Structured matrix factorization length

Abstract: Every (resp. a generic) complex $n \times n$ matrix can be expressed as a product of $2n+5$ (resp. $\lfloor n/2 \rfloor +1$) Toeplitz matrices. Motivated by this result, it is natural to ask the following question: what is the minimum number of Toeplitz matrices required to factor a given matrix? We generalize this question from Toeplitz structure to more general structures. In this paper, we introduce the notion of structured matrix factorization length when the set of matrices with a given structure is an affine variety $X \subseteq \mathbb{C}^{n \times n}$. Then we introduce the $r$-th $X$-factorization variety, defined as the Zariski closure of the set of products of $r$ matrices in $X$, and use it to define the border structured matrix factorization length. In particular, we study the cases in which $X$ is the affine variety of Toeplitz, Hankel, bidiagonal, tridiagonal, skew-symmetric or companion matrices. We calculate the dimension of the $X$-factorization varieties for all these cases, and discuss how numerical algebraic geometry can be used to obtain computational evidence for the degrees of $X$-factorization varieties with an example. In addition, we propose methods for deriving lower and upper bounds for (border) structured matrix factorization length. For lower bounds, we develop a method based on displacement rank, which can also be used to obtain some defining equations of the $r$-th $X$-factorization variety; for upper bounds, we suggest an approach using alternating minimization.

• The paper introduces an algebraic geometric framework that defines the minimal structured matrix factorization length and its border variant for various matrix classes.
• It establishes precise upper and lower bounds on factorization lengths for matrices like Toeplitz, Hankel, and tridiagonal using displacement rank and elimination theory.
• The study develops efficient alternating minimization algorithms that empirically achieve the theoretical minimal lengths, highlighting implications for computational stability.

Structured Matrix Factorization Length: An Algebraic Geometric Framework

Introduction and Problem Setting

This work introduces the concept of structured matrix factorization length in the context of algebraic geometry, providing a unified theory that extends beyond Toeplitz and Hankel matrices to a broad class of matrix structures represented as affine varieties. While prior results have shown that every complex $n \times n$ matrix can be factorized into a finite product of structured matrices (e.g., Toeplitz or Hankel), a minimal description of this “length” as an intrinsic invariant was lacking. This paper formalizes this length as the smallest integer $r$ for which a matrix admits such a factorization, generalizes the question to arbitrary algebraic matrix structures, introduces border variants (analogous to tensor border rank), and explores foundational aspects: geometry, bounds, and computational techniques.

Algebraic Geometric Formalism

Let $X \subseteq \mathbb{C}^{n \times n}$ be an affine algebraic variety encoding a matrix structure (e.g., Toeplitz, companion, tridiagonal, skew-symmetric). The $X$-factorization length of $A$, denoted $\mathbf{FL}_X(A)$, is the smallest $r$ such that $A=M_1 \cdots M_r$ for $M_i \in X$. The set of all such products (not necessarily Zariski closed) is denoted $\mu_r^0(X)$; its Zariski closure, $r$0, is the so-called $r$1-th $r$2-factorization variety. The border $r$3-factorization length $r$4 is the minimal $r$5 with $r$6. This distinction is essential in cases where the image of the multiplication map is not closed, paralleling phenomena in algebraic complexity where border rank can differ from (tensor) rank.

This geometric lens allows access to powerful algebraic tools: computation of dimension/degree, limits and closures (border variants), dominance and irreducibility, and applications of displacement structure.

Main Theoretical Contributions

1. Unified Framework and Extension to Arbitrary Matrix Structures

The paper systematically applies the above formalism to classical spaces: Toeplitz ($r$7), Hankel ($r$8), (upper/lower) bidiagonal ($r$9, $X \subseteq \mathbb{C}^{n \times n}$0), tridiagonal ($X \subseteq \mathbb{C}^{n \times n}$1), companion ($X \subseteq \mathbb{C}^{n \times n}$2), and skew-symmetric ($X \subseteq \mathbb{C}^{n \times n}$3) matrices. For each, dimension and (when feasible) degree of the corresponding factorization varieties are analyzed, and (border) lengths are bounded both above and below.

Strong statements obtained include:

• Every $X \subseteq \mathbb{C}^{n \times n}$4 matrix is a product of $X \subseteq \mathbb{C}^{n \times n}$5 Toeplitz matrices; a generic $X \subseteq \mathbb{C}^{n \times n}$6 matrix requires exactly $X \subseteq \mathbb{C}^{n \times n}$7 Toeplitz factors, and this is shown to be tight.
• The same generic/maximal lengths hold for Hankel matrices due to an algebraic reduction via permutation conjugacy.
• For tridiagonal, the minimal generic length is $X \subseteq \mathbb{C}^{n \times n}$8, and every matrix is a product of at most $X \subseteq \mathbb{C}^{n \times n}$9 such matrices.
• For companion matrices, a generic matrix requires $X$0 factors, every matrix is a product of at most $X$1.

2. Border Lengths, Closure, and Irreducibility

The gap between $X$2 and $X$3 is demonstrated with explicit examples—e.g., some diagonal or upper $X$4-diagonal matrices do not admit $X$5 term factorizations but can be written as the limit of such products. It is established that if $X$6 is irreducible (which holds for most structures considered), $X$7 is irreducible and its Zariski and Euclidean closures coincide.

3. Algebraic Geometry of Factorization Varieties

A detailed analysis is provided for the dimension of the $X$8-factorization varieties:

• For Toeplitz and Hankel: $X$9, proved via careful analysis of the Jacobian of the multiplication map at certain “special points” and block diagonalization arguments.

  Figure 1: The set at (\ref{eq:picking Lij 1}) is the union of three groups of matrix entries selected for rank calculation in the $A$0, two-factor Toeplitz analysis.

  

  Figure 2: Entry selection for the $A$1, three-factor Toeplitz factorization case, illustrating combinatorial structure in the proof for lower dimension bounds.

  

  Figure 3: Illustration of ordered index selection $A$2 for dimensional analysis in higher-order Toeplitz factorization; identifying pivot entries for block diagonal dominance.

• For tridiagonal, bidiagonal, and companion matrices, the factorization varieties are shown to be linear, with dimension expressible in closed form, confirming that all defining equations are linear.
• For skew-symmetric varieties, detailed table-based enumeration for low $A$3 is included, with dimension calculations that exploit the algebraic properties of skew-symmetric orbits under conjugation.

4. Lower and Upper Bounds via Displacement Rank and Alternating Minimization

A general method leveraging displacement operators (Sylvester type) is developed to yield lower bounds on (border) factorization length:

• For Toeplitz: rank conditions on $A$4 and $A$5 supply concise determinantal equations that vanish on $A$6—tight for many but not all cases.
• Analogous results are outlined for other structures, e.g., Hankel.

Upper bounds and explicit constructions are studied via:

• Inductive block construction arguments for tridiagonal and bidiagonal matrices.
• A practical alternating minimization algorithm (ALS/ALS-like block Gauss-Seidel) for explicit computation of Toeplitz (or other structured) factorizations, with convergence proofs for the (nonnegative) residual (not the factor sequence).

  Figure 4: Residual decay (log-scale) for 3 × 3 Toeplitz alternating minimization illustrates monotonic convergence to exact factorization.

  

  Figure 5: Residual convergence for $A$7 Toeplitz target with $A$8 factors, matching the theoretical generic upper bound.

  

  Figure 6: Visualization of the computed 15 Toeplitz factors for the $A$9 target; all factors displayed on identical color scale, illustrating factor diversity/structure.

5. Computational Investigation of Degrees and Defining Ideals

The degrees of factorization varieties, in particular for the Toeplitz case, are explored computationally using homotopy continuation and elimination theory. For instance, for $\mathbf{FL}_X(A)$0, degree 74 is numerically verified. The approach also details why, for non-linear varieties, complete algebraic computation of all defining equations is difficult, motivating this hybrid algebraic-numeric strategy.

6. Treatment of Orthogonally Invariant Structures

The paper extends analysis to matrix varieties invariant under orthogonal conjugation, notably the space of traceless symmetric matrices. For $\mathbf{FL}_X(A)$1, the maximal (generic) factorization length is shown to be 2, in contrast to $\mathbf{FL}_X(A)$2, where such factorizations cannot be surjective. Every matrix is a product of at most 10 traceless symmetric matrices, due to the ability to realize diagonal matrices as such products.

Implications

Practically, these results signify that for a wide class of matrix structures, minimal-length structured factorizations are both intrinsically algebraic and efficiently characterizable. In numerical computation, exploiting such structures yields not just computational efficiency (as in Toeplitz multiplication) but optimality guarantees tied to precise algebraic geometry invariants. The border variant distinguishes between representability and approximation (via algebraic closure), with ramifications for understanding stability under perturbation (“border rank” jump phenomena). The alternating least-squares approach provides a computational pipeline for approximate factorizations, with strong empirical performance at the theoretical minimal lengths identified.

Theoretically, this perspective unifies disparate structural results and situates structured matrix factorizations as algebraic secant/cactus varieties, inviting deeper investigation using tools from algebraic geometry, invariant theory, and complexity.

Open Problems and Future Directions

Key open questions include:

• Determination of the exact degree and complete ideal of $\mathbf{FL}_X(A)$3 (and analogous non-linear varieties).
• Stability and Euclidean distance degree (ED degree) under perturbations, linking algebraic geometry and numerical analysis.
• Computational complexity classification (e.g., NP-membership, hardness of factorization length).
• Extension to real algebraic varieties and to “multi-structure” settings (e.g., factorization into alternating types).
• Algorithmic development for efficient construction of minimal-length factorizations in large-scale numerical settings.

Conclusion

This work recasts structured matrix factorization length as an algebraic geometric invariant, providing sharp length bounds and analytic tools via dimension, degree, and displacement structure. It affirms that, for a diverse array of matrix structures, both the generic and worst-case minimal factorization lengths are now susceptible to systematic algebraic analysis. The introduction of border length parallels advances in tensor rank theory, while computational schemes like alternating minimization bridge the theory with practical matrix computations.

The mathematical code and tools developed herein constitute a foundation for future theoretical exploration and algorithmic development in structured matrix computations.

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Reference:

"Structured matrix factorization length" (2606.07407)