Computing the matrix square root: A problem-solving approach using Mathematica and Pólya's strategies

This paper introduces a novel method for solving the matrix equation X² - A = 0 by computing matrix square roots. Inspired by George Pólya's structured problem-solving strategies and leveraging Wolfram Mathematica, the approach offers a systematic, efficient, and clear solution to the problem. The method extends the computation of matrix square roots to large matrices and those with complex eigenvalues, significantly broadening its applicability in diverse fields, including control theory, quantum physics, and signal processing. The approach is demonstrated through comprehensive examples and original Mathematica code, providing a practical toolkit for solving similar mathematical challenges. The method is designed to be both intuitive and versatile, making it a valuable resource for educators, students, and researchers engaged in advanced mathematical problem-solving.