Nondiagonalizable

In linear algebra, a square matrix A is called nondiagonalizable if it is not similar to any diagonal matrix. In other words, A cannot be written as P D P^{-1} with D diagonal. A matrix is diagonalizable if it has a full set of linearly independent eigenvectors, i.e., there exist n eigenvectors that span the space.

Key concepts include eigenvalues and their multiplicities. For each eigenvalue λ, the algebraic multiplicity is its multiplicity as a root of the characteristic polynomial, while the geometric multiplicity is the dimension of the eigenspace ker(A − λI). A matrix is diagonalizable iff, for every eigenvalue, the geometric multiplicity equals its algebraic multiplicity, so the direct sum of eigenspaces has dimension n. Equivalently, the minimal polynomial of A splits into distinct linear factors over the field.

Jordan form provides a constructive view. Over the complex numbers (or any algebraically closed field), every square matrix is similar to a Jordan canonical form, a block-diagonal matrix with Jordan blocks. The matrix is diagonalizable exactly when all Jordan blocks have size 1; if any block has size greater than 1, the matrix is non-diagonalizable.

Examples help illustrate the concept. The 2×2 matrix N = [[0, 1], [0, 0]] has eigenvalue 0 of algebraic multiplicity 2 but only one linearly independent eigenvector, hence it is not diagonalizable. Similarly, A = [[1, 1], [0, 1]] has eigenvalue 1 with algebraic multiplicity 2 and only one eigenvector, so it is nondiagonalizable.

Over the real field, nonreal eigenvalues occur in conjugate pairs and correspond to real Jordan blocks of size 2.