MATLAB Polyvalm Explained: Matrix Polynomial Evaluation

In this episode, we dive into matrix polynomial evaluation in MATLAB using polyvalm, and connect it directly to one of the most important results in linear algebra: the Cayley–Hamilton Theorem. Using MATLAB’s built-in Pascal matrix, we construct a symmetric, positive-definite matrix and compute its characteristic polynomial with poly. We then evaluate that polynomial at the matrix itself using polyvalm, revealing why the result is (numerically) close to zero. This behavior is not a coincidence — it’s a direct computational demonstration of the Cayley–Hamilton theorem, which states that every square matrix satisfies its own characteristic equation. 🔍 What you’ll learn in this video: What polyvalm does and how it differs from polyval How MATLAB evaluates polynomials of matrices What the Pascal matrix is and why it’s mathematically special How poly generates a characteristic polynomial from a matrix Why polyvalm(p, X) ≈ 0 for a square matrix X A clear, intuitive explanation of the Cayley–Hamilton theorem How MATLAB Help documents these functions 📐 Key MATLAB functions used: pascal – generating a symmetric positive-definite matrix poly – forming the characteristic polynomial polyvalm – evaluating a polynomial at a matrix