Jordan normal form

Jordan normal form represents a complex square matrix, up to similarity, as a block diagonal matrix of Jordan blocks. Each block records an eigenvalue and the lengths of chains of generalized eigenvectors.

Jordan block: A square matrix with one eigenvalue on its diagonal, ones on the superdiagonal, and zeros elsewhere. These blocks are the pieces assembled along the diagonal of the form.
Characteristic polynomial: The polynomial det(tI − A) whose roots are the eigenvalues of a square matrix A. Its roots identify the eigenvalues represented in the Jordan blocks.
Matrix exponential: The matrix-valued function eᴬ defined by the power series Σ Aⁿ/n!. Jordan blocks reduce its computation to exponentials multiplied by finite polynomials.
Schur decomposition: A unitary similarity that transforms a complex square matrix into upper triangular form. It offers a numerically stable alternative that does not require generalized eigenvectors.
Diagonalizability criterion: The condition that a matrix is diagonalizable exactly when its minimal polynomial has no repeated roots over the field. Jordan block sizes show why repeated roots can prevent diagonalization.
Generalized eigenvector: A vector annihilated by some positive power of a matrix minus a scalar multiple of the identity. Chains of these vectors determine each block’s size.
Eigenspace: The subspace of vectors satisfying Av = λv for a fixed eigenvalue λ. Its dimension counts the Jordan blocks associated with that eigenvalue.
Linear differential equation: A differential equation in which a function and its derivatives enter linearly. Jordan form yields explicit solutions for constant-coefficient systems x′ = Ax.
Rational canonical form: A canonical block form for a linear operator over a field, built from companion matrices of invariant factors. It works over general fields where Jordan blocks may not exist.
Invariant subspace: A subspace mapped into itself by a given linear transformation. The spans of initial segments of Jordan chains form invariant subspaces.

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