their associated eigenvectors. It is always less than or equal to the algebraic multiplicity of the eigenvalue.
Tips
A common mistake is to confuse the dimension of the eigenspace with the total number of eigenvalues. Ensure to solve for the null space accurately.
Sources
- Algebraic and geometric multiplicity of eigenvalues - StatLect - statlect.com
- Tutorial: how to find the geometric multiplicity of an eigenvalue - statlect.com
- How to find Geometric Multiplicity? - GeeksforGeeks - geeksforgeeks.org
