💐 Eigenvalue

Eigenvalues and eigenvectors explain the properties of a 🗺️ Linear Mapping. For square matrix , is an eigenvalue if it satisfies

for some , and is an eigenvector. Furthermore, it also satisfies and .

We can find eigenvalues by solving the characteristic polynomial. is an eigenvalue if and only if it is a solution to

Note that and .

The algebraic multiplicity of is the number of times it appears as a root in the characteristic polynomial.

Note

The 📖 Determinant of a matrix is the product of its eigenvalues.

Info

If has distinct eigenvalues, its eigenvectors are linearly independent. A matrix with less than linearly independent eigenvectors is called defective.

Eigenspace §

The set of all eigenvectors associated with eigenvalue spans the eigenspace . This is the solution space of

The geometric multiplicity of is the dimension of . This value is always between and the algebraic multiplicity, inclusive.

Spectral Theorem §

The Spectral theorem states that if is symmetric, there exists an orthonormal basis consisting of eigenvectors of , and each eigenvalue is real.

Power Method §

The Power method is an algorithm for finding eigenvalues of square matrix .

Any vector can be written as a summation over scaled eigenvectors, . Then, we have the following observation.

Every time we multiply by , the eigenvector corresponding the largest eigenvalue gets bigger; multiplying by multiple times, we approach this eigenvector.

After multiplying multiple times by , we find this eigenvector , project it off, and continue to find .