Characterization of Jordan Vectors

• Jordan vectors are generalized eigenvectors defined by specific algebraic, analytic, and geometric chain conditions.
• They facilitate the analysis of non-diagonalizable operators, matrix polynomials, and operator-valued functions across various dimensions.
• Their study enables precise decompositions in algebraic geometry and provides insights into critical phenomena in statistical mechanics.

A Jordan vector is a generalization of the concept of an eigenvector and arises naturally in several settings involving linear operators, matrix polynomials, and operator-valued functions. Jordan vectors play a central role in the algebraic and geometric analysis of non-diagonalizable operators, underpinning both classical finite-dimensional structures such as Jordan canonical form and the analytic theory of operator-valued function zeros. Their systematic characterization extends across operator theory, algebraic geometry, and mathematical physics.

1. Algebraic Definition and Jordan Chains

Let $X$ be a linear operator on a (finite- or infinite-dimensional) vector space $V$, and let $\mu$ be an eigenvalue of $X$. The classical algebraic definition specifies that a vector $v$ is a Jordan vector of rank $k$ (or length $k$) for eigenvalue $\mu$ if

$(X - \mu\,\mathrm{id})^k v = 0$

but $(X - \mu\,\mathrm{id})^{k-1} v \neq 0$. A Jordan chain of length $k$ is a sequence $\{v_1, v_2, \dots, v_k\}$ in $V$ satisfying \begin{align*} (X - \mu\,\mathrm{id}) v_1 &= 0, \ (X - \mu\,\mathrm{id}) v_{j+1} &= v_j,\quad 1 \leq j < k. \end{align*} Here, $v_1$ is an ordinary eigenvector and $v_2, \dots, v_k$ are generalized eigenvectors (Morin-Duchesne et al., 2013, Abo et al., 2015).

2. Analytic Characterization for Operator-Valued Functions

Let $\mathcal{H}$ be a complex Hilbert space, and let $$Q\colon D(Q) \subset \mathbb{C} \to \mathcal{L}(\mathcal{H})$$ be a holomorphic operator-valued function. A root function $\phi(z)$ for $Q$ at a zero $\alpha$ of order at least $\ell$ is a holomorphic $\phi(z)$ with $\phi(\alpha) \neq 0$ such that

$$(Q(z)\phi(z))^{(j)}(z \to \alpha) \to 0\quad \text{for}\quad j=0,1,\dots, \ell-1.$$

A sequence $\{\phi_0, \dots, \phi_{\ell-1}\}$ is a Jordan chain of length $\ell$ for $Q$ at $\alpha$ if, for each $j = 0,1,\dots,\ell-1$,

$$\sum_{p=0}^j \frac{1}{p!} Q^{(p)}(\alpha) \phi_{j-p} = 0.$$

The main analytic result is that there is a bijection between Taylor expansions of root functions of order $\ell$ and Jordan chains of length $\ell$ for $Q$ at $\alpha$. Precisely, any root function $$\phi(z) = \sum_{s=0}^{\ell-1} (z-\alpha)^s \phi_s + (z-\alpha)^\ell O(z-\alpha)$$ yields a Jordan chain $\{\phi_0, \dots, \phi_{\ell-1}\}$ by $\phi_s = \phi^{(s)}(\alpha)/s!$, and vice versa (Borogovac, 9 Dec 2025).

3. Geometric Perspective: Eigenschemes and Tangent Bundles

In the algebraic-geometric framework, the collection of all (generalized) eigenvectors of a matrix $A$ is captured by the eigenscheme $Z_A \subset \mathbb{P}^{r-1}$, defined by the vanishing of the $2\times 2$ minors of the matrix $(A\,x\,|\,x)$. The reduced components correspond to the eigenspaces; the non-reduced scheme-theoretic structure retains information about higher-rank Jordan vectors.

At a point $[v_1] \in Z_A$ representing an eigenvector, the tangent directions in the scheme at $[v_1]$ correspond to vectors $w$ such that $(A-\lambda I) w \in \mathrm{Span}(v_1)$, i.e., $w$ is a length-2 generalized eigenvector. Higher-order nilpotents in the stalk of the local ring at $[v_1]$ correspond to longer Jordan chains. The structure of the eigenscheme, via its primary decomposition, encodes the block sizes and multiplicities of the Jordan canonical form (Abo et al., 2015).

4. Generalization: Matrix Polynomials and Operator-Valued Functions

For a matrix polynomial $L(z) = \sum_{i=0}^\ell A_i z^i$, the classical chain relations,

$$\sum_{p=0}^j \frac{1}{p!} L^{(p)}(\alpha) \phi_{j-p} = 0,\quad j=0,\dots,k-1,$$

exactly characterize those chains $\{\phi_0, \dots, \phi_{k-1}\}$ that produce solutions to differential equations of the form

$$u(t) = [\phi_0 + t\phi_1 + \cdots + t^{k-1}/(k-1)! \phi_{k-1}] e^{\alpha t}.$$

The holomorphic operator-valued function framework strictly extends this finite-dimensional theory: if $Q(z)$ is a general operator-valued function, the analytic characterization by Taylor series and derivational conditions recovers the full Jordan structure without reliance on generalized inverses or resolvent expansions. This analytic approach encompasses both the finite-dimensional and infinite-dimensional settings (Borogovac, 9 Dec 2025).

5. Physical and Algebraic Context: Jordan Vectors in Loop Models

In statistical mechanics, Jordan vectors and cells govern the indecomposable structure of transfer matrices and Hamiltonians in critical lattice models. For the transfer matrix $T_N$ in the periodic Temperley-Lieb algebra $\mathrm{EPTL}_N(\beta,\alpha)$, Jordan cells arise when the eigenvalues of certain central elements collide, indicating logarithmic conformal field theory (LCFT) behavior in the scaling limit.

Jordan vectors within and between defect sectors have explicit algebraic constructions. For example, in the fixed-defect sector of the periodic loop model, using the Martin–Saleur intertwiner, one can construct explicit Jordan chains of length two at critical parameter values. The existence criteria for such cells depend on combinatorial congruence conditions and representation theory over $\mathrm{EPTL}_N(\beta,\alpha)$, with explicit vector formulas in terms of quantum group generators (Morin-Duchesne et al., 2013).

6. Analytical and Algebraic Interplay: Proof Strategies and Primary Decomposition

The characterization of Jordan vectors and chains via analytic (Taylor expansion) or algebraic (ideal-theoretic) methods is exact and constructive:

• Analytically, all chain relations are obtained from the Taylor expansion of $Q(z)\phi(z)$ and equating coefficients of $(z-\alpha)^j$.
• Algebraically, for matrices $A$ over a splitting field, primary decomposition of the ideal generated by $2\times 2$ minors of $(A\,x\,|\,x)$ recovers block sizes and multiplicities directly (Abo et al., 2015).

No use of generalized inverses, resolvent expansions, or ad hoc algebraic factorization is necessary in either approach—linearity and holomorphic expansion suffice (Borogovac, 9 Dec 2025).

7. Summary Table: Jordan Chain Characterizations

Context	Jordan Chain Condition	Main Reference
Matrix $A \in \mathbb{C}^{n\times n}$	$(A-\lambda I) v_1 = 0$, $(A-\lambda I) v_{j+1} = v_j$	(Morin-Duchesne et al., 2013, Abo et al., 2015)
Matrix Polynomial $L(z)$	$$\sum_{p=0}^j \frac{1}{p!} L^{(p)}(\alpha) \phi_{j-p} = 0$$	(Borogovac, 9 Dec 2025)
Operator-Valued $Q(z)$	$$\sum_{p=0}^j \frac{1}{p!} Q^{(p)}(\alpha) \phi_{j-p} = 0$$	(Borogovac, 9 Dec 2025)

This table summarizes the progression from finite matrices to general operator-valued functions and their respective Jordan chain conditions.

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These characterizations of Jordan vectors unify and extend the theory of generalized eigenvectors from classical linear algebra to broad analytic, algebraic, and physical frameworks, enabling a precise description of operator singularities, the analytic structure of solutions to associated differential equations, and the algebraic geometry of eigenschemes. Their occurrence in statistical mechanics models and operator theory underlines their foundational role across mathematics and physics (Borogovac, 9 Dec 2025, Abo et al., 2015, Morin-Duchesne et al., 2013).