Jordan Normal Form

The canonical form for operators that cannot be diagonalized.

The idea

Jordan normal form. Every operator on a finite-dimensional complex vector space has a basis in which its matrix is block-diagonal, each block being a Jordan block: one eigenvalue $\lambda$ repeated down the diagonal with $1$s on the superdiagonal, as in $J_3(\lambda) = \begin{pmatrix} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{pmatrix}.$ The operator determines the eigenvalues and sizes of the blocks, so the form is unique up to the order of the blocks.

Jordan form replaces diagonalization when diagonalization fails. Diagonalizing requires a basis of eigenvectors, and some operators have too few: an eigenvalue can repeat $k$ times and still govern fewer than $k$ independent directions. A Jordan block of size $k$ needs only one genuine eigenvector $v_1$, extended by a chain $v_2, \ldots, v_k$ with $(T - \lambda I)v_j = v_{j-1}$; the superdiagonal $1$s record precisely those links, and a suitable collection of chains always exists.

Each block contributes exactly one eigenvector, so the number of blocks belonging to an eigenvalue is its geometric multiplicity, while the sizes of those blocks sum to its algebraic multiplicity. A diagonal matrix is the case in which every block has size $1$, so an operator is diagonalizable exactly when the two multiplicities agree for every eigenvalue.

Ways to work on it

• Walkthrough. Jordan blocks, eigenvector counts, and when diagonalization fails.
• Practice. Count Jordan blocks from algebraic and geometric multiplicities.
• Hardest. Recover exact block sizes from kernel dimensions of powers.

Not sure where to start? Take the ten-question placement test.