Given a square matrix $A\text{,}$ it is likely similar to many, many other matrices. Of all these possibilities, which is the best? “Best” is a subjective term, but we might agree that a diagonal matrix is certainly a very nice choice. Unfortunately, as we have seen, this will not always be possible. What form of a matrix is “next-best”? Our goal, which will take us several sections to reach, is to show that every matrix is similar to a matrix that is “nearly-diagonal” (Section Section 3.3). More precisely, every matrix is similar to a matrix with elements on the diagonal, and zeros and ones on the diagonal just above the main diagonal (the “super diagonal”), with zeros everywhere else. In the language of equivalence relations (see Theorem SER), we are determining a systematic representative for each equivalence class, where the relation is similarity. Such a representative for a set of similar matrices is called a canonical form.