Talk:Characteristic polynomial

Secular function section

This section is completely incomprehensible to me. What on earth is "matrix tearing and gruing" and "determinant of a matrix with a shift"? Where do the poles of the secular function come from? A polynomial has no poles.

Most of this section looks like something from uncyclopedia and is completely confusing. I doubt it belongs on this page at all. HugoBarnaby (talk) 08:55, 20 October 2010 (UTC)

I agree. Everything after the first paragraph, particularly the middle of this section, is very poorly written. I can only assume it was written by a non-native speaker of English or, like you said, as a joke. It would be great if someone familiar with secular functions could re-write this section more clearly. 155.41.17.62 (talk) 18:42, 14 October 2011 (UTC)

Note that the source 10 "secular equation". Retrieved January 21, 2010. is down. — Preceding unsigned comment added by Anonime17 (talk • contribs) 13:11, 16 December 2020 (UTC)

Some pointers

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

Extended content

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)

Overhaul needed

Article mentions two conventions, justifies settling on one, then uses the other in the tail of the article. Perhaps settle on one convention (the one argued for at the beginning) used throughout except in one section which explains the other, gives corresponding forms, show how to interconvert, so that all "alternative" convention is restricted to that section.

An explicit expansion of the polynomial (with the alternative form in the "alternative" section) showing the terms/coefficients referred to later might help clear things up. — Preceding unsigned comment added by 70.72.144.66 (talk) 17:13, 8 May 2013 (UTC)

$\bigwedge$ instead of $\Lambda$?

In the entry text I read:

"$p_{A}(t)=\sum _{k=0}^{n}t^{n-k}(-1)^{k}\operatorname {tr} (\Lambda ^{k}A)$"

After consulting the entry about exterior algebra (which is referred to by the text), I noticed that the symbol $\bigwedge$ is used there.

Is this actually the symbol that was intended / should be used instead of capital lambda in the current entry?Redav (talk) 13:42, 28 November 2020 (UTC)

Characteristic of what?

The current entry text contains:

"When the characteristic is 0 it may alternatively be computed as a single determinant, that of the k×k matrix, [...]"

Could someone tell us the characteristic of what is meant here? (I did follow the link to "characteristic" (algebra) but that seems to deal with rings and fields, and the connection to the quoted statement is not clear to me.Redav (talk) 14:01, 28 November 2020 (UTC)

The connection is that the facor 1/k! makes no sense in non-zero characteristice. However I have edited the article for clarification. D.Lazard (talk) 14:43, 28 November 2020 (UTC)

Thanks a lot! I consider that helpful.Redav (talk) 16:17, 29 November 2020 (UTC)

Notation

This article uses p_A(t), but I’ve also seen \chi_A(t). Perhaps a brief note on common notations is needed.