Jordan form describes exactly how diagonalization can fail over an algebraically closed field. The failure is controlled by nilpotent chains attached to eigenvalues.
Minimal polynomial
Definition. The minimal polynomial \(m_T(t)\) of an operator \(T\) is the monic polynomial of least degree such that \(m_T(T)=0\). It divides every polynomial \(p\) with \(p(T)=0\), and it divides the characteristic polynomial.
Proposition. The operator \(T\) is diagonalizable over \(F\) if and only if its minimal polynomial is a product of distinct linear factors over \(F\). The largest exponent of \((t- \lambda)\) in \(m_T\) measures the largest Jordan block size for \(\lambda\).
Generalized eigenspaces
Definition. For an eigenvalue \(\lambda\), the generalized eigenspace is \[ G_\lambda=\ker(T- \lambda I)^N \] for \(N\) large enough, for example \(N=\dim V\). Vectors in \(G_\lambda\) eventually become genuine eigenvectors after repeatedly applying \(T- \lambda I\).
Theorem. If the characteristic polynomial of \(T\) splits as \(\prod_\lambda(t- \lambda)^{a_\lambda}\), then \[ V=\bigoplus_\lambda G_\lambda, \qquad \dim G_\lambda=a_\lambda. \]
Proof. The powers \((t- \lambda)^{a_\lambda}\) are pairwise coprime. Bezout identities among these factors produce polynomial projection operators whose images are the generalized eigenspaces. The direct-sum decomposition follows from coprimeness, and dimensions agree with algebraic multiplicities by triangularizing \(T\).
Jordan chains
Definition. A Jordan chain of length \(r\) for \(\lambda\) is a list \(v_1, \ldots,v_r\) satisfying \[ (T- \lambda I)v_1=0, \qquad (T- \lambda I)v_j=v_{j-1}\quad (j\ge2). \] In this basis, the restriction of \(T\) has a Jordan block with \(\lambda\) on the diagonal and ones on the superdiagonal.
Theorem. Over an algebraically closed field, every linear operator has a basis formed by Jordan chains. In that basis, the matrix is block diagonal with Jordan blocks \[ J_r(\lambda)= \begin{pmatrix} \lambda&1&0&\cdots&0\\ 0&\lambda&1&\cdots&0\\ \vdots&\vdots&\ddots&\ddots&\vdots\\ 0&0&\cdots&\lambda&1\\ 0&0&\cdots&0&\lambda \end{pmatrix}. \]
Reading a Jordan form
Calculation. The number of Jordan blocks for \(\lambda\) is \(\dim\ker(T- \lambda I)\). The number of blocks of size at least \(k\) is \[ \dim\ker(T- \lambda I)^k-\dim\ker(T- \lambda I)^{k-1}. \] These dimensions determine the block sizes.
Powers and functions of Jordan blocks
Proposition. If \(J=\lambda I+N\) with \(N^r=0\), then \[ J^m=\sum_{k=0}^{r-1}\binom{m}{k}\lambda^{m-k}N^k. \] This formula explains polynomial factors in powers of non-diagonalizable matrices.
Remark. Jordan form is fragile numerically: small perturbations can split repeated eigenvalues. Conceptually, however, it is the cleanest finite-dimensional classification of one operator over an algebraically closed field.
A single Jordan block
Example. For \[ J=\begin{pmatrix}\lambda&1\\0&\lambda\end{pmatrix}, \qquad N=J-\lambda I=\begin{pmatrix}0&1\\0&0\end{pmatrix}, \] one has \(N^2=0\) but \(N\neq0\). Hence \((J-\lambda I)^2=0\) while \(J-\lambda I\neq0\), so the minimal polynomial is \((t-\lambda)^2\). The extra superdiagonal entry is exactly the data that prevents diagonalization.
Calculation. The powers are \[ J^k=(\lambda I+N)^k=\lambda^k I+k\lambda^{k-1}N \] when \(\lambda\neq0\), because all terms containing \(N^2\) vanish. This formula explains why Jordan blocks introduce polynomial factors in matrix powers and matrix exponentials.
Construction of Jordan chains
Theorem (nilpotent Jordan decomposition). If \(N^m=0\) on a finite-dimensional space, there is a basis consisting of chains \[ v,\ Nv,\ \ldots,\ N^{r-1}v \] written in reverse order so that \(N\) has nilpotent Jordan blocks.
Proof. Put \(K_j=\ker N^j\). Choose a basis of \(K_1\), extend it to a basis of \(K_2\), and continue through \[ 0\subset K_1\subset K_2\subset\cdots\subset K_m=V. \] For each new class in \(K_j/K_{j-1}\), choose a representative \(w\). Its vectors \(N^{j-1}w,\ldots,Nw,w\) form a chain. They are independent: in a linear relation, apply the highest useful power of \(N\) and use the independence of the selected classes in successive quotients. Their total number is \[ \sum_{j\ge1}j\bigl( \dim K_j-2\dim K_{j-1}+\dim K_{j-2} \bigr)=\dim V, \] so the chains form a basis.
Apply this theorem to \(N=(T-\lambda I)|_{G_\lambda}\) on each generalized eigenspace. The primary decomposition then combines the resulting chain bases and proves the full Jordan theorem rather than merely asserting it.
Recovering block multiplicities
Let \(d_k=\dim\ker(T-\lambda I)^k\), with \(d_0=0\). Then \[ d_k-d_{k-1} \] is the number of blocks of size at least \(k\), while \[ 2d_k-d_{k-1}-d_{k+1} \] is the number of blocks of size exactly \(k\). These invariants are basis-free, so two Jordan matrices are similar exactly when they have the same block data for every eigenvalue.