Strassen's 2x2 matrix multiplication algorithm: A conceptual perspective

Despite its importance, all proofs of the correctness of Strassen's famous 1969 algorithm to multiply two 2x2 matrices with only seven multiplications involve some more or less tedious calculations such as explicitly multiplying specific 2x2 matrices, expanding expressions to cancel terms with opposing signs, or expanding tensors over the standard basis. This is why the proof is nontrivial to memorize and why many presentations of the proof avoid showing all the details and leave a significant amount of verifications to the reader. In this note we give a short, self-contained, easy to memorize, and elegant proof of the existence of Strassen's algorithm that avoids these types of calculations. We achieve this by focusing on symmetries and algebraic properties. Our proof combines the classical theory of M-pairs, which was initiated by B\"uchi and Clausen in 1985, with recent work on the geometry of Strassen's algorithm by Chiantini, Ikenmeyer, Landsberg, and Ottaviani from 2016.