How to find the algebraic multiplicity of an eigenvalue?

Although the geometric multiplicity is easy to determine (it equals the number of linearly independent eigenvectors), the algebraic multiplicity of an eigenvalue is not as straightforward. The algebraic multiplicity of an eigenvalue is the number of linearly independent eigenvectors that span the eigenspace. So, although two eigenvectors can be linearly independent, if they span the entire eigenspace, they are not linearly independent. The algebra

How to find all the algebraic multiplicities of an eigenvalue?

If you know that there is a single eigenvalue, you can find the algebraic multiplicity in the following way. Let’s say we have the following matrix: A=[a1 b1 a2 b2 a3 b3], where a1, a2, a3 are the first three columns of A, and b1, b2, b3 are the first three rows of B. The first step consists on taking the product of A and B, which results

How to find the quadratic algebraic multiplicities of an eigenvalue?

If you are able to find the eigenvectors of the given matrix, you can use them to find the quadratic algebraic multiplicity of each eigenvalue. To do so, take one eigenvector and compute the quadratic form $v^TAv$ where $A$ is the given matrix and $v$ is the given eigenvector. If the value of $v^TAv$ is zero or a negative number, then the given eigenvalue

How to find the algebraic multiplicities of eigenvalues?

The algebraic multiplicity of an eigenvalue is defined as the number of linearly independent eigenvectors for that eigenvalue. In other words, if you have an eigenvalue $\lambda$, then the algebraic multiplicity of the eigenvalue is the number of linearly independent eigenvectors which also span the associated eigenspace.

How

If you are trying to find the algebraic multiplicity of an eigenvalue of a square matrix, the easiest way is usually to use the property that the sum of the algebraic multiplicities of the eigenvalues of a square matrix equals the total number of rows in its eigenvalue equation. That is, the sum of the algebraic multiplicities of the eigenvalues of a square matrix equals the number of pivot rows in its reduced row echelon form.