Introduction to a group of articles in tribute to Vlastimil Ptak
This tribute consists of a group of six short articles of
differing kinds and a bibliography. We shall briefly describe
these articles and then add comments.
In 1956 Vlastimil Ptak published in German a beautiful proof of the
existence of the Jordan Form of a complex square matrix by means of
duality theory, [P1]. Some 20 years later, he first suggested to me
that LAA should publish an English translation of his article. He sent
me a translation by another person, which however had deficiencies. I
then proposed that he should translate the article himself as his
English was as perfect as his German. I received no response. Years
passed, my copy of the original translation was lost and in 1998 Ptak
repeated his suggestion, to which I made the same reply. Several
months later, somewhat to my surprise, I received his English
translation which, with very small changes, is presented here, [P2],
as the core of the group of articles. At the same time I also received
his commentary on the unusual circumstances of the original submission
and publication of the article in a Hungarian journal in 1956, the
final version of which is also included here, [P3].
As Ptak's commentary explains, the current translation [P2]
contains two theorems and their proofs, and omits the subsequent
derivation of the usual Jordan Form form contained in the 1956
article [P1]. This was written by an editor (L. Redei) and was
added by Ptak at his urging. The current package of articles
contains such a derivation in an excellent separate article by
Carl de Boor [dB], since Ptak wished to publish his article in
English in the short and simple form in which he had first
submitted it.
In an interesting talk at the SIAM Applied Linear Algebra Meeting
in Minneapolis in 1991, Irving Kaplansky presented the same proof
which he had independently rediscovered and then found in Ptak's
1956 note. I discovered that there was an unpublished note of
Kaplansky's on this topic which also contained a generalization of
the Jordan Form to a canonical form for pairs of matrices under
contragredient equivalence. Kaplansky's contribution was
acknowledged by Horn and Merino [HM] in a paper on this topic. The
contragredient canonical is presented here by Olga Holtz [H]. In
her note, she also gives additional references, and generalizes
the form to matrices over an arbitrary field with a proof based on
Ptak's duality method and de Boor's derivation.
Mirek Fiedler has added a short appreciation of his colleague
Vlastimil Ptak. He has also compiled a list of Ptak's publications
supplementary to that published by LAA in [V].
We now discuss the somewhat philosophical issues behind Redei's
addition to the paper. The extra part begins with the observation "As
is known (see for example [M]), Theorems 1 and 2 [of Ptak's paper] are
the only essential parts of the theory of the Jordan Normal Form,
however for the convenience of the reader a derivation is sketched of
the usual form from these two theorems". Obviously Redei was repeating
Ptak's view in the first part of the remark, and surely the two
theorems are the essence of the structure theory of a finite
dimensional vector space considered as a module over an algebra
generated by a single linear transformation over the complex numbers.
They can be stated and proved in an entirely co-ordinate free manner,
as they are in [P1] and [P2]. But linear algebra has many faces, and
structure theory is a very important one, but only one. To my mind,
part of the essence of the Jordan Form is that you can actually write
down a matrix which is canonical for similarity and that this matrix
can be fully specified by a list of eigenvalues and corresponding
block sizes. The frequent use of the Jordan Form in other branches of
mathematics hinges on this. Furthermore, it is possible to organize a
certain part of linear algebra by considering various equivalence
relations on sets of matrices and the corresponding canonical forms,
and then ask what do these forms have in common, for example: what are
the common features of the reduced row echelon form (canonical for row
equivalence) and the Jordan Form (canonical for similarity). One might
note that both are "near combinatorial", that is both are described by
a zero-nonzero pattern, though the value of some of the elements plays
a role. The usual proof of the existence (if not the uniqueness) of
the reduced row echelon form is near combinatorial, and one could then
ask is there a proof of the same type of the existence of the Jordan
Form, and yes, there is. It is to be found in the book by Turnbull and
Aitken [TA] and it is well presented, with some gaps filled in, by
Brualdi [B] who called this proof marvelously simple (and see this
article for other proofs of a similar nature). In the field of linear
algebra, and perhaps in other areas of mathematics, what forms the
essence of an argument may depend on the questions that a
mathematician finds interesting.
Observations of this kind led to an interesting discussion between
Ptak and myself and, though Ptak agreed that not every problem was
suitable for co-ordinate free treatment, I do not think I
succeeded in persuading him of my point of view. However, I sent
him de Boor's article [dB] and he did not object to its inclusion
here. Lest there be any misunderstanding of what I am saying, it
should be noted that I agreed in principle to the publication of a
translation of Ptak's paper many years ago, that this is a breach
of the normal policy of LAA of publishing original research or
substantial surveys, and that in the thousands of papers published
by LAA the number of translations of previously published articles
may be counted on the fingers of one hand. Ptak's proof is a gem.
Throughout our correspondence over a period of several months Ptak
was obviously hampered by illness. Soon, I noticed with some
surprise that his email messages were coming from a hospital and
once he told me that he was facing a life threatening operation.
This he survived, and in late March he wrote optimistically "I can
move somewhat more freely; I don't have an oxygen tube...". But
only a short time was left. He kept writing on any topic I
desired, for example once he enlightened me on a point of the
Czech language. His last message, less than two weeks before he
died, dealt with an LAA paper he was handling as editor which he
feared he had neglected because of his illness. Ptak was a
dedicated mathematician.
References
[B] R.A. Brualdi, The Jordan Canonical Form: an Old Proof, 94
(1987) Amer. Math. Monthly 94 (1987), 257 - 267.
[dB] C. de Boor, On Ptak's derivation of the Jordan Normal Form,
LAA (this issue)
[F] Miroslav Fiedler, Vlastimil Ptak, 8 November 1925 - 5 May
1999.
[H] Olga Holtz, Applications of the duality method to
generalization of the Jordan Canonical Form, LAA (this
issue).
[HM] R. Horn and D. Merino, Contragredient equivalence: a
canonical form and some applications, Lin. Algebra Appl.
214 (1995), 43 - 92.
[M] A.I. Maltsev, Foundations of Linear Algebra (in Russian),
Moscow-Leningrad, 1948.
[P1] Vlastimil Ptak, Eine Bemerkung zur Jordanschen Normalform von
Matrizen, Acta Sci. Math. (Szeged), 17 (1956), 190 - 194.
[P2] Vlastimil Ptak, A remark on the Jordan Normal From of
Matrices, translation of part of [P1], LAA (this issue)
[P3] Vlastimil Ptak, Cirumstances of the submission of my paper in
1956, LAA (this issue).
[TA] H.W.Turnbull and A.C.Aitken, An Introduction to the theory of
Canonical Matrices, Blackie, 1932.
[V] Z. Vavrin: Miroslav Fiedler and Vlastimil Ptak: Life and
Work, Lin. Algebra Appl. 223/224 (1995), 3 - 29.
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