How did Werner Heisenberg use matrices to formulate quantum mechanics?

In 1925, Werner Heisenberg sought to resolve the limitations of existing atomic models by insisting that quantum mechanics be based solely on observable quantities, such as spectrum intensities. While studying the electron as an anharmonic oscillator, he developed a formalism rooted in noncommutative multiplication.12

Heisenberg did not initially frame his work using matrix algebra; rather, his mathematical approach relied on these noncommutative relations. Shortly after, Max Born and Pascual Jordan recognized that Heisenberg’s formalism could be expressed through matrix algebra. Together with Heisenberg, they published a series of papers—most notably the "three-man paper"—that established matrix mechanics as a foundational framework for the new quantum theory. This discovery, for which Heisenberg was awarded the Nobel Prize, provided a rigorous conceptual foundation for physics, distinct from the later wave mechanics proposed by Erwin Schrödinger.123