Canonical Forms
7.1 The Jordan Canonical Form
Let \(\mathsf T\) be a linear operator on a finite-dimensional vector space \(\mathsf V\) over \(F\), and assume its characteristic polynomial splits into linear factors over \(F\). A Jordan canonical basis \(\beta\) puts its matrix into block-diagonal form:
\[ [\mathsf{T}]_\beta = \begin{pmatrix} A_1 & O & \dots & O \\ O & A_2 & \dots & O \\ \vdots & \vdots & & \vdots \\ O & O & \dots & A_k \end{pmatrix}, \]
where each \(O\) is a zero matrix, and each \(A_i\) is a square matrix of the form \((\lambda)\) or \[ \begin{pmatrix} \lambda & 1 & 0 & \dots & 0 & 0 \\ 0 & \lambda & 1 & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & 0 & \dots & \lambda & 1 \\ 0 & 0 & 0 & \dots & 0 & \lambda \end{pmatrix} \]
for some eigenvalue \(\lambda\) of \(\mathsf{T}\). Such a matrix \(A_i\) is called a Jordan block corresponding to \(\lambda\), and the matrix \([\mathsf{T}]_\beta\) is called a Jordan canonical form of \(\mathsf{T}\). We also say that the ordered basis \(\beta\) is a Jordan canonical basis for \(\mathsf{T}\). Observe that each Jordan block \(A_i\) is “almost” a diagonal matrix—in fact, \([\mathsf{T}]_\beta\) is a diagonal matrix if and only if each \(A_i\) is of the form \((\lambda)\).
Let \(\mathsf{T}\) be a linear operator on a vector space \(\mathsf{V}\), and let \(\lambda\) be a scalar. A nonzero vector \(x\) in \(\mathsf{V}\) is called a generalized eigenvector of \(\mathsf{T}\) corresponding to \(\lambda\) if \[ (\mathsf{T} - \lambda\mathsf{I})^p(x) = 0 \] for some positive integer \(p\).
Notice that if \(x\) is a generalized eigenvector of \(\mathsf{T}\) corresponding to \(\lambda\), and \(p\) is the smallest positive integer for which \((\mathsf{T} - \lambda\mathsf{I})^p(x) = 0\), then \((\mathsf{T} - \lambda\mathsf{I})^{p-1}(x)\) is an eigenvector of \(\mathsf{T}\) corresponding to \(\lambda\). Therefore \(\lambda\) is an eigenvalue of \(\mathsf{T}\).
Let \(\mathsf{T}\) be a linear operator on a vector space \(\mathsf{V}\), and let \(\lambda\) be an eigenvalue of \(\mathsf{T}\). The generalized eigenspace of \(\mathsf{T}\) corresponding to \(\lambda\), denoted \(\mathsf{K}_\lambda\), is the subset of \(\mathsf{V}\) defined by \[ \mathsf{K}_\lambda = \{x \in \mathsf{V} : (\mathsf{T} - \lambda\mathsf{I})^p(x) = 0 \text{ for some positive integer } p\}. \]
Note that \(\mathsf{K}_\lambda\) consists of the zero vector and all generalized eigenvectors corresponding to \(\lambda\).
7.2 Generalized Eigenspace Properties
Theorem 7.1 Let \(\mathsf{T}\) be a linear operator on a vector space \(\mathsf{V}\), and let \(\lambda\) be an eigenvalue of \(\mathsf{T}\). Then:
\(\mathsf{K}_\lambda\) is a \(\mathsf{T}\)-invariant subspace of \(\mathsf{V}\) containing \(\mathsf{E}_\lambda\).
For any scalar \(\mu \neq \lambda\), the restriction of \(\mathsf{T} - \mu\mathsf{I}\) to \(\mathsf{K}_\lambda\) is one-to-one.
Theorem 7.2 Let \(\mathsf{T}\) be a linear operator on a finite-dimensional vector space \(\mathsf{V}\) such that the characteristic polynomial of \(\mathsf{T}\) splits. Suppose that \(\lambda\) is an eigenvalue of \(\mathsf{T}\) with multiplicity \(m\). Then:
\(\dim(\mathsf{K}_\lambda) \le m\).
\(\mathsf{K}_\lambda = \mathsf{N}((\mathsf{T} - \lambda\mathsf{I})^m)\).
Theorem 7.3 Let \(\mathsf{T}\) be a linear operator on a finite-dimensional vector space \(\mathsf{V}\) such that the characteristic polynomial of \(\mathsf{T}\) splits, and let \(\lambda_1, \lambda_2, \dots, \lambda_k\) be the distinct eigenvalues of \(\mathsf{T}\). Then, for every \(x \in \mathsf{V}\), there exist vectors \(v_i \in \mathsf{K}_{\lambda_i}\), \(1 \le i \le k\), such that \[ x = v_1 + v_2 + \dots + v_k. \]
Theorem 7.4 Let \(\mathsf{T}\) be a linear operator on a finite-dimensional vector space \(\mathsf{V}\) such that the characteristic polynomial of \(\mathsf{T}\) splits, and let \(\lambda_1, \lambda_2, \dots, \lambda_k\) be the distinct eigenvalues of \(\mathsf{T}\) with corresponding multiplicities \(m_1, m_2, \dots, m_k\). For \(1 \le i \le k\), let \(\beta_i\) be an ordered basis for \(\mathsf{K}_{\lambda_i}\). Then the following statements are true.
\(\beta_i \cap \beta_j = \varnothing\) for \(i \neq j\).
\(\beta = \beta_1 \cup \beta_2 \cup \dots \cup \beta_k\) is an ordered basis for \(\mathsf{V}\).
\(\dim(\mathsf{K}_{\lambda_i}) = m_i\) for all \(i\).
Let \(\mathsf{T}\) be a linear operator on a vector space \(\mathsf{V}\), and let \(x\) be a generalized eigenvector of \(\mathsf{T}\) corresponding to the eigenvalue \(\lambda\). Suppose that \(p\) is the smallest positive integer for which \((\mathsf{T} - \lambda\mathsf{I})^p(x) = 0\). Then the ordered set \[ \{(\mathsf{T} - \lambda\mathsf{I})^{p-1}(x), (\mathsf{T} - \lambda\mathsf{I})^{p-2}(x), \dots, (\mathsf{T} - \lambda\mathsf{I})(x), x\} \] is called a cycle of generalized eigenvectors of \(\mathsf{T}\) corresponding to \(\lambda\). The vectors \((\mathsf{T} - \lambda\mathsf{I})^{p-1}(x)\) and \(x\) are called the initial vector and the end vector of the cycle, respectively. We say that the length of the cycle is \(p\).
Theorem 7.5 Let \(\mathsf{T}\) be a linear operator on a finite-dimensional vector space \(\mathsf{V}\) whose characteristic polynomial splits, and suppose that \(\beta\) is a basis for \(\mathsf{V}\) such that \(\beta\) is a disjoint union of cycles of generalized eigenvectors of \(\mathsf{T}\). Then the following statements are true. (a) For each cycle \(\gamma\) of generalized eigenvectors contained in \(\beta\), \(\mathsf{W} = \operatorname{span}(\gamma)\) is \(\mathsf{T}\)-invariant, and \([\mathsf{T}_{\mathsf{W}}]_\gamma\) is a Jordan block. (b) \(\beta\) is a Jordan canonical basis for \(\mathsf{V}\).
Theorem 7.6 Let \(\mathsf{T}\) be a linear operator on a vector space \(\mathsf{V}\), and let \(\lambda\) be an eigenvalue of \(\mathsf{T}\). Suppose that \(\gamma_1, \gamma_2, \dots, \gamma_q\) are cycles of generalized eigenvectors of \(\mathsf{T}\) corresponding to \(\lambda\) such that the initial vectors of the \(\gamma_i\)’s are distinct and form a linearly independent set. Then the \(\gamma_i\)’s are disjoint, and their union \(\gamma = \bigcup_{i=1}^q \gamma_i\) is linearly independent.
Theorem 7.7 (Existence and Uniqueness of Jordan Form) If the characteristic polynomial of a linear operator on a finite-dimensional vector space splits over the base field, then a Jordan canonical basis exists. Its Jordan blocks are unique up to their order.
In particular, every complex square matrix is similar to a Jordan matrix. For an eigenvalue \(\lambda\), the number of Jordan blocks equals the dimension of its eigenspace, and the sum of their sizes equals its algebraic multiplicity.
Reference: MIT 18.700: Jordan Normal Form Notes.