diagonalizability criterion
Diagonalizing a matrix means finding coordinates in which it acts as pure scaling — A = P D P^-1 with D diagonal. Not every matrix admits this. The diagonalizability criterion is the clean checklist that tells you, before you do any heavy computation, whether the dream is achievable.
There are three equivalent tests, and any one settles it. First: A is diagonalizable iff for every eigenvalue the geometric multiplicity equals the algebraic multiplicity. Second (equivalent): iff the eigenvectors span the whole space, i.e. an eigenbasis exists. Third: iff the minimal polynomial of A splits into distinct linear factors — no repeated roots in the minimal polynomial. Each captures the same fact: enough independent eigenvectors.
A handy sufficient (but not necessary) shortcut: if an n-by-n matrix has n distinct eigenvalues, it is automatically diagonalizable, because distinct eigenvalues force independent eigenvectors. The converse fails — [3, 0; 0, 3] has a repeated eigenvalue yet is already diagonal. And the spectral theorem guarantees every real symmetric (or complex Hermitian) matrix is diagonalizable, in fact by an orthonormal eigenbasis.
Distinct eigenvalues guarantee diagonalizability; repeated eigenvalues require checking each multiplicity.
The minimal-polynomial test is the most practical for theory: A is diagonalizable iff its minimal polynomial is a product of distinct linear factors. Repeated factors in the minimal polynomial are exactly the signature of Jordan blocks bigger than 1x1.