To my way of thinking, it's a marvelously simple proof. It's over 50 years old, and when one uses the modem language of graph theory, it can be made very visual. The most difficult part of the proof is in achieving triangular form. After that the details are extremely easy to follow: elementary row operations followed by the corresponding elementary column operations. Thus for students who are more familiar with Gaussian elimination than with other aspects of linear algebra, it is a good way to introduce the Jordan canonical form. So where is this proof and why isn't it a standard proof? (I think it should be and I hope to convince you.) It's in a classical book [8] by H. W. Turnbull and A. C. Aitken entitled An Introduction to the Theory of Canonical Matrices. This book is well known to matrix theorists, so that one cannot claim that it's in an obscure book that has long been forgotten. The reason that it has been ignored may be due to the changes in mathematical rigor that have occurred since 1932 and the lack of the formalism provided by graph theory at that time. (Konig's classical book entitled Theorie der Endlichen und Unendlichen was published in 1936, although his classical paper [6] Graphen und Matrizen was published in 1931.) At first reading, Turnbull and Aitken's proof is somewhat obscure, and it is not clear that their proof is general. They refer to chains of nonzero elements in a matrix which now we would call paths in the digraph associated with the matrix. Whatever the reason, I hope-to revive their proof by publication of this article. In a recent article [2] in this journal, Fletcher and Sorensen describe a proof for the existence of the Jordan canonical form which is algorithmic in nature. This proof was adopted in [5]. The proof, like the one to be given here, proceeds in three steps: (I) reduce to upper triangular form; (II) further reduce to the case in which all eigenvalues are equal; (III) use induction to reduce an upper triangular matrix with equal eigenvalues to Jordan canonical form. As pointed out in [2], the only nonconstructive step is I. In [2] step II is accomplished by solving, using induction, a linear matrix equation of the form AX - XA = S, while step III is accomplished by induction and matrix factorizations. The Turnbull-Aitken approach accomplishes