Talk:Characteristic polynomial

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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 Nov 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 Dec 2003 (UTC)

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 Jul 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 Jul 2004 (UTC)

[edit] 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 Jul 2004 (UTC)

[edit] 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 ? — MFH: Talk 22:41, 24 May 2005 (UTC)

[edit] Why use 't' for the indeterminate?

't' just seems like such a random choice, 'X' is much clearer.

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