Polynomial eigenvalue problems arise in diverse areas of science and engineering, including structural vibration analysis, computer-aided geometric design, robotics, and machine learning. Their solution often relies on constructing suitable matrix linearizations that allow the use of established algorithms for the generalized eigenvalue problem.
We introduce a new class of companion matrices for structured scalar polynomials, which in many instances possess smaller and better-scaled entries than those in classical Frobenius or Fiedler forms. This construction connects to several notable polynomial families, including Euclid and Narayana–Mandelbrot polynomials, for which we obtain companion matrices of minimal integer height.
These ideas are then extended to matrix polynomials through systematic methods that combine known linearizations into new structured ones. A central component is the formulation of generalized standard triples, applicable to a broad range of polynomial bases—including monomial, Bernstein, Lagrange, Hermite, and orthogonal bases—and accommodating shared similarity transformations.
Finally, we address the numerical stability of these algebraic linearizations. Experimental investigations using pseudospectra indicate that reusing structured component linearizations frequently yields better-conditioned problems than approaches based on expanded polynomial coefficients.
Dr. Eunice Y. S. Chan is an Assistant Professor in the School of Medicine at The Chinese University of Hong Kong, Shenzhen. She holds a Ph.D. in Applied Mathematics from Western University, Canada, where she specialized in numerical linear algebra, matrix polynomial theory, and computer algebra. Her current research integrates these mathematical foundations with machine learning for healthcare applications, including health data science, AI‑enabled clinical decision-making, and disease burden modeling. Dr. Chan has published in leading journals such as SIAM Review and Linear Algebra and its Applications, and co‑authored the SIAM textbook Computational Discovery on Jupyter.
