From: israel@math.ubc.ca (Robert Israel)
Subject: Re: Theorem of Frobenius and Perron
Date: 23 Feb 2000 19:44:55 GMT
Newsgroups: sci.math
Summary: [missing]

In article <88vrh6$6l7$1@nnrp1.deja.com>,
 not_quite@msn.com writes:
 
>  I suspect it was a case of independent discovery, but could someone
> please tell me when what today's textbooks typically refer to as "the"
> Theorem of Perron and Frobenius was first published? (A website that
> had that kind of information would also be nice to know about.)

I got the following references from Gantmacher, "The Theory of Matrices".

Perron's original theorem dealt with positive matrices (i.e. those with
all entries > 0): 
O. Perron, "Uber Matrizen", Math. Ann. 64 (1907) 248-63.

Frobenius generalized the theorem to irreducible non-negative matrices:
G. Frobenius, "Uber Matrizen aus positiven Elementen", S.-B. Deutsch.
Akad. Wiss. Berlin Math-Nat. Kl. 1908, 471-76; 1909, 514-18.
G. Frobenius, "Uber Matrizen aus nicht negativen Elementen", S.-B. Deutsch.
Akad. Wiss. Berlin Math-Nat. Kl. 1912, 456-77.

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2
==============================================================================

From: bumby@lagrange.rutgers.edu (Richard Bumby)
Subject: Re: Theorem of Frobenius and Perron
Date: 23 Feb 2000 18:46:24 -0500
Newsgroups: sci.math

not_quite@msn.com writes:

> Hello.

> I suspect it was a case of independent discovery, but could someone
>please tell me when what today's textbooks typically refer to as "the"
>Theorem of Perron and Frobenius was first published? (A website that
>had that kind of information would also be nice to know about.)

I'm sure there is a website, but I haven't recorded its location.
Since I have a number of books on the subject within close reach, I
can give you a few old-fashioned twentieth century references.  You
are sure to please your local librarian by requesting something in
printed format.

There are some books dealing entirely with nonnegative matrices, but I
don't have one handy.  Such a book would be useful for additional
details and references to current work on the topic.

The first book I reached for was Horn & Johnson, "Matrix Analysis".
The topic is discussed in chapter 8.  There were some references to
recent work, but I didn't see references to the primary sources.  It
did give enough information to suggest that your suspicion was wrong.
Perron proved the theorem for matrices with all entries positive in
1907.  Frobenius generalized the result to apply to *nonnegative*
matrices, introducing the key notion of primitivity in 1912.  For more
details, the reader is referred to the book by Berman and Plemmons or
the book by Senata.

Then, I looked at Gantmacher, "Matrix Theory".  Here, the topic
doesn't appear until chapter XIII, which is in volume two in the
version I have.  This includes references to the original sources, and
reveals that Frobenius did write on *positive* matrices in 1908.
There may have been some independent discovery since Frobenius was
about 60 at the time and Perron appears to have been at an early stage
of his career (I don't have any biographical data on him, but his book
on Continued Fractions, published in 1929 claims to be based on
lectures given in 1909).

-- 
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bumby@math.rutgers.edu||  Math. Dept. Computer Coordinator 1998--NOW
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