2.2.4—Eigenvalues and eigenvectors

Eigenvalues satisfy det(A−λI)=0. For each root λ, solve (A−λI)v=0 to find the non-zero eigenvectors in its eigenspace.

The characteristic polynomial may have repeated roots. A repeated algebraic root does not automatically provide the same number of independent eigenvectors; solve the nullspace and count its dimension.

For a diagonal matrix with diagonal entries 3 and −1, det(A−λI)=(3−λ)(−1−λ), giving λ=3 and −1; the corresponding eigenspaces are the coordinate axes.

Do not solve det(A−λ)=0 without the identity matrix, and do not treat λ itself as an eigenvector.