Eigenvalue Cheatsheet
Definitions and identities for eigenvalue problems.
For a square matrix A, a nonzero vector v is an eigenvector if Av = λv. The transformation keeps v on the same line: λ scales it, and a negative λ reverses its direction.
Counting algebraic multiplicity, the eigenvalues sum to the trace and multiply to the determinant. A triangular matrix has its eigenvalues on the diagonal. Similar matrices have the same eigenvalues.
A matrix is diagonalizable when it has a basis of eigenvectors. Real symmetric matrices have real eigenvalues and an orthonormal eigenbasis. For a general matrix, repeated eigenvalues require checking the dimensions of the eigenspaces.