Distance to nearest skew-symmetric matrix polynomials of bounded rank

TL;DR

This work addresses the problem of nearest skew-symmetric matrix polynomials of bounded rank to a given $P(\lambda)$ by exploiting a simple, constructive parametrization of generic sets via factorizations $P(\lambda)=L(\lambda)E(\lambda)L(\lambda)^T$, where $E(\lambda)$ encodes rank-$2r$ structure. It develops an alternating-least-squares approach (GEARS) based on this parametrization to compute the distance and the nearest polynomial, with a variant (GEARS-SVD) tailored for skew-symmetric pencils to boost efficiency. The authors establish a principled framework for both the distance problem and its solution, including a specialized treatment for pencils that yields substantial speedups. Numerical experiments show that GEARS matches or surpasses existing methods in accuracy while offering significantly faster performance and scalability, enabling practical rank-$2r$ approximations in control and differential-algebraic equation contexts.

Abstract

We propose an algorithm that approximates a given matrix polynomial of degree $d$ by another skew-symmetric matrix polynomial of a specified rank and degree at most $d$. The algorithm is built on recent advances in the theory of generic eigenstructures and factorizations for skew-symmetric matrix polynomials of bounded rank and degree. Taking into account that the rank of a skew-symmetric matrix polynomial is even, the algorithm works for any prescribed even rank greater than or equal to $2$ and produces a skew-symmetric matrix polynomial of that exact rank. We also adapt the algorithm for matrix pencils to achieve a better performance. Lastly, we present numerical experiments for testing our algorithms and for comparison to the previously known ones.

Figures (7)

• Figure 1: Comparison of the computed minimal distances to singularity for random skew-symmetric matrix polynomials of given size, degree, and with the entries in the underlying field (real or complex). All the algorithms find a skew-symmetric matrix polynomial at essentially the same distance from a given skew-symmetric matrix polynomial.
• Figure 2: Comparison of the "singularity" of the output for random skew-symmetric matrix polynomials of given size, degree, and the entries in the underlying field (real or complex). To be exact, we follow GnGu25 and plot smallest singular value $\sigma_{\min}$ of the constructed matrix polynomial evaluated at the points ${x_j + \sqrt{-1} y_j}$ given by $[X,Y] = meshgrid(-1000:40:1000)$. In all the performed experiments GEARS produces the outputs with significantly lower singular values.
• Figure 3: Comparison of the speed for random skew-symmetric matrix polynomials of given size, degree, and the entries in the underlying field (real or complex). In all the performed experiments GEARS is the fastest.
• Figure 4: GEARS algorithm for computing the minimal distance to singularity for random skew-symmetric matrix polynomials of sizes $20\times20$, $30\times30$, $40\times40$, and $50\times50$, of degree $2$, with entries over the real or complex field (4 examples for each setting).
• Figure 5: Illustration of the time scaling of the GEARS algorithm as the input size increases. The plot shows the runtime for random skew-symmetric matrix polynomials of given sizes and degree $2$, with entries over the real or complex field (4 examples for each setting).

Theorems & Definitions (19)

• Theorem 2.1
• Theorem 2.2
• Theorem 3.1
• Lemma 3.2
• proof
• Definition 3.3
• Lemma 3.4
• Theorem 3.5
• proof
• Definition 3.6