Talk:Characteristic polynomial/Archive 1

Murky beginning
This description gets a bit murky to the non-mathematician at the definition of the polynomial. What is t? What is a "polynomial ring"? Is there a simpler way to describe this concept without bringing in so many other mathematical areas, or at least a way to make them optional? Perhaps a good example is required to help ground the definition, although that might damage the generality. Brent Gulanowski 19:01, 30 November 2003 (UTC)


 * Well, it's dense rather than murky. But I agree, really. I've added some initial comments that are intended to clarify what is happening. We'll see if these are to others' taste. Charles Matthews 17:22, 1 December 2003 (UTC)

Motivation
Well, I'm sorry to say that I wasn't too happy with the motivational part, since it didn't really say what the goal was (to get a polynomial whose zeros are the eigenvalues) and it used the fact that every matrix can be approximated by diagonalizable ones, which is not intuitively clear. I tried to write some other motivational intro.

Also I added an example and I changed the definition to det(tI-A), since that is a monic polynomial and it works better with the companion matrix article. AxelBoldt 18:45, 11 July 2004 (UTC)


 * The approximation business - it may not be intuitive, but it certainly helps a great deal to understand linear algebra if one has this concept. I was once told that my proof of the Cayley-Hamilton theorem using it was the 'worst ever'. But that was by a functional-analyst; while to an algebraic geometer it is just a good way to use the Zariski topology, and then use the fact that identities hold on closed sets.


 * So, I wonder where it belongs in the WP articles. Charles Matthews 07:34, 15 July 2004 (UTC)

Roots and zeroes
I think this is a somewhat pedantic point. But given the edit comment The values for which a polynomial has a value of zero are called 'roots' and not 'zeros, I think it should be pointed out that in P. M. Cohn's Algebra, it is polynomials that have zeroes and equations P = 0 that have roots. Charles Matthews 09:10, 15 July 2004 (UTC)

another 'WP' style formal definition
Once again, a page starting with "In [ some area of mathematics ]..." with a "Formal definition" going like "...you can think of..." and "This [ det(tI-A) ] is indeed a polynomial, since determinants are defined in terms of sums of products."


 * Nowhere is said what t is and how it can multiply I
 * I don't even dare to ask what the author calls a polynomial (imho, "in mathematics" (*sigh*), this should be a map from N into some group)
 * same question for determinants (at least, I know another definition than the above...)

Now, if you accept this definition, then you also must accept the usual "student's proof" of the Cayley-Hamilton theorem:
 * P(A) = det( AI-A ) = det( A-A ) = det O = 0.

Easy ! What's all that fuzz about ? &mdash; MFH: Talk 22:41, 24 May 2005 (UTC)

Why use 't' for the indeterminate?
't' just seems like such a random choice, 'X' is much clearer.

Is there a reason for the change from writing the polynomial as a function of 'λ', to one of 't'? Pscholl 19:45, 7 June 2007 (UTC)

Merging
I saw the header on Characteristic equation, and wished to indicate my strong support. The content, at least that part of it regarding linear algebra, appears to be very much the same. Golwengaud 06:22, 23 August 2007 (UTC)
 * I feel that these articles could be merged, so long as both terms are mentioned. Some lecturers refer to it as Characteristic Equation and some as Characteristic Polynomial, so both need to be incorporated.  —Preceding unsigned comment added by 172.213.84.207 (talk) 23:00, 7 January 2008 (UTC)


 * I redirected Secular determinant here. It was previously a dead-end orphaned article whose content was a description of the characteristic polynomial of a matrix.  Some google searching shows this is a chemistry term.  I think the "secular equations" are a matrix, and the "secular determinant" is their characteristic polynomial.  If someone knows more, then they could add a paragraph to this article.  JackSchmidt (talk) 22:21, 14 February 2008 (UTC)


 * I noticed actually secular function and secular equation were also complete synonyms (according their articles) for characteristic polynomial. I think having five articles on the same concept is a bit extreme.  I added merge notices to secular equation and secular function (which have content), and made secular determinant redirect to secular equation for now.  In case someone opposes the merge, the redirect will take them to a synonymous page using the language they expect. JackSchmidt (talk) 22:31, 14 February 2008 (UTC)


 * I agree that the articles should be merged. As far as I know, the characteristic polynomial is never used by itself. To be useful, it must equal zero. --Emok (talk) 20:51, 25 March 2008 (UTC)

Needs work!
The relationship between the zeros of the polynomial and the eigenvalues of the matrix is very poorly described. It is only shown that an eigenvalue is a zero, but there is the other way around and the matter of multiplicities. The case of non-symmetric matrices is non-trivial. McKay (talk) 13:43, 31 March 2009 (UTC)

Product of two matrices
Can someone give a reference (or even a proof) for the given identity p(BA)=p(AB)*t^(m-n) for two non-square matrices? --Roentgenium111 (talk) 18:22, 4 November 2008 (UTC)
 * Something like http://www.math.sc.edu/~howard/Classes/700/charAB.pdf would probably suffice. It proves the square result for singular matrices as well. &mdash;TedPavlic (talk/contrib/@) 16:37, 26 May 2010 (UTC)

Some pointers
The term "secular equation" is used widely in planetary theory and in quantum mechanics. See, e.g.

Hyperspherical Harmonics and Generalized Sturmians Series: Progress in Theoretical Chemistry and Physics, Vol. 4 Avery, John S. 2000, ISBN 978-0-7923-6087-2,

The Nuclear Many-Body Problem Series: Theoretical and Mathematical Physics Ring, Peter, Schuck, Peter 1980, ISBN 978-3-540-21206-5,

Theory of Orbits Volume 2: Perturbative and Geometrical Methods Series: Astronomy and Astrophysics Library Boccaletti, Dino, Pucacco, Giuseppe 1999, ISBN 978-3-540-60355-9, Atomic clusters and nanoparticles. Agregats atomiques et nanoparticules Les Houches Session LXXIII 2-28 July 2000 Series: Les Houches - Ecole d'Ete de Physique Theorique, Vol. 73

Guet, C.; Hobza, P.; Spiegelman, F.; David, F. (Eds.) 2001, ISBN 978-3-540-42908-1,

Also text by Brown.

Classical statement about transfer of term from astronomy to quantum theory in Pauling and Wilson Introduction to Quantum Mechanics p. 171. Note that secular equations occur in applications of perturbation methods and variation methods to Schrodinger equation for ANY system -- atoms, molecules, ions, solids -- and these are NOT restricted to electronic properties -- see any undergraduate text on physical chemistry.

Secular equations are solved by variety of methods. See e.g. Golub and van Loan, section 7.7 for traditional matrix approaches. For several decades starting in 1930s most work expanded secular determinant into explicit characteristic polynomial and solved that. Algorithm widely used published by Hicks in J.Chem.Phys. in 1940. Rediscovered (invented) by Markov -- see Fadeyev and Fadeyeva 1963 -- same method used to expand of symbolic determinant to characteristic polynomial e.g. by Collins et al in SACLIB package. Full citations in my paper with Decker and Krandick, J. Chem. Phys. 114, 23, 10265, 2001.

Bit curious why this article has been criticized and other articles on related topics that contain more serious mathematical faux pas seem unassailable. Michael P. Barnett (talk) 20:36, 3 May 2011 (UTC)