Talk:Cayley–Hamilton theorem

Proof can be simplified

The proof using polynomials with matrix coefficients (3.5) can be significantly simplified. The remark that evaluating such a polynomial by replacing the “unknown” (I would prefer the term “indeterminate”) by a polynomial is not a morphism of rings is essential.

But for a given product of such polynomials, it is not necessary for the matrix in whith the evaluation is done to be in the center of the ring generated by the coefficients of the two polynomials. It is sufficient that this matrix commutes with the coefficients of the left factor if evaluation is done with coefficients on the right – or with the right factors if evaluation is done with coefficients on the left.

To put it in formulas, we can write this lemma:

Let $M(t)$ and $N(t)$ be two polynomials in $t$ with matrix coefficients

Let us use evaluation with coefficients on the right, that is $M(P)\,:={\displaystyle \sum _{i}P\cdot M_{i}}$

Then, if $P$ commutes with all the coefficients $M_{i}$, $(M\cdot N)(P)={\displaystyle M(P)\cdot N(P)}$

The proof of this lemma is quite obvious: it is based on the equalities

$P^{i+j}\cdot M_{i}\cdot N_{j}=P^{i}\cdot M_{i}\cdot P^{j}\cdot N_{j}$

which hold when $P$ commutes with all the $M_{j}$

One can apply this lemma to the identity used for the proof, ${\displaystyle (tI_{n}-A)\cdot B=p(t)\cdot I_{n}}$

Obiously, $A$ commutes with the left factor $(tI_{n}-A)$

It is then legitimate to replace $t$ by $A$ in this identity, and we are done.

It is true that $A$ commutes also with the matrix coefficients in the right factor, $B$, but this is not necessary for the proof.

I let it to the author to double-check the reasoning above and to simplify the demonstration acccordingly.

With this simplification, this proof using polynomials with matrix coefficients seems to me the nicest and simplest proof.

Silvain.dupertuis (talk) 14:27, 9 March 2018 (UTC)

Request

Can somebody add a proof outline for the commutative ring case? The proofs I know work only over fields (or integral domains). AxelBoldt 00:52 Mar 30, 2003 (UTC)

General case

I was going to add this to the article, but it may be a bit too technical to include there. Furthermore, it is poorly written. (PLEASE correct and include in the article.)

I don't think there's such a thing as "too technical", as long as the more technical stuff is at the end of an article. The first few sentences should be written for a high school student, and then progressively more advanced. PAR 03:21, 4 May 2005 (UTC)

I agree. One should not try to knock the heck out of college students by starting with a terribly technical thing. But an article can (and should in many circumstances) gradually develop the subject and end up being quite advanced. Oleg Alexandrov 03:27, 4 May 2005 (UTC)

Okay, I'll move it into the article. It still may need some corrections, though.

I've corrected it (I'm a professional mathematician): the proof was incomplete, so I have filled in the gaps, and I tried to reflect the development from the elementary to the advanced.Geometry guy 12:51, 7 February 2007 (UTC)

Can somebody check on the corecctnes of the example? i think that when you substract the matrix {[t,0][0,t]} from the original one, that you DO NOT negate the second and third number!

I do not think the given proof works

I will buy the Adj(A-tI)(A-tI) = det(A-tI)I = p(t)I
This is proved using matrices over B = R[t].
However it is an equation in S = all such matrices.
You then apply this to m in M. However you have given no definition of such an action.
Also you have given no evaluation homomorphism for such a thing.
~reader

problem

The action definition is not given. You have to assign some meaning to Am. If you want to imitate the proof in Atiyah-MacDonald you need to operate on a vector with components from m. You can then drop all reference to the evaluation operation.

~reader

I may be stupid but...

I don't understand why this theorem is not a triviality...

$p(A)=det(A-AI)=det(A-A)=det(0)=0$

Isn't it?

I think the introduction to this article should show why this is not so. 131.220.68.177 09:47, 9 September 2005 (UTC)

Your "proof" only holds for 1×1 matrices. If you read carefully example 1, the idea is that you start with p(t) where t is a number, and then do the calculation of det(A-tI) according to the properties of the determinant. Then, plug in formally instead of t the matrix A, and you will get zero.

The error in your proof is the following. The quantity tI is supposed to be interpretted as

${\begin{bmatrix}t&0&0\\0&t&0\\0&0&t\end{bmatrix}}$.

If you plug right in here instead of the number t the matrix A, you get a matrix of size n². But then the subtraction A-AI can't be done in your proof, as the sizes don't match. Oleg Alexandrov 16:58, 9 September 2005 (UTC)

Yes I understood that already. In fact that's not that easy. I really see the point. Wouldn't it be possible to make this point clear from the beginning of the article. So that nobody fall in this trap again.131.220.44.10 14:49, 12 September 2005 (UTC)

I don't see a good way of writing that while still maintaining the nice style of the article. I think things are most clear if you look at the first example, they show how to interpret that t. Oleg Alexandrov 15:56, 12 September 2005 (UTC)

I patched up the proof for you. It is just a matter of defining an evaluation map between modules over the polynomial ring. I also thought it would be nicer (given the idea to proceed from the novice to the advanced) to give the proof for matrices before discussing the generalizations. This provides an opportunity to discuss the famous incorrect proof as well.Geometry guy 12:52, 7 February 2007 (UTC)

I agree that the intro is poorly done. I'd love to see someone who understands the intro, alone, who hasn't already seen this theorem. Regardless, changing "The Cayley–Hamilton theorem states that substituting the matrix A in the characteristic polynomial (which involves multiplying its constant term by In, since that is the zeroth power of A) results in the..." to "The Cayley–Hamilton theorem states that substituting the matrix A in the characteristic polynomial after the determinant has been taken (which involves matrix exponentiation instead of scalar exponentiation) results in the..." The current parenthesis are a bit pointless, and imply what I've written. "after the determinant has been taken" gets around the point above. 67.158.43.41 (talk) 10:07, 22 December 2009 (UTC)

I agree that the formulation was not optimal, but I do not agree with the suggested replacement, since (1) there is a real difficulty with the constant term, and (2) "after the determinant has been taken" is confusing to me (the characteristic polynomial is already the determinant, so you cannot "take the determinant" when you've already got the characteristic polynomial). I've changed to formulation to address (IMHO) most issues. The important thing is that one must view that characteristic polynomial as a true polynomial, not a polynomial function, and that the substitution is a somewhat unusual one in that the result is not a value in the same set as where the polynomial coefficients live (but instead in the set of matrices). In formal algebraic terms, the matrices form a k-algebra M (where k is the base field, say the complex numbers) and an element of M can be substituted into a polynomial in k[λ] to produce another element of M. This involves embedding k into M, and thus interpreting constants (in k) as the corresponding multiples of the identity. All this need not be said in the lede, but the reinterpretation of the constant term is a real change every reader should be aware of (one could maintain that the constant term is already multiplied by λ⁰ which changes into I_n by the substitution, but I'd bet very few readers will implicitly realize that). Marc van Leeuwen (talk) 18:07, 23 December 2009 (UTC)

Accessibility

I fixed up this article about a month ago, but I would like to edit it again to make the proof (for matrices) more accessible (for instance to undergraduate math students). If anyone has comments or suggestions, please let me know. Geometry guy 21:19, 21 February 2007 (UTC)

Undefined term

Hi, can the term "N-partitioned matrices" be explained or be linked to an appropriate article? Thanks. Randomblue (talk) 20:43, 17 January 2008 (UTC)

Link introduced. User:PhilDWraight 11:30, 11 February 2008 (UTC)

Fun with page names

Hmm, this is neat: http://en.wiki.x.io/wiki/Cayley-Hamilton_theorem vs. http://en.wiki.x.io/wiki/Cayley%E2%80%93Hamilton_theorem But maybe you'll like to change it... —Preceding unsigned comment added by 141.84.69.20 (talk) 00:25, 30 January 2008 (UTC)

Yes, I see the former redirects to the latter one, which has a bizarre page name. I'm not aware of when or how such a thing happened, or how it can be undone, but obviously it should be. Marc van Leeuwen (talk) 10:55, 30 January 2008 (UTC)

Nope, by now I found found that this "bizarre" page name just contained an "en dash", and that the wikipedia manual of style says this is the right symbol to use here, since Cayley and Hamilton are separate persons (the manual gives the Michelson–Morley experiment as an example of correct use of an "en dash"). Marc van Leeuwen (talk) 11:34, 30 January 2008 (UTC)

What's that about N-partitioned matrices?

The lead says the C-H theorem also holds for N-partitioned matrices. Can anybody explain what that means and why it is any different from the usual case? Or is somebody claiming the theorem makes sense (and holds) for matrices over another ring of matrices? If not, I'm tempted to delete that mention. Marc van Leeuwen (talk) 10:47, 7 March 2008 (UTC)

C-H is important in assessing system stability - re: position of eigenvalues etc - in multidimensional systems this becomes a complex issue. Partitioning the A/system matrix can make the problem much simpler - essentially looking at the problem as sub-systems - particularly useful for on-line analysis. Proof the C-H theorem holds for such matrices is vital. User:PhilDWraight 10.22, 22 April 2008 (UTC)

I understand (somewhat) the concern, but I still don't understand what the statement that C-H holds for partitioned/block matrices means. Suppose for concreteness that I have a block matrix

$M={\begin{pmatrix}A&B\\C&D\end{pmatrix}}$

where A, B, C, D are 10×10 matrices, say. First question: is there a characteristic polynomial of this matrix that is not of degree 20. The usual characteristic polynomial of a 2×2 matrix would be

$P=X^{2}-(A+D)X+AD-BC=X^{2}-(A+D)X+AD-CB,$

but as polynomials with matrix coefficients, the two versions are different (and in any case the unwritten coefficient 1 of $X^{2}$ should be interpreteed as a 10×10 identity matrix). So what exactly would the Cayley–Hamilton theorem say for this block matrix? Marc van Leeuwen (talk) 07:33, 25 April 2008 (UTC)

Changes to proof section in July 2008

On 4 July 2008 the section on proofs was considerably rewritten. The following discussion is copied from respective talk pages.

(from User talk:Marc van Leeuwen)

Marc, I've completely changed the section (originally mostly due to you, it looks like) in Cayley-Hamilton theorem concerning proofs. I've tried to retain the philosophical points you made, but some of what you said was simply not true, and it was overly opinionated. I'm afraid a lot of the pedagogy of comparing and correcting incorrect proofs has gone by the wayside as a result of the last one. Since you seem to feel strongly about the issue, I thought you would like to know so you could take a look. By the way: do you know of a published source for the proof you wrote (now tidied up a bit and included as the "First proof")? I gave a second proof from Atiyah-MacDonald, a third proof based on one which had appeared on that page some time ago, and a fourth based on some of the comments you made, but these latter two are likewise unreferenced (unreferenceable?). Ryan Reich (talk) 14:22, 4 July 2008 (UTC)

(from User talk:Ryan Reich)

Ryan, I've seen your edits on the Cayley–Hamilton article, and I appreciate that you informed me on my talk page, since indeed I had invested quite a bit of time in the part that you replaced. The text you replaced it with is interesting, but in my opinion not an improvement, even though it seems essentially correct (I have a few gripes but these are not so serious). But I'm probably not the most neutral person to judge, so we'll see how your change fares by other editors opinions. Let me just say a few things that come to mind.

• Your text frequently (at least four times) mentions the "defining property" of adjugates. It is important, but not "defining". The definition of the adjugate is that its entries are certain minors, and the mentioned property follows from that. If the property were "defining" any zero matrix could have any matrix of the same size as adjugate.
• You say "a lot of the pedagogy of comparing and correcting incorrect proofs has gone by the wayside as a result of the last one". I don't understand the phrase, which one is the last one?
• Your first edit summary mentions a didactic diatribe, but I did not want to push any didactic point. It is just that I think it is really a singular property of the Cayley–Hamilton theorem to inspire false proofs, and it is valid for the article to say so. I've seen many false proofs, some in print; before I edited the article all the proofs given there were false (rest assured, yours are not). Most false proofs are more sophisticated than bluntly substituting A for t in the defining equation of the characteristic polynomial, but they are false anyway. The thing that worries me most about the text you substituted is not that it it will mislead people, but that it will soon get replaced by people honestly convinced that they can do better than this.
• While interesting, none of the proofs given gives me the impression that it touches the essence of why Cayley–Hamilton holds (well the final argument invoking Euclidean division comes close, but phrases like the one starting "This incorporates the evaluation map" put me off (frankly I don't understand what it is affirming) and seem to have little to do with Euclidean division). The first three proofs are all quite long, and apart from the first one I doubt there are many wikipedia users that can actually understand them (given that their average level seems to be high school). The second took me long to absorb, and the essential point, that the determinant of B is p(A) needs more thought than is suggested.
• You say some of what I wrote is simply not true. I would appreciate if you were more specific. And then, you could have corrected (and added the proofs you did) without throwing away everything.
• The proof I wrote was loosely based on a book I found in our library (being dazzled by all the wrong proofs I looked for some solid ground), which happened to be an algebra course by Patrice Tauvel (in French); I undid it of some of what seemed to me unnecessary generality for the context at hand (it arrives at the Cayley–Hamilton theorem as a corollary to something more general). The consideration of Euclidean division was a result of discussions with colleagues at our math institute. But later I found much of it also on the French wikipedia (in some indirectly related article I cannot trace right now), so there is no point in claiming (or being accused of) original research here. In fact somebody sent me a paper reviewing some 20 different proofs... It seems like that many people have been thinking about this.
• I regret the disappearence of some points
  • the observation that the inital naive method not only gives a wrong argument, but also leads to the wrong conclusion (a scalar rather that matrix 0).
  • the observation that confusion arises from confusing unwritten (matrix and scalar) multiplication operations
  • the example that shows how naive substitution leads to genuinely false identities
  • the (before last) expression that shows that the adjugate of A is in fact a polynomial in A (with coefficients taken from the characteristic polynomial of A). This is an inportant and very general fact, which implies Cayley–Hamilton immediately, without being as easily implied by it.

Marc van Leeuwen (talk) 20:21, 4 July 2008 (UTC)

Thanks for replying! My response:

The reason I keep saying "defining property" is that it's hard to number equations in Wikipedia, and I need some other memorable device to refer back to them. Perhaps I will do as you did and insert (*) next to this one.

"The last one" referred to the last item in the previous sentence (which you didn't copy here), namely, that your text was sometimes opinionated. Basically, it seemed to me that your main goal was to correctly instruct the reader in the art of proving Cayley-Hamilton, and in particular, to push the point that trying to use the evaluation map directly could never work. All of your examples, including one example false proof, made this point; this is the thing that I thought was not right in what you wrote, since in fact it is possible to formulate a proof correctly using the evaluation map (you just have to be more careful, as in the third proof I wrote). The whole effort seemed "didactic" in that it was primarily concerned with correcting a misconception and instructing the reader through numerous but subtly different arguments that the only path to a proof of the Cayley-Hamilton theorem is through "real work" (that phrase really did have to go, by the way).

Concerning the loss of these subtly different arguments: looking back at the last version of the page before I edited it, the two big points you made were: there is no evaluation homomorphism from M[t] to M (in my notation), because A is not in the center of M; and, direct evaluation of the equation

$p(t)=\det(tI_{n}-A)\,$

at A leads to equating a matrix, p(A), with a scalar, det(0); a sub-point of this is that even replacing t by A on the right requires one to reconcile three things:

1. That t I_n is a diagonal matrix with t on the diagonal, so substituting A gives a matrix with matrix entries;
2. That A itself, as it appears in that expression, is a matrix with scalar entries;
3. That if we interpret t I_n as the "quantity" t multiplied by the matrix I_n, substitution of A for t transforms a scalar multiplication into a matrix multiplication.

You also think that these are among your main points. The one about there being no evaluation homomorphism is the one I think is wrong (given the proper context, and this distinction did make it into my version). As for the others, I actually think that they have been partially retained in my text, although perhaps in an excessively terse form. The first numbered point and its comparison with the second are explicitly in my text right before the first proof. I did miss the opportunity to give the "matrix equals scalar" dichotomy, but there is an obvious place to do so also right before the first proof, and I will make that correction. The point about multiplication is also implicit in the juxtaposition of the second and third proofs, though as you observe, the proofs as a whole are long and detailed, and perhaps extracting their "meaning" is not easy. I will expand the discussion before the proofs in order to reincorporate these points.

The reasons I replaced your analysis of erroneous arguments with just some proofs are that first, I felt that the existence of my third proof invalidated your frequently-expressed assertion that there could be no proof based on the evaluation homomorphism; second, although the above points are worthy ones, they could be easily expressed more briefly in the context of correct proofs, rather than as criticisms of incorrect ones; and third, that what was there concerned itself at least as much with educating the reader as with informing them. The second and third reasons are both related to the nature of the medium here: since an article is not a discussion, the false arguments you shoot down are more of the nature of a straw man than a real opposing position; and since it is also not a page in a textbook, the instruction you provide doesn't reside in the proper context for it to be received as intended.

I didn't mention original research because I think that any attempt to be philosophical about the proof of any theorem borders on it (comparison of proofs is not a major mathematical activity, although you say that for this theorem, it may be). I believe that a discussion such as you wrote is a good idea, but that to have the discussion in full requires a more scholarly medium.

This also goes for the proofs I included. I liked the third one much more before I started to write it up, at which time I realized that the story for Z is not as nice as I had thought, since it is not commutative. The whole thing ends up being a little technical, whereas the concept, which is to restrict the evaluation homomorphism to a context in which it is a homomorphism but still does the job intended for it, is quite simple. The fourth proof based on your Euclidean division idea is much more elegant. You say that you don't feel like any of the proofs gets at the "why" of the theorem, except maybe the fourth, but I think that the second one is really the best (this is somehow to be expected, given the author). My reason is that since (as we both agree is essential) p(A) is to be the zero matrix, and since matrices are naturally endomorphisms of vector spaces, this should be verified by considering its action on a vector space, and not simply by having p(A) = 0 pop out of a piece of algebraic machinery. Using the evaluation homomorphism is an elegant trick, but polynomial algebras are at their core a piece of algebraic machinery, a formal device; in the second proof, the matrix B is literally the matrix-with-matrix-entries which is t I_n - A, evaluated at A, and with the A already appearing considered as having matrix entries (which are all scalar matrices). This interpretation is consistent with the idea of using "actions on vector spaces" in the proof. That det(B) = p(A) is clear once this point is made; perhaps it needs to be made better, but I think this proof (which, unlike the last two, is sourced) is an important part of the philosophy of this theorem.

As for the fact that Adj(A) is a polynomial in A, of course, I did make that point in the fourth proof. I didn't mention that the coefficients are those of the characteristic polynomial, though that would of course give still a third way of using the Euclidean division technique to prove the theorem (the first two are: do division, observe that the remainder is p(A), and also that the remainder must be zero; do division, observe that the quotient is in k[A][t], and then show that the evaluation map is a homomorphism).

The main thing I'm getting out of this discussion is that the theorem is even more interesting than I had thought. What is this paper with the 20 proofs in it? I'd like to see it. Ryan Reich (talk) 17:13, 6 July 2008 (UTC)

(end of copied discussion) Marc van Leeuwen (talk) 12:31, 8 July 2008 (UTC)

I think we have sufficient common ground to cooperate productively. I may do some editing to restore some of the previous version without pushing any point you don't like; I see that version had plenty of defects, among which being too verbose. But I think we more or less agree that most algebraic proofs have the curious property of looking like attempts to justify by some elaborate arguments a simple but incorrect argument. I don't think though that the incorrect argument is necessarily the straight substitution into the definition of the defintion of the characteristic polynomial. The second proof does seem to try to make sense out if that, but at also has to invoke adjugate matrices and the original punchline seems to get lost in the details. But I think a greater temptation is to proceed as in the third proof, and cut short by using evaluation without the necessary precaution. The current text does warn against that, but the formulation is still somewhat confused: "At this point, it is tempting to consider...likewise not equal" makes it seem like there is one natural map ev_A (which fails to be a homomorphism); however there are two equally natural maps, left and right evaluation, neither of which is a homomorphism. The given ev_A is right evaluation, probably due to the fact of thinking of M[t] as polynomials in t with matrix coefficients written to the left, but one could eqully well write the powers of t on the left before replacing them by A, giving left evaluation. But anyway, I think the reason it is so tempting to think evaluation is well defined (and a homomorphism) is that most of us learn about polynomials in a commutative context, and worse, learn about polynomial functions before learning about polynomials as algebraic objects; the very notation p(A) illustrates the mental identification of polynomials with polynomial functions, and all of this makes a very bad preparation for handling polynomials with non-commuting coefficients with the proper caution. Note that even though one could define "polynomial functions" of a non-commutative ring to itself as those defined by left- or right-evalution of a polynomial with coefficients in the ring, such functions do not behave algebraically as the polynomials do (and in fact a product of such "polynomial functions" is usually not a polynomial function), so this is really an idea to put out of ones thoughts when considering polynomials over a non-commutative ring. I think some warning about this danger before setting up a proof would be in place, thought maybe not with as much emphasis on possible wrong proofs as in the old version of the article.

Actually I would prefer changing the order of the proofs to first, third, second, as I think the (current) second proof is really the hardest on the neurons, with all the respect I have for its authors. Here are some reasons why I think it is actually the most abstract of all. It considers a matrix with matrix entries, but this is not to be considered as a block matrix, since the determinant of a block matrix is not the usual determinantal expression in terms of the blocks (which does not even have a well defined meaning). So the matrix B must be thought of as a matrix in abstract indivisible objects (say endomorphisms of our vector space), so that the deteminant is just the determinant of those abstract values. However that only makes sense if the values live in a commutative ring, so one must observe that coefficients live in a commutative subring of endomorphisms before the notion of determinant even makes sense (the current text does make this point, but for the existence of the adjugate; what I would want to add is that in order for det(B) to even make sense this is a prerequisite). In fact I have the impression the motivation for introducing this strange matrix B is not so much to give a justification to the apparent bizarre operation of substituting a matrix into a matrix coefficient, as to define some abstract matrix whose determinant evaluates to p(A). Then the relation $\sum _{i}B_{i,j}e_{j}=0$ looks like it is a 1-line array (e₁ ... e_n) of abstract vectors right-multiplied by a column of B, that is a 1-column array of endomorphisms, through the usual matrix multiplication rule, but with products of "coeffcients" e_i and B_i,j evaluated by applying (the endomorphism) B_i,j to (the vector) e_i, which is quite a mental exercise. I know, one does not have to interpret it like that, it is just a sum of endomorphism-applied-to-vector values, but in that view the fact that the B_i,j are arranged into a matrix seems irrelevant. In any case I think trying to see the "sense" of the manipulations of that proof is not easy. Marc van Leeuwen (talk) 04:15, 9 July 2008 (UTC)

The second proof (and the third, for that matter) turned out quite a bit more involved when I wrote it than I had anticipated. The version given in Atiyah-MacDonald negelcts to mention the commutativity controversy, and therefore gives something of a false impression of completeness (I guess it's something they can assume any experienced mathematician would fill in automatically). The current third proof really does work better right after the first one, since it employs the same basic object. As for the erroneous arguments: it would probably work well to place a brief discussion of what could go wrong and how it is averted before each proof, or within it (for example, I think the existing discussion of the evaluation map in the third proof works well where it is). I may not have time to work on this page for a few days, though. Ryan Reich (talk) 02:35, 10 July 2008 (UTC)

The topological proof

given here is actually valid for matrices over any field, not just the complex numbers, since over any Algebraically closed field, the diagonalizable matrices always form a dense subset in the zariski topology. Liransh Talk 15:30, 10 April 2009 (UTC)

"Determinant-free" proof

Sheldon Axler, in his (very nice, it won an award) American Mathematical Monthly article "Down with Determinants!", and presumably also in his later book Linear Algebra Done Right, gives a short proof of the Cayley-Hamilton theorem using a definition of the characteristic polynomial that avoids determinants entirely. Is it equivalent to one of the proofs here? If not, I wonder (but doubt) whether it's possible to incorporate the proof somehow, without having to re-prove all the five pages of theorems beforehand. Shreevatsa (talk) 08:41, 29 September 2009 (UTC)

I've just rapidly read the article. While the paper is interesting, I cannot consider the result it calls "Cayley-Hamilton theorem" to be the same as the one this wikipedia article is about. That is because the notion of characteristic polynomial it uses, being defined without mentioning determinants, is very hard to match with the formal algebraic object defined here to be the characteristic polynomial. For one thing the article depends quite essentially on the complex numbers. It would be rather hard to deduce from its approach what the characteristic polynomial of a matrix with entries in for instance the a finite field (or in the integers, or in a commutative ring with zero divisors) would have to be, and why it takes coefficient in the same field (or commutative ring) as the matrix; all this is immediately obvious from the "determinant" definition. Also one has to read all the way to the end of the article, where the usual definition of determinants is finally "deduced", to understand why the coefficients of the characteristic polynomial depend in a polynomial (or even continuous) fashion on the coefficients of the matrix. In fact this is the fundamental reason why the characteristic polynomial, rather than the minimal polynomial (which the article naturally comes upon first), is of crucial importance for the theory. In the paper the characteristic polynomial comes across as a somewhat artificially "blown up" version of the minimal polynomial (by replacing the numbers $\alpha _{i}$ by the dimensions of generalized eigenspaces), making the fact that it is divisible by the minimal polynomial (i.e., the "Cayley-Hamilton theorem") more or less built into the definition. So in short, the answers to your questions are are no and no (even if one does re-prove those five pages). Marc van Leeuwen (talk) 21:02, 29 September 2009 (UTC)

Hmm, good point. Elsewhere, he says:

Another of the comments on this blog asks how to deal with linear operators on vector spaces over non-complete fields. There are two ways to do this: (1) Embed the non-complete field in its algebraic completion, as I do in the paper or (2) Give up on factoring a polynomial into linear factors and work directly with the non-complete field; the techniques for doing this are illustrated in Chapter 9 (titled “Operators on Real Vector Spaces”) of my book.

So it may be possible to extend the approach, but it doesn't seem utopian anymore. :-) Shreevatsa (talk) 02:28, 30 September 2009 (UTC)

one more proof?

One proof that is easy to follow is based on the similarity of any complex matrix to an upper-triangular one. This is the one I would choose to lecture engineers or scientists on this subject.

Thanks! Gold gerry (talk) 18:53, 18 January 2010 (UTC)

Diagonalisable matrices *are* dense in GL(n,R)

In the Preliminaries section, it is said that "for matrices with real entries the diagonizable ones do not form a dense set". This is wrong. If $M=PJP^{-1}$ is a Jordan decomposition of a real square matrix $M$, one can always perturb $J$ by $D={\textrm {diag}}(\epsilon ,\epsilon ^{2},...,\epsilon ^{n})$ for an arbitrarily small $\epsilon >0$ to obtain a matrix $J+D$ with distinct eigenvalues, and obtain in turn a diagonalizable real square matrix $P(J+D)P^{-1}$.The suffocated (talk) 14:09, 22 April 2010 (UTC)

This is not true. Rotation of R² by ninety degrees gives an example of a real matrix that cannot be approximated by real diagonalizable matrices. Algebraist 16:03, 22 April 2010 (UTC)

But it is complex-diagonalisable. I mean, the original text is confusing because the set of complex-diagonalisable real matrices is indeed dense in GL(n,R). You just cannot perform diagonalisation over the real field, but the proof can still work without modification. Even if you insists doing arithmetics on R, you can still follow the same idea and approximate the matrix by a direct sum of real scalars and 2x2 blocks (modulo a similarity transform). So you just need an additional step to prove the theorem for the 2x2 case.The suffocated (talk) 19:31, 22 April 2010 (UTC)

That's what that line in the article says, isn't it? You can prove Cayley–Hamilton for real matrices by extending to the complexes and diagonalizing, but it's a rather unsatisfactory proof. Algebraist 20:03, 22 April 2010 (UTC)

What I'm saying are: (1) the line is imprecise as one may (actually I did) mistake it as saying "the subset of all complex-diagonalisable matrices in GL(n,R) is not dense in GL(n,R)"; (2) the proof is unsatisfactory, but not because it does not work on R. It does work on R without extending to the complex field. All you need is to add one small step to verify the theorem for scalar multiples of 2x2 rotation matrices (which is easy). On a second thought, I also think that dismissing the proof for complex matrices as "strange" is ... strange. I′ll bet most people will find this proof more natural and intuitive than the others, because it reduces the whole theorem to the (trivial) scalar case. Also, continuity arguments are very useful and common tricks in matrix analysis. Transforming one problem into another easier one is also a usual practice in mathematics. If one solves a differential equation by using Fourier transform, shall we say that working on the frequency domain is strange? ;-D The suffocated (talk) 21:57, 22 April 2010 (UTC)

A `Non-proof'?

What is this ludicrous thing doing in the article? —Preceding unsigned comment added by 95.181.12.52 (talk) 09:29, 22 May 2010 (UTC)

I am not sure if `ludicrous' refers to the phrase `non-proof' or the wrong proof itself. If you are asking what purpose does the wrong proof serve, I believe it is here because its validity is the most confusing part for novices. See, for example, the discussion thread ″I_may_be_stupid_but...″ in the above.The suffocated (talk) 13:09, 26 May 2010 (UTC)

Unclear explanation of the bogus proof

There seem to be people who have already spent a lot of time on this, so I won't change anything and set off an edit war, but the explanation of the bogus "proof" seems to be needlessly obfuscated. This is really a simple issue with a simple solution: people don't understand the notation p(A), so we should write out p(A) explicitly. As currently written, this section never points out what p(A) actually is. This seems absurd to me. Note that p(A) is currently the fifth displayed line of this section, but the accompanying text gives the impression that this line is some sort of error. (More precisely, that line would be p(A), if the given text didn't re-interpret the matrix as a 4x4 matrix instead of a 2x2 matrix with entries in R[A].) Some of the other stuff here is worthwhile -- I like pointing out that the bogus proof tries to compare a scalar to a matrix, since I've seen too many of my students try to equate things that aren't even the same type of object -- but we should lead with the simple explanation "p(A) is THIS, not THAT." 165.123.215.110 (talk) 04:36, 10 February 2013 (UTC)

I suppose it is always hard to clearly steer the reader to take the received wrong steps correctly! The correct 2x2 matrix definition of p(A) is already in the last displayed eqn of the Example, section 1, at the top of the article, indeed, the takeout message of the whole article. I would argue against encouraging the confused reader to take the "right two left turns" to get to the proper p(A) here--it might be a good idea to not mix the right with the wrong in the bogus section. If the reader knew how to properly and safely compute and interpret non-scalar-component matrices, as you assume, she/he need not be here, in the first place! Any discussion of entries in R[A] would compound the confusion. Cuzkatzimhut (talk) 13:12, 10 February 2013 (UTC)

Claim about sufficient condition for a matrix to be diagonal is false as stated.

The article claims that: "(for a matrix to be diagonalizable it suffices for instance that its characteristic polynomial not have multiple roots)" Here is a simple counter example: A = [[1,1],[0,1]]. A has charpoly (t-1)*(t-1), but it is already in nondiagonal Jordan form and thus is not diagonalizable.

Now, perhaps the author here intended to say that the matrix have distinct roots. Perhaps this could be stated (although not clearly IMO) that the characteristic polynomial not have repeated roots, but the claim as stated is very misleading.

I don't understand your problem with the cited formulation. In your example (t-1)*(t-1) does have a multiple root, namely 1; are you making a fine semantic distinction between "multiple roots" and "repeated roots" (and if so which)? Maybe you just read "multiple roots" as saying "more than one root" (in which case to "not have multiple roots" would be more simply expressed as "have at most one root", which is entirely different from what is meant). If that is the case, than I will just modify the language to avoid this confusion. Marc van Leeuwen (talk) 13:02, 9 March 2014 (UTC)

Quaternions

If Hamilton really did prove that the theorem holds for quaternions it is quite a feat, especially if this was done for matrices of arbitrary order. Note that the quaternions constitute a non-commutative ring. This seems to me to strongly suggest that it holds for the complex numbers because they are embedded in the quaternions and all its subrings. It also ought to hold for rings isomorphic to subrings of the quaternions in general.

Something is not right, either in my reasoning above or that Hamilton's quaternion proof was in full generality for them. Is it the case that there are plenty of rings not isomorphic to a subring of the quaternions? In any case, this needs to be cleared up in the article.

(The statement that it holds for quaternions was buried in the reference section before my recent edits. The name of the reference itself clearly indicates that it really is quaternions we are talking about.) YohanN7 (talk) 21:41, 29 January 2015 (UTC)

See #wanted: location of the theorem in Hamilton's treatise. Hamilton's proof corresponds to certain 4 × 4 matrices. The theorem is true for general quaternionic matrices. YohanN7 (talk) 03:41, 14 February 2015 (UTC)

"Bogus proof" should be deleted

Unless somebody comes up with a convincing rationale for keeping the "bogus proof", I will delete it. You could construe the same sort of nonsense proof for whatever theorem, so why report this one? -- Julia Abril (talk) 13:54, 5 February 2015 (UTC)

Please do not! It is obviously not for you. The C-H theorem is exceptionally prone to such nonsense proofs, and they must be addressed. The section has a long history, and controversy, and lots of people do not like it; but this is a particularly pervasive and recurring question and this is the place to put it to rest. If you can improve it, cf. items 4,14,15 above, go ahead. But this is an educational section for the novice and the teacher dealing with him---a thinking expert need not be here! Recall this is not a treatise, but an educational resource. Dismissing an obvious common canard efficiently is not pointless. Besides, as you can see, there is a tangent on which it can be actually be salvaged. In any case, if you delete it, sooner or later, somebody will perceive a gap and reinsert something like it. If it offends you, and you cannot improve it, skip it in reading. Cuzkatzimhut (talk) 15:28, 5 February 2015 (UTC)

Not convinced, but since you insist that boldly, I will certainly not touch the section. -- Julia Abril (talk) 20:09, 13 February 2015 (UTC)

wanted: location of the theorem in Hamilton's treatise

The reference "Hamilton 1853" is insufficient. We should refer to a specific section in that treatise.

Only then will be able to definitely resolve the question whether Hamilton was dealing with matrices of quaternions or just with quaternions. From a cursory reading of the table of contents of the treatise (https://archive.org/details/lecturesonquater00hami), I think matrices appear nowhere, and the recent revert (https://en.wiki.x.io/w/index.php?title=Cayley%E2%80%93Hamilton_theorem&oldid=646712995) is plain wrong. -- Julia Abril (talk) 20:08, 13 February 2015 (UTC)

As you can see two threads up, I want to know as well what exactly Hamilton actually proved.

The Cayley–Hamilton theorem is stated in terms of matrices. If Hamilton proved it for "quaternions but not matrices of quaternions", the only way I can interpret it is as if he proved it for 1 × 1-matrices for quaternions, a k a quaternions. From the article,

For a 1×1 matrix A = (a), the characteristic polynomial is given by p(λ) = λ − a, and so p(A) = (a) − a(1) = 0 is obvious.

As you can see, the 1 × 1-version of the Cayley–Hamilton theorem then essentially asserts the equality of a quaternion with itself. This is not much to become famous for. YohanN7 (talk) 22:45, 13 February 2015 (UTC)

I have found the relevant references. Hamilton's version is about the inversion of linear quaternion valued functions of a quaternion and the existence of a "symbolic equation" (depending on the linear function) satisfied by the linear function (or operator). On the face of it, it seems to be equivalent to the usual formulation of Cayley–Hamilton theorem in the special case of certain 2 × 2 complex matrices or 4 × 4 real matrices since quaternions can be represented as such. I'll put in the references (soon) and reformulate the text (in due time).

It is clear that Hamilton didn't prove it for matrices of quaternions in general. But it seems plausible that the Cayley–Hamilton theorem does hold for n × n quaternion matrices, by appealing to the embedding of M_n(ℍ) in M_2n(ℂ) or M_4n(ℝ). For that, if true, other references will be needed. YohanN7 (talk) 23:59, 13 February 2015 (UTC)

I think I did sort most of it out. The claim that Hamilton's version corresponds to certain 2 × 2 complex matrices or 4 × 4 real matrices is without reference, but is supported in this: On Hamilton's Proof of the Cayley-Hamilton Theorem (not published, hence not quotable). The case for general quaternionic matrices seems plausible by the above, but I left it out. YohanN7 (talk) 02:57, 14 February 2015 (UTC)

It is now all sorted out. The referenced paper (for the full quaternion case) is by the way open access. Disclaimer: I haven't verified the page number in Hamilton's book (got it from third party). Don't have time to browse the online version a t m. YohanN7 (talk) 03:38, 14 February 2015 (UTC)

CAYLEY-HAMILTON THEOREM FOR MATRICES OVER AN ARBITRARY RING

CAYLEY-HAMILTON THEOREM FOR MATRICES OVER AN ARBITRARY RING by Jeno Szigeti

Serdica Math. J. 32 (2006), 269–276. Communicated by V. Drensky

Abstract. For an n × n matrix A over an arbitrary unitary ring R, we obtain the following Cayley-Hamilton identity with right matrix coefficients: (λ_0I + C_0) + A(λ_1I + C_1) + · · · + A^(n−1)(λ_(n−1)I + C_n−1) + A^n(n!I+C_n) = 0, where λ_0 +λ_1x+· · ·+λ_n−1x^(n−1)+n!x^n is the symmetric characteristic polynomial of A in R[x], I ∈ M_n(R) is the identity matrix and the entries of the n × n matrices C_i, 0 ≤ i ≤ n are in [R, R]. If R is commutative, then C_0 = C_1 = · · · = C_(n−1) = C_n = 0 and our identity gives the n! times scalar multiple of the classical Cayley-Hamilton identity for A.

We can probably list it in the references with an inline citation. Does the publication have a DOI (Digital object identifier)? (It is accessible via the journal site.) YohanN7 (talk) 17:17, 1 February 2016 (UTC)

J. Szigeti: CAYLEY-HAMILTON THEOREM FOR MATRICES OVER AN ARBITRARY RING Serdica Math. J. 32 (2006), 269–276.

No doi, free download from the journal`s homepage at: http://www.math.bas.bg/serdica/2006/2006-269-276.pdf — Preceding unsigned comment added by 98.202.15.145 (talk) 20:41, 5 February 2016 (UTC)

An other related paper is J. Szigeti: NEW DETERMINANTS AND THE CAYLEY-HAMILTON THEOREM FOR MATRICES OVER LIE NILPOTENT RINGS, PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 8, August 1997, Pages 2245–2254 Free access at: http://www.ams.org/journals/proc/1997-125-08/S0002-9939-97-03868-9/S0002-9939-97-03868-9.pdf — Preceding unsigned comment added by 98.202.15.145 (talk) 20:48, 5 February 2016 (UTC)

This seems okay too. Why don't you get an account by the way? You can also edit yourself (unless perhaps you are J. Szigeti in which case it is prunent to ask someone else to do it). YohanN7 (talk) 13:00, 6 February 2016 (UTC)

"Bogus proof" section has no citations, to be removed as original research

The (apparently controversial) section entitled "Bogus proof" has 0 citations, and despite years of existence has never had a single citation. Moreover, I have not been able to find any treatment of this "proof" in the mathematical literature (of reputable algebra textbooks and papers), aside from a few lecture slides for introductory classes that probably derive from this article via citogenesis. As an infinite number of spurious or logically unsound "bogus proofs" can be supplied for any theorem, it is incumbent upon advocates for inclusion to show by appropriate citations that this particular bogus proof is notable, in the senses that (1) it is a common pitfall frequently erroneously cited by students or instructors, and (2) that its inclusion is of unusual educational value, going beyond the educational value of explicitly debunking any incorrect method (where again there are infinitely many). As the section is uncited, I can only assume that this "proof" is an example of original research, i.e. a deliberately erroneous proof that a Wikipedian came up with and thought to include, and will be removed unless (1) and (2) can be addressed. 2001:630:12:2E23:FD65:BC80:EA1E:CDFC (talk) 20:02, 27 February 2016 (UTC)

I can tell it has been bugging you for years, so you opted for going anonymous. Sections 4, 14, 15, 18, above, at the very least, document what you call "apparent controversy". Of course no reputable book or article would publish it---it is a common pitfall of the undeserving and the confused. They too need to get something out of this article, if , at least some of them, bothered to ask the question. I do not understand your hostility to them. But it is a bit bizarre to call it "original research"---unless you have no personal professional experience with the pedagogical problem involved...

If you had a better way to protect the clueless from the fallacy, and you truly cared for that, sure, go ahead, supplant it. But giving short shrift to the confusable in an article that I would assume you'd never visit again, just because it offended you there are untutored readers that need it, well... In any case, I strongly believe this is the most pervasive slip among novices, and both your (1) and (2) points above, hidebound citations being at hand or not, appear eminently warranted. Over 2000 readers in this page deem the point worth reading about. How do you argue for cytogenesis in P J Morandi's course notes?

If you thought you could be constructive, seek more suitable protective citations, or else improve on the point made. But, no, I would assume most readers would distinctly not agree that this is just yet another one of an infinity of stupid things one can say about the theorem. It commands place of honor among stupidities, and, as such, it commands attention. If you really deemed that salutary, maybe it could be relegated to a monster footnote, but then it would be missed by the readers who need it most. Perhaps other page watchers could have suggestions on this. Cuzkatzimhut (talk) 21:21, 27 February 2016 (UTC)

I should hope adducing Garrett's standard book dispels the odd "original research" notion. Cuzkatzimhut (talk) 01:22, 28 February 2016 (UTC)

Power series

The exposition in the section on power series via Cayley-Hamilton can certainly be improved. The way it stands, it feels like a physicist's dump. Manoguru (talk) 03:23, 9 March 2016 (UTC)

Of all categories of talk page comments, the stupidest of all may be this article can be improved. YohanN7 (talk) 08:33, 9 March 2016 (UTC)

Simpler proof

It seems to me that there is an easier proof than the ones given. Since I don't have a source for it, I want to get confirmation before adding it to the article.

Say we are working over a field F. The proof is by induction on n. The base case n = 0 (or n = 1) is straightforward. Suppose we know the theorem for (n – 1) × (n – 1) matrices. Let λ be an eigenvalue of our n × n matrix A. Because the characteristic polynomial remains unchanged when we change the basis, we may choose a basis e₁, e₂, ..., e_n such that e_n is an eigenvector of A with eigenvalue λ. Then A looks like

$A={\begin{pmatrix}&&&0\\&B&&\vdots \\&&&0\\*&\cdots &*&\lambda \end{pmatrix}}$

where B is an (n – 1) × (n – 1) matrix. By cofactor expansion, we have p_A(t) = p_B(t)(t − λ). Moreover, by the induction hypothesis, p_B(B) = 0.

If we view F^{n – 1} as a subspace of Fⁿ with basis e₁, e₂, ..., e_{n – 1}, then B is just the restriction of A to a linear transformation F^{n – 1} → F^{n – 1}. Thus for any vector w ∈ F^{n − 1},

$p_{B}(A)w=p_{B}(B)w=0w=0$.

For any vector v ∈ Fⁿ, we see from the above form of A that (A − λI_n)v ∈ F^{n − 1}. Applying the above equality with w = (A − λI_n)v, we get

$p_{A}(A)v=p_{B}(A)(A-\lambda I_{n})v=0$.

Since this holds for all v ∈ Fⁿ, we conclude that p_A(A) = 0.

Enigmatologist (talk) 23:47, 29 May 2019 (UTC)

@Enigmatologist: we may choose a basis e₁, e₂, ..., e_n such that e_n is an eigenvector of A with eigenvalue λ – this is generally false; the theorem still holds for non-diagonalizable matrices. The proof for the diagonalizable case is trivial and doesn't even require induction; the proof given does not assume the existence of an eigenbasis (or indeed, whether any eigenvectors exist at all; we may not even have eigenvalues in the ring in question).--Jasper Deng (talk) 00:37, 30 May 2019 (UTC)

I think you misunderstood. I'm not saying that every basis vector is an eigenvector, just the last one. This can be done as long as there is an eigenvalue. Though I should have assumed that the field is algebraically closed, as there might not be an eigenvalue otherwise. Enigmatologist (talk) 23:23, 31 May 2019 (UTC)

I did not misunderstand. To be clear, that last part is the problem. The ring in question need not even be a field, let alone an algebraically closed one. Rings that aren't integral domains cannot have a field of fractions so a proof for the case where R is a field obtained by extending F to its algebraic closure is not powerful enough for the general case. In the algebraically closed case, the Jordan normal form suffices without induction.--Jasper Deng (talk) 23:31, 31 May 2019 (UTC)

Ridicdulously unclear sentence

"For such cases, for an eigenvalue λ with multiplicity m, the first m – 1 derivative of p(x) vanishes at the eigenvalues."

What? "The first m - 1 derivative of p(x) vanishes at the eigenvalues."???

If you don't know how to express plurals, and their accompanying verbs, in English, then it would be better to learn how to do this, instead of confusing people on Wikipedia with ridiculous sentences like the one above. Especially in mathematics.

I'm guessing that the intended sentence is

"For such cases, for an eigenvalue λ with multiplicity m, the first m – 1 derivatives of p(x) vanish at the eigenvalues." (The boldface is mine, for emphasis.)

But it's hard to be sure.50.205.142.50 (talk) 16:45, 13 March 2020 (UTC)

Grammar fixed to salutary approximant. Cuzkatzimhut (talk) 01:00, 14 March 2020 (UTC)

Rewrote the lede, extra notes

I have rewritten the lede to try to make it clearer what exactly the C-H theorem says. I think this makes the "bogus proof" section significantly less necessary, since (to my mind) it stems from an easy misinterpretation of simply defining the analogous matrix equation as simply "consisting of the replacement of the scalar variable λ with the matrix A" from the characteristic polynomial det(λI_n - A). That said, I won't immediately take the initiative to remove it from this page. I also completely removed the sentence "The powers of A, obtained by substitution from powers of λ, are defined by repeated matrix multiplication; the constant term of p(λ) gives a multiple of the power A^0, which is defined as the identity matrix.", because it seems bizarrely elementary for the intended audience of this article, who is presumably already familiar with algebraic operations on matrices. A Lesbian (talk) 17:01, 30 October 2020 (UTC)

Short proof for all commutative rings

The proof I propose has three short stages:

1. Prove that the theorem is true for matrices over $\mathbb {C}$. This can be done by observing that the diagonalisable matrices are dense in the set of all square matrices. The proof of this for $\mathbb {C}$ is very easy.

2. Prove that the theorem is true for free commutative rings by using the fact that it's true for $\mathbb {C}$. This is easy to do since the elements of a free commutative ring can be interpreted as polynomials over $\mathbb {C}$.

3. Prove that the theorem is true for all commutative rings using the fact that it's true for the free commutative rings.

I'm not an algebraist, and I don't have a citation for the above. --Svennik (talk) 22:16, 27 July 2021 (UTC)

Do you mean "free commutative algebra"? Perhaps I am being pedantic, but there is only one free commutative ring which is $\mathbb {Z} [X]$. The other polynomial rings $R[x]$ (for a commutative ring $R$) are free $R$-algebras (that is, free in the category of $R$-algebras) and not free rings (except for when $R=\mathbb {Z}$, since commutative $\mathbb {Z}$-algebras are commutative rings, which gives $\mathbb {Z}$ as the free commutative ring) – for this reason I prefer the term "polynomial algebra" to "polynomial ring", but unfortunately the latter is too deeply entrenched for the former to become standard.

The above potential pedantry aside, could you give an outline of Step 3? I have heard about such "universal arguments" a few times before but I am not familiar with how they go – is it just a simple application of some universal property? Joel Brennan (talk) 01:57, 3 February 2022 (UTC)

Infinitely many free commutative rings do exist. They take the form $\mathbb {Z} [X_{1},X_{2},\dotsc ]$ where $\{X_{1},X_{2},\dotsc \}$ are the elements of an arbitrary (and arbitrarily large) set. See here for more.

Every commutative ring $R$ should be the result of quotienting a free commutative ring $A$ by some ideal. Let $\phi :A\to R$ be the canonical projection. Let $M$ be a matrix over $A$, and let $p$ be its characteristic polynomial. From $0=p(M)$ (because we're assuming that the theorem is true for free commutative rings), we have that $0=\phi (p(M))=\phi (p)(\phi (M))$. Since $\phi (p)$ is also the characteristic polynomial of $\phi (M)$, the Cayley-Hamilton theorem follows.

I'm not an algebraist, so I don't know how to dot the eyes and cross the tees in the above argument. --Svennik (talk) 15:31, 4 February 2022 (UTC)

immediate consequence of Jordan normal form...

This proof section begins with "the Cayley–Hamilton theorem is an immediate consequence of the existence of the Jordan normal form for matrices over algebraically closed fields" without further clarification. This is the proof that I would be most interested in. Does it mean that A=V J V^-1 so that P(lambda) = |A - I lambda| = |J - I lambda|, ie the characteristic equation is invariant, and thus P(A) = (j1 I - J)(j2 I - J)..(jn I - J), which has a 0 at every diagonal position in at least one factor, and thus equals the 0 matrix? I think a fleshed out version of this would be great to include right after that statement, especially about the invariance of the characteristic equation, and therefore det, tr, and everything in between. And a reference to Invariants of tensors. Chris2crawford (talk) 03:59, 9 February 2022 (UTC)

Oh that is sketched out in here. Added a direct reference so others are aware of it.Chris2crawford (talk) 13:27, 14 February 2022 (UTC)