eigenstuff

An eigenvalue (or characteristic value) is a scalar value that represents how a linear transformation changes a vector. More specifically, given the transformation matrix , an eigenvalue is a scalar such that there exists a non-zero eigenvector

There are infinitely many eigenvectors with the same eigenvalue. If we know one, we may always rescale by a scalar . For example, both and has the eigenvalue of

Find eigenvalues and eigenvectors

Summary

Given the matrix encoding the linear transformation To find the eigenvalue, we find the roots of the characteristic polynomial of

After the eigenvalue are found, we can substitute into . Eigenvectors are the basis vectors of its null space

Solving the linear system

is equivalent to the following homogenous system

To find eigenvector, we’re finding non-zero vectors that makes the above system true i.e. find the null space of characteristic polynomial . Since there are infinitely many solutions of eigenvectors, this matrix must be singular

After solving for an eigenvalue , find a solution to the homogeneous system. Any solution will be the eigenvector for that specific eigenvalue

which be rearranged to

Manual & Geometric Intuition