How to find the geometric multiplicity of an eigenvalue?

To find the geometric multiplicity of an eigenvalue, first note that the geometric multiplicity of the eigenvalue is equal to the algebraic multiplicity of the eigenvalue plus the number of linearly independent eigenvectors associated with the eigenvalue. For example, if we have an $n$ by $n$ square matrix $A$, the geometric multiplicity of the eigenvalue $0$ is $n$, since this eigenvalue has $n$ lin

How to find

If you are trying to understand the geometric multiplicity of an eigenvalue, it is often helpful to use a graphical approach. Let's say we have a square matrix A with an eigenvalue λ. If you graph each of the columns of A as a vector in a three-dimensional space, an eigenvector of A associated with λ would be a vector that lies in that column and pointing towards the point where the column is located in the three-dimensional space. To find

How to find the geometric multiplicity of an eigenvalue in MATLAB?

The geometric multiplicity of an eigenvalue is the number of linearly independent eigenvectors for the associated eigenvalue. If you have an eigenvalue with geometric multiplicity one, you can find an eigenvector by solving the system of equations. If you have an eigenvalue with geometric multiplicity greater than one, you can find an eigenvector by solving the system of equations for each eigenvalue and then taking the first column of the results.

How to find the geometric multiplicity of an eigenvalue in sympy?

SymPy has a function named eigvals which returns an array of generalized eigenvectors and eigenvalues of a matrix. There’s also a function called sympy.linalg.eveig which does the same thing, but only for real symmetric matrices. Using either function, you can find the geometric multiplicity of an eigenvalue. The algorithm works by computing the rank of the matrix formed by the eigenvectors that correspond to the eigenvalue

How to find the algebraic multiplicity of an eigenvalue?

For complex eigenvalues, the geometric and algebraic multiplicities are the same. However, for real eigenvalues, the algebraic and geometric multiplicities can be different. The algebraic multiplicity is the number of eigenvectors associated with the eigenvalue. You can find the algebraic multiplicity of an eigenvalue by solving for the nullspace of the matrix A - λI, which is the eigenvalue problem for A - λI.