In a recent paper 1 I stated that von Neumann had originated the suggestion for the use of Schur's canonical form for arbitrary matrices. I have since learned that the suggestion actually is due in the first instance to John Greenstadt, who brought it to von Neumann's attention. The history of this is rather interesting and was communicated to me in a letter from John Greenstadt, which I quote below. “The full story is, that the triangularization occurred to me early in 1953, after trying in vain to find a general iterative diagonalization procedure, even where one knew that it was possible to diagonalize (defective degeneracy being the impossible case). It seemed to me that one thing that made for the stability of the Jacobi method was the fact that all the elements in the transformation matrix were less than 1. A natural generalization embodying this requirement was to consider unitary transformations. Then, a quick check of Murnaghan's book showed that one could hope only to triangularize, but that this was always possible. “I did some hand calculations on this, and lo and behold! it converged in the few cases I tried. I then programmed it for the CPC and tried many other cases. For several months thereafter, Kogbetliantz, John Sheldon, and I tried to prove convergence, when the algorithm involved the sequential annihilation of off-diagonal elements. We (particularly Sheldon) tried many approaches, but with no hint of success. Finally, in the latter part of 1953, we decided to ask von Neumann, who was then a consultant for IBM, when he was in New York at our offices. “I had prepared a writeup describing the procedure, but von Neumann (rightly) didn't want to bother reading it, so I explained it to him in about two minutes. He spent the next 15 minutes thinking up all the approaches we had thought of in three or four months, plus a few ones—all, however, without promise. “At this point he decided that it was a nontrivial problem, and perhaps not worth it anyway, and immediately suggested minimizing the sum of squares of subdiagonal elements, which is, of course, the truly natural generalization of the Jacobi method. For the next 15 minutes he investigated the case when it would be impossible to make an improvement for a particular pivotal element and found that these cases were of measure zero. “I recoded my procedure for the 701 and tried many other matrices of various sizes. I myself never had a failure, but it has since been demonstrated that the method will indeed fail for a class of matrices. Hence, a proof is clearly impossible. However, I think a statistical proof is possible, along lines suggested by Kogbetliantz, which, however, I have not been able to find. I do not think von Neumann's variation of the method would fail. (However, it is more complicated and time consuming.)”

The so-called Jacobi Procedure is extended to the case of normal matrices. A stable iterative procedure is described utilizing plane unitary transformations for such matrices which yield both the characteristic values and their associated vectors. Generally, the technique consists of minimizing at each stage the sum of the squares of the off-diagonal elements of the given matrix; however, there is one case in which this leads to no improvement; i.e. the lowest value for the change is non-negative. In this case, it is shown that a convergent procedure is still possible.

article Free AccessThe Jacobi Method for Real Symmetric Matrices Authors: H. H. Goldstine International Business Machines Corporation, Yorktown, N.Y. International Business Machines Corporation, Yorktown, N.Y.View Profile , F. J. Murray Columbia University, New York, N.Y. Columbia University, New York, N.Y.View Profile , J. von Neumann View Profile Authors Info & Claims Journal of the ACMVolume 6Issue 1Jan. 1959 pp 59–96https://doi.org/10.1145/320954.320960Published:01 January 1959Publication History 50citation1,202DownloadsMetricsTotal Citations50Total Downloads1,202Last 12 Months140Last 6 weeks20 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF