Rank-One Factorization of Matrix Polynomials

• Rank-one factorization of matrix polynomials is a decomposition approach that expresses a matrix polynomial as a product of two low-dimensional polynomial factors, uncovering its algebraic structure.
• The method offers canonical forms and degree classifications that ensure unique minimal decompositions, which are crucial for practical applications in spectral analysis and systems theory.
• Explicit algorithms based on polynomial GCD computations and minimal bases enable efficient manipulation of polynomial eigenstructures, enhancing applications in signal processing and algebraic system theory.

A rank-one factorization of matrix polynomials is a decomposition of a matrix-valued polynomial into products involving rank-one matrix factors, revealing algebraic and structural properties that underpin applications in algebraic systems theory, signal processing, and spectral analysis. The theory encompasses canonical forms, uniqueness and parametrization results, explicit factorization algorithms based on polynomial GCD computations, and perturbation formulas that allow algebraic manipulation of polynomial eigenstructures.

1. Basic Concepts and Definitions

Let $P(\lambda) \in \mathbb{C}[\lambda]^{m \times n}$ denote an $m \times n$ matrix polynomial of degree $d$. A rank-one factorization is an expression of the form

$P(\lambda) = L(\lambda) R(\lambda)$

where $L(\lambda) \in \mathbb{C}[\lambda]^{m \times 1}$, $R(\lambda) \in \mathbb{C}[\lambda]^{1 \times n}$, and both are nonzero. For rank-one auto-correlation matrix polynomials $R(z)$ of length $N$, each entry $R_{ij}(z)$ is typically Hermitian and admits representation $R_{ij}(z) = X_i(z)\,\widetilde{X_j}(z)$, where $$\widetilde{X_j}(z) \equiv z^{N-1} \overline{X_j(z^{-1})}$$ and $X_j(z)$ is a univariate polynomial signal of bounded degree (Usevich et al., 2023). For polynomial matrices of generic normal rank one, the factorization is called minimal if $L(\lambda)$ and $R(\lambda)$ are minimal bases of the range and co-range, respectively (Dmytryshyn et al., 2023).

2. Canonical and Minimal Rank-One Factorizations

For generic $P(\lambda)$ of normal rank one and degree $d$, minimal rank-one factorizations exist and are characterized by degree patterns and the minimal basis property (Dmytryshyn et al., 2023):

• $L(\lambda)$ (column) and $R(\lambda)$ (row) are column- and row-reduced, respectively, with $\deg P = \deg L + \deg R$.
• There are generically $d+1$ degree classes, indexed by $a \in \{0,1,\ldots,d\}$, with

  $\deg L = d-a, \quad \deg R=a$

• Each degree class is a nonempty Zariski-open and dense subset of the algebraic manifold of rank-one degree-$d$ $m \times n$ matrix polynomials.

The uniqueness of the factorization (up to unimodular scaling) is established: if $$P(\lambda) = L_1(\lambda) R_1(\lambda) = L_2(\lambda) R_2(\lambda)$$ are minimal rank factorizations, then $L_2(\lambda) = L_1(\lambda) u$ and $R_2(\lambda) = u^{-1} R_1(\lambda)$ for a nonzero constant $u$ (Dmytryshyn et al., 2023).

3. Uniqueness Criteria and Enumeration: GCD Structure

For Hermitian auto-correlation matrix polynomials $R(z)$, necessary and sufficient uniqueness of rank-one factorization relies on the greatest common divisor (GCD) structure:

• Define $$H(z) = \gcd\{R_{ij}(z)\}_{i,j=1}^K = Q(z)\,\widetilde{Q}(z)$$, with $Q(z) = \gcd\{a_1(z), \ldots, a_K(z)\}$.
• The factorization $R(z) = a(z) a(z)^*$ is essentially unique (up to multiplication by $\beta \in \mathbb{T}$) if and only if all roots of $Q(z)$ lie on the unit circle; equivalently, $H(z)$ has no zeros in $\mathbb{C} \setminus \mathbb{T}$ (Usevich et al., 2023).

If $Q(z)$ has off-unit-circle roots, each such root yields independent choices between the root and its conjugate inverse, giving rise to multiple non-trivially different factorizations. The number of distinct factorizations is $\prod_{i=1}^P (\mu_i + 1)$, where $\mu_i$ are root multiplicities off the unit circle (Usevich et al., 2023).

4. Explicit Algorithms for Rank-One Factorization

In the uniqueness regime, explicit algorithms construct rank-one factorizations:

1. Choose any nonzero row in $R(z)$.
2. Compute the GCD $A_j(z)$ of that row’s entries.
3. Compute $\widetilde{A_j}(z) = z^{N-1} \overline{A_j(1/z)}$.
4. For each $k$, determine $\widehat{a}_k(z) = R_{kj}(z) / \widetilde{A_j}(z)$.
5. Normalize by the $\ell^2$-norm at lag $N-1$ to obtain $a_k(z)$.

The resulting $a_k(z)$ satisfy $R_{ij}(z) = a_i(z) \widetilde{a_j}(z)$ and are unique up to a global unimodular coefficient (Usevich et al., 2023).

For general polynomial matrices, existence proof and parameterization follow by Smith decomposition and minimal basis reduction, leading to explicit families in each degree class (Dmytryshyn et al., 2023).

5. Spectral and Structural Effects of Rank-One Perturbations

Rank-one perturbations play a significant role in spectral manipulation. For a matrix polynomial $A(z) = \sum_{i=0}^d A_i z^i$, a rational rank-one perturbation,

$$\tilde{A}(z) = A(z)\left(I + \tau(z) Q\right),\quad \tau(z) = \frac{\gamma}{z-\lambda_0},~ Q = u v^*$$

with $u$ a right eigenvector for eigenvalue $\lambda_0$ and $v^* u = 1$, modifies only the eigenvalue $\lambda_0 \to \mu$ and leaves the others unchanged (Bini et al., 2015).

This perturbed polynomial $\tilde{A}(z)$ remains of degree $d$, with explicit coefficient correction: $$\tilde{A}_i = \begin{cases} A_i + \gamma \sum_{k=0}^{d-i-1} \lambda_0^k A_{k+i+1} u v^*, & 0 \leq i < d \ A_d, & i = d \end{cases}$$ The correction $\Delta A(z) = \tilde{A}(z) - A(z)$ is manifestly rank-one in each coefficient (Bini et al., 2015).

For polynomials admitting canonical Wiener–Hopf factorization $A(z) = U(z) L(z^{-1})$, the outer factor $U(z)$ remains unchanged under the rank-one perturbation (when $|\lambda_0|, |\mu| < 1$), while the inner factor is updated by a rank-one correction (Bini et al., 2015): $$\tilde{L}_i = L_i - \gamma \sum_{j=1}^i \lambda_0^{j-1} L_{i-j} u v^*,\qquad i \geq 1$$

6. Absence of Eigenvalues and Predictable Degree Phenomenon

For generic complex rank-one matrix polynomials $P(\lambda)$ (with generic $m, n, d$), the complete eigenstructure is trivial: there are no eigenvalues, as the GCD of scalar entries is generically one, and the leading coefficient has full rank (Dmytryshyn et al., 2023). The degree of $P$ is strictly determined by the minimal degrees of $L$ and $R$: $\deg P = \deg L + \deg R$.

7. Computational and Structural Ramifications

Rank-one factorizations offer maximal data compression for matrix polynomials, expressing all information through two low-dimensional polynomial factors. For finite degree $d$, there are $d+1$ Zariski-open degree classes for minimal rank-one factorizations, and transitions between them can be induced by small-degree modifications to $L$ or $R$ (Dmytryshyn et al., 2023). In settings such as structured Markov chains and quadratic matrix equations, rank-one shifts can be implemented with $O(d n^2)$ effort, and existing factorizations may be reused with a low-rank correction, incurring no cost increase asymptotically (Bini et al., 2015).

Enumeration of non-equivalent rank-one factorizations for Hermitian auto-correlation polynomials is governed by the GCD structure, with explicit formulae for the number of nontrivial decompositions depending on off-unit-circle root multiplicities (Usevich et al., 2023). All roots on the unit circle guarantee essential uniqueness. This structure also underpins algorithm design for efficient factor computation in practical applications.

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References:

• (Bini et al., 2015) D. A. Bini & B. Meini, “Generalization of the Brauer Theorem to Matrix Polynomials and Matrix Laurent Series”
• (Usevich et al., 2023) “On factorization of rank-one auto-correlation matrix polynomials”
• (Dmytryshyn et al., 2023) “Minimal rank factorizations of polynomial matrices”