Positive Bidiagonal Factorization

Updated 25 January 2026

Positive bidiagonal factorization is an explicit matrix decomposition that expresses totally positive or oscillatory matrices as products of bidiagonal factors with strictly positive off-diagonal entries.
It enables stable, subtraction-free computations for spectral analysis, system solves, and the construction of orthogonal polynomial families.
Its explicit construction via Neville elimination or moment-based methods connects total positivity with classical continued fraction theory and the spectral Favard theorem.

A positive bidiagonal factorization is an explicit decomposition of a totally positive or oscillatory (often structured, banded) matrix as a product of bidiagonal matrices—typically lower and upper triangular—whose off-diagonal entries are strictly positive. This structure enables stable, high-accuracy computations of spectral data, direct inversion, and explicit construction of associated families of orthogonal or multiple orthogonal polynomials. It provides a deep link between total positivity, the spectral Favard theorem, and algorithmic linear algebra for structured matrices (Delgado et al., 21 Jan 2025, Martinez, 2018, Higham, 2023, Branquinho et al., 2024, Branquinho et al., 2022, Branquinho et al., 2022).

1. Algebraic and Structural Definition

Given a nonsingular, totally positive or oscillatory $n\times n$ matrix $A$ , its positive bidiagonal factorization (PBF) expresses $A$ or related structured matrices in the form

$A = F_{n-1} F_{n-2} \cdots F_1 \; D \; G_1 \cdots G_{n-2} G_{n-1}$

where:

$F_k$ is lower unit-bidiagonal with strictly positive (often parameterized) subdiagonal entries,
$G_k$ is upper unit-bidiagonal with strictly positive superdiagonal entries,
$D$ is a positive diagonal matrix (often reducible to identity under special conditions).

In banded settings, for example a Hessenberg with $p$ subdiagonals,

$H = L_1 L_2 \cdots L_p U$

each $L_j$ is unit lower-bidiagonal, $U$ is unit upper-bidiagonal, and all factor entries are strictly positive. More generally, semi-infinite or block settings require $p$ lower and $q$ upper bidiagonals, with diagonal factors separating them (Branquinho et al., 2022, Branquinho et al., 2024, Branquinho et al., 2024).

The decomposition is tightly linked to Neville elimination: all strictly totally positive matrices admit subtraction-free Neville elimination, with multipliers and pivots forming the bidiagonal data (Martinez, 2018). In certain families (e.g., generalized Pascal matrices), the pivot structure simplifies further to $D=I$ (Delgado et al., 21 Jan 2025).

2. Existence Criteria and Underlying Total Positivity

The existence of a positive bidiagonal factorization is governed by total positivity and oscillation theory:

For general, finite matrices: strict total positivity (all minors $>0$ ) is necessary and sufficient (Martinez, 2018, Higham, 2023, Branquinho et al., 2024).
For banded or Hessenberg operators (finite or infinite/semi-infinite): oscillatory matrices (totally nonnegative, invertible, irreducible; or matrices for which all leading minors of powers are strictly positive) admit PBF (Branquinho et al., 2022, Branquinho et al., 2023, Branquinho et al., 2022).

For finite $(p,q)$ -banded matrices, explicit initial minor criteria or continued fraction conditions (via LU or Gauss–Borel theory, or branched continued fractions) characterize existence (Branquinho et al., 2024, Branquinho et al., 2022, Branquinho et al., 2024). In the case of oscillatory tetradiagonal (p=2) Hessenberg matrices, the PBF exists if and only if an explicit continued-fraction inequality holds (Branquinho et al., 2022).

For structured matrices like generalized Pascal, Bessel, or certain Hankel or banded Toeplitz matrices, explicit parameter regions can be described where all Neville multipliers and hence all bidiagonal factor entries are strictly positive (Delgado et al., 21 Jan 2025, Delgado et al., 17 Jan 2025, Branquinho et al., 2022, Branquinho et al., 2023).

3. Explicit Construction and Algorithms

The explicit construction of a positive bidiagonal factorization proceeds via extraction of Neville elimination parameters, ratio-based recurrences in minors, or moment-based approaches:

Neville elimination route: Direct computation of bidigonal multipliers and pivots from ratios of contiguous minors or via recurrence on minors yields explicit formulas, often with closed form for classical matrices (Vandermonde, Cauchy, Pascal, Bessel) (Martinez, 2018, Higham, 2023, Delgado et al., 21 Jan 2025, Delgado et al., 17 Jan 2025).
Moment and production matrix route: For recurrence or moment matrices associated with orthogonal or multiple orthogonal polynomial systems, block (Gauss–Borel) LU factorization provides bidiagonal factors. Darboux/Christoffel transforms lead to systematic construction of the individual bidiagonals (Branquinho et al., 2024, Branquinho et al., 2024).
Algorithmic implementation: All efficient algorithms proceed by forming the bidiagonal multipliers through subtraction/addition and multiplication operations solely on the positive input data or prescribed parameters, ensuring no catastrophic cancellation and high relative accuracy. This is exemplified in Pascal and Bessel matrix cases where factor entries depend only on $x$ and $\lambda$ via sums like $x + (i-2k)\lambda$ (Delgado et al., 21 Jan 2025), or integer-polynomial expressions (Delgado et al., 17 Jan 2025). See, for instance:

for k = 1:n
  for i = k+1:n+1
    m_{i,k} = x + (i - 2k) * lambda
  end
end

(Delgado et al., 21 Jan 2025)

For mixed/multiple orthogonality, the explicit bidiagonal entries reflect ratios of leading coefficients in Christoffel-perturbed polynomials (Branquinho et al., 2024).

4. Spectral and Polynomial-Theoretic Consequences

Positive bidiagonal factorization is the central tool enabling the constructive spectral theory of banded oscillatory (or totally positive) matrices:

Spectral Favard Theorem: Matrices admitting PBFs correspond exactly to those for which there exists a system of scalar or matrix-valued positive Borel measures such that the recursion polynomials (scalar, vector, or matrix) form a family of (mixed) multiple orthogonal polynomials biorthogonal with respect to these measures (Branquinho et al., 2022, Branquinho et al., 2024, Branquinho et al., 2023, Branquinho et al., 2024, Branquinho et al., 2022).
Eigenvalue and singular value computation is expressible via the bidiagonal data; in finite truncations, the spectrum is real, positive, simple, and exhibits strict interlacing (Higham, 2023, Higham, 2023, Branquinho et al., 2022). For unbounded or semi-infinite settings, limiting arguments relying on weak convergence of Christoffel-weighted discrete measures recover the full operator spectral measure (Branquinho et al., 18 Jan 2026, Branquinho et al., 2023).
In the special case of tridiagonal (Jacobi) and higher-band operators, PBF gives rise to multiple orthogonal polynomial systems and associated Gauss–quadrature rules, with explicit degree of precision (Branquinho et al., 2024, Branquinho et al., 2022).

5. Applications and Numerical Impact

Positive bidiagonal factorization enables highly accurate and stable linear algebra for structured matrices:

Accurate spectral computations: All eigenvalues, singular values, and condition numbers for totally positive matrices can be computed with high relative accuracy, independent of conditioning in the usual normwise sense (Delgado et al., 21 Jan 2025, Martinez, 2018, Higham, 2023, Delgado et al., 17 Jan 2025).
Stable system solves: Solving systems $Ax = b$ is backward stable componentwise provided $A$ or $A^{-1}$ has a positive bidiagonal factorization (see error bounds in (Higham, 2023, Martinez, 2018)). Intermediate computations are subtraction-free or have strictly bounded relative error.
Markov chains and probabilistic models: Positive bidiagonal factorizations for stochastic (transition) matrices underlie the spectral theory of birth–death and banded Markov chains, yielding explicit Karlin–McGregor-type representations of transition probabilities, formulas for stationary distributions, and sharp recurrence/ergodicity criteria linked to the spectral measure's mass at $x=1$ (Branquinho et al., 15 Jan 2026, Branquinho et al., 2022).
Orthogonal polynomials and combinatorics: The existence and explicitness of positive bidiagonal factorizations are central for identifying parameter ranges where systems like Hahn, Jacobi–Piñeiro, or Laguerre multiple orthogonal polynomials admit totally positive recurrences—and for deriving their continued fraction or lattice path interpretations (Branquinho et al., 2023, Branquinho et al., 2024, Branquinho et al., 2022).

6. Generalizations, Parameter Spaces, and Uniqueness

The positive bidiagonal factorization theory extends across a wide variety of matrix classes:

Explicit parametrization of the (semi)group of banded totally positive matrices is in terms of the strictly positive parameters of the bidiagonal factors and the diagonal (Branquinho et al., 2024).
For banded and block cases (e.g., matrices arising in mixed multiple orthogonality), PBF involves a prescribed number of left and right bidiagonal elements together with a positive diagonal, with all parameters continuous and parametrizing a full cell (Branquinho et al., 2024, Branquinho et al., 18 Jan 2026, Branquinho et al., 2022).
Uniqueness is typically guaranteed, up to trivial rescalings associated with the Gauss–Borel or LU procedure (Branquinho et al., 2024, Higham, 2023). In the context of continued fraction expansions, PBF corresponds to the unique expansion with all positive coefficients.
Regions of parameter space (e.g., for classical matrix families) where PBF exists are fully characterized and connected to domains where all relevant minors or continued-fraction coefficients are strictly positive (Delgado et al., 21 Jan 2025, Branquinho et al., 2023).

7. Further Developments and Theoretical Connections

Positive bidiagonal factorization unifies and extends classical areas:

Provides a direct, structure-exploiting bridge between operator theory, total positivity, linear algebra, analytic function theory (via continued fractions), and algebraic combinatorics (via lattice paths or determinants) (Branquinho et al., 2024, Grover et al., 2022).
The operational and spectral consequences of PBF are applicable in, and extend to, unbounded operator settings, with suitably adapted truncation-and-compactness tools (Branquinho et al., 18 Jan 2026).
Parameterization and semigroup results connect PBF with the infinite divisibility and closure properties of structured matrix cones (Branquinho et al., 2024, Grover et al., 2022).

In conclusion, positive bidiagonal factorization provides a foundational and ubiquitous framework for the theory and computation of structured totally positive and oscillatory matrices, essential in spectral theory, multiple orthogonal polynomials, and high-accuracy numerics (Delgado et al., 21 Jan 2025, Branquinho et al., 2024, Branquinho et al., 2022, Branquinho et al., 15 Jan 2026, Branquinho et al., 2022, Higham, 2023, Branquinho et al., 2024).