How to find the algebraic multiplicity of an eigenvalue?

We will use the fact that the algebraic multiplicity of an eigenvalue equals the number of linearly independent eigenvectors for that eigenvalue. This method works well when the matrix is square. But, for rectangular matrices, we will use the rank of the matrix instead. The rank of a square matrix is equal to the number of linearly independent rows or columns.

How to find the multiplicity of an eigenvalue in MATLAB?

The matlab function eig returns the eigenvalues of a square matrix and the eigenvectors. However, it does not tell you the algebraic multiplicities. You can use the function eigvals to find the algebraic multiplicities of the eigenvalues. For example, consider the following matrix A:

How to find the algebraic multiplicity of an eigenvalue of a matrix?

If there are $n$ linearly independent eigenvectors associated with an eigenvalue $\lambda$, the algebraic multiplicity of $\lambda$ is $n$. The geometric multiplicity of an eigenvalue is the number of Jordan blocks associated with the eigenvalue. Geometric multiplicity is equal to the number of eigenvectors associated with the eigenvalue that are not in the span of the other eigenvectors.

How to find the algebraic multiplicity of an eigenvalue in Maple?

The function eigenvals and eigenspaces can give you the list of eigenvalues and their algebraic multiplicities for a square matrix. If you want to see the number of eigenvalues, just use the keyword count. For example, here we find the number of eigenvalues of a positive-definite matrix A:

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If we find the multiplicity of each eigenvalue of A, then the algebraic multiplicity of the eigenvalue is the sum of the multiplicities of all the eigenvalues which have this eigenvalue as a root. If we find these multiplicities for each of the eigenvalues of A, then we have the algebraic multiplicity of each e