Eigenvalues and eigenvectors describe invariant directions of linear maps. Diagonalization simplifies powers and functions of matrices when a basis of eigenvectors exists.
Details
Concepts
Characteristic polynomial det(A - \lambda I); eigenvalues are its roots.
Eigenvectors satisfy (A - \lambda I)v = 0; eigenspace is Null(A - \lambda I).
Algebraic vs geometric multiplicity; diagonalizable iff sum of eigenspace dimensions equals n.
Diagonalization: A = PDP^{-1} with D diagonal of eigenvalues if eigenvectors form a basis.
Applications: computing A^k, solving linear recurrences/ODEs, spectral decompositions (symmetric A).
Spectral theorem: real symmetric matrices are orthogonally diagonalizable.
Exercises
Hands-on
Find eigenvalues/eigenvectors of a 3×3 matrix and determine if it is diagonalizable.
Compute A^k using diagonalization or Jordan form when applicable.
For a symmetric matrix, show orthogonality of eigenvectors corresponding to distinct eigenvalues.