Talk:Eigenvalues and eigenvectors/Archive 1

message: Now that we have all this information in one place we need to figure out the correct structure. --MarSch 3 July 2005 12:20 (UTC)
:I agree, this page is too haphazard and it was hard to find what I was looking for (specifically how to find them). Fresheneesz 19:15, 10 December 2005 (UTC)
thread_title:  Fixing up this page 

message: For the matrix having 2 on the diagonal and 0 off diagonal, the power method as explained in this article will fail to converge to any eigenvector, since applying the matrix is the same as multiplying by two, and this is done an infinite number of times. Am I getting something wrong? Oleg Alexandrov 4 July 2005 00:06 (UTC)
:Uhm, yes, well, I have to admit that "converge" is used in a very loose sense here. I tried to amend the text. It is still work in progress anyway. Somewhere on Wikipedia it should also be mentioned that the method fails with matrices like
:: \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, 
:which has two eigenvalues of equal magnitude. Sigh, so much to do ... -- Jitse Niesen (talk) 4 July 2005 00:37 (UTC)
The claim that insolvability of the quintic by radicals implies no algorithm to solve polynomials in general is wrong. I have rephrased that section. Stevelinton 09:56, 29 August 2005 (UTC)
thread_title:  [[Power method]] 

message: I find the article to be fairly accessible to those who are not math gurus but too much emphasis in the article is on the abtruse language generally employed by mathematicians to describe what I colloquially term "eigenstuff". What may help is if someone adds some of the physical significances of eigenvalues and eigenvectors, such as:
* The moment of inertia tensor always produces orthogonal eigenvectors which correspond to principal rotational axes about which the equations of rotational motion become very simple. The eigenvalues are the scale factors of the moment of inertia equation in such a case.
* A matrix which describes an approximation of a conjugated pi system (The Hueckel Approximation, if anyone is curious) often appears as a determinant, and setting it equal to zero yields the eigenvalues (allowed energies) and the eigenvectors (normalized but not necessarily mutually orthogonal in whatever vector space it spans, not that chemists care about the vector space involved) yield the contribution of each atom in the pi system to the associated molecular orbital represented by the eigenvalue.
* In heat transfer the eigenvectors of a matrix can be used to show the direction of heat flow. I am not entirely sure in this case what the eigenvalues represent. My thermodynamics didn't extend to matrix versions of the Fourier equation, so... (???)
* Someone mentioned stress-strain systems where the eigenvectors represent the direction of either greatest or least strain or stress, depending, IIRC, on the way the matrix is set-up. Again, not having done engineering in detail I can't be more specific.
* More generally someone mentioned to me once that in spectroscopy you can set up a matrix which represents the molecule under study, and in Infrared in particular, because it effectively acts as a series of coupled springs which vibrate at discrete energies, the eigenvalues here give the energies (wavenumbers) for the peaks and the eigenvectors tell what kind of motion is happening. Again, without quantum spectroscopy in detail I can't be more specific than this.
I would like to encourage Wikipedians in general, if they are editing math-specific entries, to try and relate the significance, either mathematically, or physically, of the concept or method in less abtruse language than is typical in linear algebra texts. :)
--142.58.101.46 4 July 2005 16:23 (UTC)
::Well, I'm a math person, and I find the majority of what you just said far more "abstruse" than the math jargon. It's all subjective!! 216.167.142.30 20:40, 23 October 2005 (UTC)
thread_title:  Physical significance of eigenvectors 

message: Yes? No? --HappyCamper 06:12, 11 August 2005 (UTC)
thread_title:  Eigenfunctions belong here too? 

message: A reference to singular values and singular value decomposition could be of interest too ? --Raistlin 16:16, 13 August 2005 (UTC)
thread_title:  Singular Values ? 

message: Oleg Alexandrov's edit summary (full disclosure: complaining about my subsubsubsection):
:I'd rather not have this as a subsubsubsection. That could be correct, but does not look good in the toc
I agree that it looks bad in the TOC, but it seems wrong that we should be compromising the article in order to improve the æsthetics of a standardised TOC.  Has this problem been solved satisfactorily elsewhere on WP?  (The obvious suggestion would be an option for subsubsubsections (or whatever) not to be mentioned in the TOC at all.) —Blotwell 09:26, 18 August 2005 (UTC)
replies: text: There's a (very short) discussion about the possibility of limiting the table of contents to display only up to a certain level of headings at Wikipedia talk:Section#Choosing TOC level, but not much more than wishful thinking at the moment. I agree with User:Blotwell that we shouldn't disrupt structural tags in the article in order to make the table of contents look prettier. —Caesura(t) 18:05, 7 December 2005 (UTC)
thread_title:  Table of contents 

message: projection onto a line: vectors on the line are eigenvectors with eigenvalue 1 and vectors perpendicular to the line are eigenvectors with eigenvalue 0.
Isn't that true only of orthogonal projections?  Consider
A=\begin{bmatrix}0 & 0\\ 1 & 1\end{bmatrix}.
Doesn't that qualify as a projection onto a line?  Yet it has no eigenvectors perpendicular to the line. Josh Cherry 14:35, 20 August 2005 (UTC)
replies: text: You are right, I fixed it.--Patrick 09:16, 21 August 2005 (UTC)
thread_title:  Projections 

message: A little while ago I suggested a sentence in the introduction to give some motivation, something really simple to tell the reader why they should care. It was along the lines of:
: eigenvectors/eigenvalues etc allow us to replace a complicated object (a linear transformation or matrix) with a simpler one (a scalar).
This simple idea, expressed in very plain english, explains why we care about eigenvalues/vectors, without confusing the uninitiated with terminology like bases, diagonal entries, etc. The sentence was deleted at some point, and I feel the omission with some pain. Does anyone else agree that such a sentence would be a Good Thing? Dmharvey  Talk 02:30, 23 August 2005 (UTC)
replies: text: Me, I agree.  Though I'm no pedagogue.  —Blotwell 05:52, 23 August 2005 (UTC)
thread_title:  motivation? 

message: In the section on "Identifying eigenvectors", it was not immediately clear where the vector (1, 1, -1) came from. It turned out it was the solution for  (A - \lambda I) x = 0 , and this turned out to be pretty easy. But it would be nice to explicitly state where to find the "sets of simultaneous linear equations" to be solved.
WillWare 23:14, 25 August 2005 (UTC)
thread_title:  Where eigenvectors come from 

message: I have done a lot of work with respect to the intro and definition.  I think some of the things I wrote are not optimized and are redundant with infos somewhere else on the page.  Now I am tired and would like to see other authors contribute to a better structured article. 131.220.68.177 09:31, 8 September 2005 (UTC)
thread_title:  Improving further 

message: There's a recently spawned article Symbolic computation of matrix eigenvalues which I don't much like; it seems to be a spinoff of this project, maybe? It needs attention or cleanup or something.  linas 01:01, 14 September 2005 (UTC)
replies: text: Yes indeed.  This info was previously within this article.  I put it in a separate article because I thought this info maybe interesting for grad students fighting with matrix diagonalization but not for someone interested in the general topic of this article.  I think this is the same for eigenvalue algorithm.  Maybe the good idea would be to merge Symbolic computation of matrix eigenvalues with eigenvalue algorithm. Vb 10:13, 14 September 2005 (UTC)
thread_title:  a duplicate/related article 

message: I think now the article is getting quite good.  However I don't think this is enough for getting featured.  I think the following points have to be addressed before submitting for peer review:
# Copy edit : I think the English is not the best.  I am no native speaker and I cannot manage it.  That's the reason why I put the copy-edit flag on the page.
# Too technical : Some info on the page are too technical.  In particular many properties are stated without explaining the reader why they are important and in which context they are used.  In particular, the sections Decomposition theorem, Other theorems, Conjugate eigenvector, Generalized eigenvalue problem and Eigenvalues of a matrix with entries from a ring should be re-written to be more accessible to a broader audience.
I think much of the info here is good and interesting but sound a bit too much like a grad book for mathematicians.  Vb 16:16, 21 September 2005 (UTC)
thread_title:  Featured article? 

message: The following passage occurs as an explanatory example:
 If one considers the transformation of the rope as time passes, its eigenvectors (or eigenfunctions if one assumes the rope is a continuous medium) are its standing waves well known to musicians, who label them as various notes. The standing waves correspond to particular oscillations of the rope such that the shape of the rope is scaled by a factor (the eigenvalue) as the time evolves.
A string of random length stretched at random tension can produce a sound that corresponds to a particular frequency. But depending on various factors the string can perform a simple vibration in which the entire length of the string moves in the same direction at the same time, or it can produce more complex vibrations (like the one illustrated in the movie). The musical qualities of these various possible variations are emphasized in the reader's mind by the mention of musicians because a pure sine wave sound vibration is not musically beautiful. Not only will the reader's mind potentially be sidetracked by that line of thought, but it is not exactly true that a musician will label any frequency as a "note" -- particularly if the musician has perfect pitch and the frequency being prduced is somewhere midway between a A and an A flat.  I don't want to change this passage without being aware of what the original writer was trying to accomplish. P0M 05:37, 23 September 2005 (UTC)
replies: text: I am no musician.  This example I wrote is a typical physicist's example which has not much to do with music.  Here the objective was only to provide an example of eigenvector in an infinite dimensional space which could be understood by nonspecialists.  If someone is able to make a better link with music, I would be very happy if he could do so.  I would learn something!  However I think it is important to keep in mind that one should not go into deep details here.  This is not the place to present the whole theory of vibrating strings. Vb07:51, 23 September 2005 (UTC)

text: Yes that's true I had also higher harmonics in mind.  However, I believe it is not worth to tell more than a line about it or maybe just a link.  What you wrote is from my point of view well done.  I would replace "the things that," by "which" but my English is for sure not as good as yours.  My problem is also that I am a bit afraid of telling nonsense about music:  I have no idea about  Martin and Sears guitar.  But if you do why not making a footnote about such details: they are for sure interesting (at least to me) even if they would disturb a bit from the main topic of this article. Vb09:04, 24 September 2005 (UTC)
thread_title:  Problem with one of the examples 

message: I am very sorry but I don't see why all that info about involutions and projections belong here.  OK they have eigenvalue -1, 0 or 1.  And?... Do you really think it must be put on the same level as Spectral theorem or Applications?  I believe not.  If nobody provides here any convincing argument, I'll remove this section quite soon. Vb 17:57, 10 October 2005 (UTC)
thread_title:  Involutions and projections 

message: "Eigenvalue, eigenvector, and eigenspace" has been moved to "Eigenvalue, eigenvector and eigenspace".  I don't understand why.  I am not a native speaker but I know that BibTeX use this rule for the coma when citing a paper with more than one author and that the editors of Phys. rev. A always use that rule for a list of more than two items.  Though this change may be correct I believe it was not necessary.  This change has some influence because many links to this page are getting now grandsons of the article instead of sons and this has influence!  Could someone tell me whether one should keep the current spelling.  If noobdy can I will make the mv back.  Vb 09:00, 11 October 2005 (UTC)
replies: text: Why don't you ask User:MarSch who did the move (if I read the history correctly)? The second comma in "Eigenvalue, eigenvector, and eigenspace" is a serial comma, and, as that article says, opinion is divided on whether one needs it or not (I am not a native speaker either). -- Jitse Niesen (talk) 12:56, 11 October 2005 (UTC)
replies: text: Thank you Jiste for your explanation.  I understood something: English is not that easy! :-)  I don't do anything and let the native speakers quarrel about that! Vb 09:55, 12 October 2005 (UTC)
thread_title:  change of title 

message: But what does that make the blue vector?
:the blue vector is not an eigenvector Vb 09:54, 14 October 2005 (UTC)
thread_title:  Mona Lisa 

message: Could the mathematical and physical sections be a little more distinct? And also the linear/nonlinear. Much of the things about the linear case don't hold when you move to nonlinear differential operators, e.g. and it seems ridiculous to worry about qualifying everything. While technically the linear case is just a "special case" of the nonlinear case, this is a bit like saying analytic functions are a "special case" of R^2-R^2 real-differentiable functions, so why waste time on them. Most people will mean the linear case when they think of "eigenvector", and there is so much you can say in just the mathematical context of vector spaces and linear transformations. 216.167.142.30 20:40, 23 October 2005 (UTC)
replies: text: I don't understand your point.  Many applications of the concept are non linear and the concept of eigenvectors can easily be introduced to lay persons in the most complex context.  I believe the choice which has been made is more pedagogic.  Many who know what an eigenvector is think it appears in linear algebra only (I was thinking this after my first lecture in algebra): this is untrue and I think this is one of the reason why the article was recognized as a featured article.  You are speaking about a clearer distinction of physics and math.  Physics appear in the examples and the applications only.  So why do you bother? Vb 20:51, 23 October 2005 (UTC)
replies: text: Well, I have a ph.d. in math and had never heard of "nonlinear eigenvalues", so don't feel bad. My point is, when the article attempts to present a formal definition of things. It's one thing to introduce a touchy-feely idea of the concept to a general audience; it's another to try to introduce a formal definition later. My gripe is that the general intuitive concept is introduced as being inclusive of both linear and nonlinear, but then when the formal definition comes around, the linear definition is given, and then later on down, it's mentioned "oh, yes, it can be nonlinear" but no definition of what this might mean is given. So, as it stands, as a mathematician, I have been made aware of the fact that such a thing as "nonlinear eigenvalue problems" exist, but I have no idea what they are or what they mean. Does this makes sense? 216.167.142.30 21:02, 23 October 2005 (UTC)

text: I'm beginning to wonder if the article isn't becoming too technical altogether. Compare to fractal. At that article, after a great deal of work and compromise, the article settled into a general readable form for non-math or non-science types, but without avoiding or sidestepping important issues relating to math or science. Most of the technical details about theory and applications got shuttled off to other articles (e.g. those defining dimension, relationship to chaos theory and dynamical systems, applications to science, etc, etc) The current article for EEE seems to start off attempting to emulate the fractal article, but then quickly goes into a myriad of technical issues, even algorithm ones (how to compute complex eigenvalues), and the general big picture seems to be lost. There is more than enough relatively non-technical stuff to talk about to fill out a whole article, without avoiding technical considerations, and there is certainly enough to fill out several technical articles (e.g. the theoretical presentation of EEE in the sole context of vector spaces and linear transformations; the practical and matix algorithms in finite-dimensional case; the application to linear differential equations; the nonlinear eigenvalue problem; applications to physics; etc, etc) ALL of these could form separate articles. As it is, each of these has a "stub" here, so the article reads like several stubs following each other. 216.167.142.30 20:58, 23 October 2005 (UTC)

text: I suppose my point is, the term "EEE" have probably achieved a certain cultural currency outside of mathematics proper, in the way that "fractal" has achieved a cultural currency outside of mathematics proper. So, in these cases, the main article for such a term (such as "fractal") will have to address the large non-math audience, while other articles address other things. This article will have to address the large non-math audience, but as it is, there is no article set up anywhere for writing in a style for a mathematical audience. At the least, the article starts talking to one audience, and quickly starts talking to another, and ends up succeeding with neither. 216.167.142.30 21:06, 23 October 2005 (UTC)

text: Vb, it's also not quite clear to me what is meant with nonlinear eigenvalues. I suppose that even if T is a nonlinear operator, you can define λ to be an eigenvalue whenever T(v) = λv. But can you explain in more detail why such a definition would be useful? -- Jitse Niesen (talk) 16:46, 25 October 2005 (UTC)
thread_title:  Separating different parts 

message: The article currently has a group of words that appears to be intended as a sentence:
Sometimes, when \mathcal{T} is a nonlinear transformation, or when it is not possible or difficult to provide a basis set. 
All that is there is a "when" clause.
Assuming that this is the desired clause and it's the end of the sentence that is missing, then it would be better to change "or difficult to" to "or is difficult to." P0M 06:21, 26 October 2005 (UTC)
thread_title:  Something missing 

message: I would prefer if it was horizontal in the left picture and then in the transformed picture it would be parallel to the top and bottom sides still. This would be slightly clearer since it is easier to see what exactly happened. --MarSch 09:51, 26 October 2005 (UTC)
thread_title:  blue arrow in mona lisa 

message: I think one of the best example of the use of eigenvectors and values in finding normal modes for complex oscillations.  Its a decently intuitive use of linear algebra.  What do you think?
:True.  Be bold : do it! Vb 09:34, 1 November 2005 (UTC)
thread_title:  perhaps include normal modes 

message: The use of bolding and script is non-useful. Vectors are not bolded, except by some physicists sometimes, but even they tire of it. Furthermore the script T for a simple transformation is odd. Notation should be as simple as possible without frills that don't mean anything. What do you think? --MarSch 11:33, 1 November 2005 (UTC)
replies: text: As far as I know (I am physicist) bolded vectors v are still standard even if I prefer the notation with an arrow \vec{v}.   Here in this article, the notation v has been used for the vectors and v for their representation as vertical arrays in a particular basis set (an alternative notation used is \underline{v}).  The same has been done for the transformations \mathcal{T} opposed to their matrix representations T in a particular basis set (an alternative notation used is \underline{\underline{T}}).  I am well aware one can forget about this nuance.  I am however persuaded this is not pedagogic.  In particular I think it is important to use distinct notations for the general definition \mathcal{T}(\mathbf{v}_\lambda)=\lambda\mathbf{v}_\lambda and its representation in the linear and finite dimensional case T(v_\lambda)=\lambda v_\lambda.  Both objects are really different \mathcal{T} is a transformation, a function or a functional and T is an two-dimensional array of scalars.  Using a common notation for both would be IMO misleading. Vb 12:43, 1 November 2005 (UTC)

text: There is no reason to go out of our way and use calligraphic T here. The transformation and the matrix representation are equivalent (in certain basic circumstances), and the noncalligraphic T is only used once as to make a notational distinction rather useless. I am going to change these back when I have the time. Dysprosia 12:31, 16 April 2006 (UTC)
thread_title:  notation 

message: I find the bottom two footnotes should be in the text and not footnotes at all.--MarSch 11:35, 1 November 2005 (UTC)
replies: text: There are three footnotes.  The first one is a citation which has nothing to do with the article but only with the source of one picture.  The second is there to precise which kind of transformation we are intending to speak about.  This precision is very technical and IMO does not belong to the lead.  The third one is about the fact that zero vectors are not considered as eigenvectors.  These both very technical statements must be said but not so early in the article.  Historically it replaced parentheses because some reviewers in the FAC pointed out that these comments were diminishing the text flow. Vb 12:58, 1 November 2005 (UTC)
replies: text: In parenthesis they would also be clumsy, but at least they would be in the right place. I think the fact that the transformation is from a vector space to itself is pretty essential to understanding the concept of an eigenvector. A transformation from one space to another would land you non-sense. The non-zero claim should be explained. Since the zero vector is always an eigenvector of a linear transformation it is excluded explicitly. The flow is now very bad, since when you see the note, you have to click it to see what it says and then back again, only to discover that the note should really be in the text. --MarSch 13:20, 1 November 2005 (UTC)
replies: text: Well it breaks the flow.  You are right.  But the ones who look at these are usually assumed to be expert and therefore able to cope with is inconvenient.  But specifying things which are relevant to experts only in the body of the text is also breaking the flow.  I think the Mona Lisa pictures shows which kind of transformations we are speaking about.  Moreover in the non linear case zero vectors could be also eigenvectors even if one does not really need to mention this in this article. -- Vb 14:03, 1 November 2005 (UTC)
replies: text: Aah, the nonlinear zero eigenvector izzz convincing. Thanks. I'm sticking in the other though. --MarSch 13:02, 3 November 2005 (UTC)
thread_title:  footnotes 

message: Is the relation of the German word eigen to Old English really in the scope of this article? To me it's distracting and just a piece of trivia in this context.
--doerfler 15:35, 10 November 2005 (UTC)
replies: text: I don't mind if you remove this. Vb 16:07, 10 November 2005 (UTC)
thread_title:  Old English? 

message: The reasons for my reverts are as follows:
# Only square matrices have eigenvalues.
# The operator norm is not the least upper bound for the moduli of its eigenvalues; the spectral radius is the least upper bound. 
# Even for normal matrices, different vector norms give different operator norms. For instance, the matrix
:: R = \begin{bmatrix} \frac12\sqrt{2} & \frac12\sqrt{2} \\ -\frac12\sqrt{2} & \frac12\sqrt{2} \end{bmatrix} 
:(rotation over 45 degrees) is orthogonal, hence normal. Its spectral norm is \|R\|_2 = 1 and its maximum column sum norm is \|R\|_1 = \sqrt{2}.
-- Jitse Niesen (talk) 02:13, 5 April 2006 (UTC)
replies: text: Shoot, you're right, sorry Jitse. Something might be wrong with your notation there tho. The operator norm induced by the L^{\infty} norm of that matrix is indeed sqrt(2). Thanks. Mct mht 02:33, 5 April 2006 (UTC)
thread_title:  Connection between spectral radius and matrix norm for normal matrices 

message: Not a bad article, but I am surprised that Wikipedia calls it "a featured article, (...) one of the best articles produced by the Wikipedia community." Just looking at it for one minute one can spot a serious error: The statement that "a matrix is diagonalizable if and only if it is normal" is absolutely false. I did not read the rest of the article after a mistake of this caliber. Grand Vizier 20:46, 18 April 2006 (UTC)
replies: text: You're right, it should be "a matrix is unitarily diagonalizable if and only if it is normal". Thanks for mentioning this. However, I hope that in the future, you will be more constructive and correct it yourself; otherwise it might appear that you just want to show off how smart you are. -- Jitse Niesen (talk) 03:33, 19 April 2006 (UTC)

text: "Diagonalizable" has a specific meaning in matrix theory, which is explained in diagonalizable matrix. With this meaning, not every matrix is diagonalizable. As it says in the article, "a matrix is diagonalizable if and only if the algebraic and geometric multiplicities coincide for all its eigenvalues". -- Jitse Niesen (talk) 05:28, 20 April 2006 (UTC)
thread_title:  Normal matrix 

message: It looks like an error got inserted. Take for example matrices A=\begin{bmatrix}1 & 0 \\ 0 & -1 \end{bmatrix} and B=\begin{bmatrix}0 & 1 \\ 1 & 0 \end{bmatrix}, then B^{-1}=\begin{bmatrix}0 & 1 \\ 1 & 0 \end{bmatrix}, B^{-1}A=\begin{bmatrix}0 & -1 \\ 1 & 0 \end{bmatrix}, and the solutions to the generalized eigenvalue problem Av=\lambda Bv are the pure imaginary eigenvalues i and -i.
Therefore, there exists a pair of real symmetric matrices such that the solution of the corresponding eigenvalue problem are not real ... the contrary is stated on the page ... and I'm about to take it out ... Actually, what is stated here is true for each of the matrices independently and for their own simple eigenvalue problem, but the generalized eigenvalue problem it is not true.  tradora 00:51, 9 May 2006 (UTC)
thread_title:  Error in the generalized eigenvalue problem 

message: The current version of the article claims that the eigenvalue for a standing-wave problem is the amplitude.  This is an absurd and totally nonstandard way to formulate the problem, if it even can be made to make sense at all.  For vibrational modes, one writes down the equation for the acceleration in the form:
:\frac{\partial^2 \mathbf{x}}{\partial t^2} = - \hat{A} \mathbf{x}
where \mathbf{x} is the amplitude and -\hat{A} is the operator giving the acceleration = force/mass (i.e. from the coupled force equations).  In a linear, lossless system, this is a real-symmetric (Hermitian) linear operator.  (More generally, e.g. if the density is not constant, one writes it as a generalized Hermitian eigenproblem.)  Ordinarily, \hat{A} is positive semi-definite too.
To get the normal modes, one writes \mathbf{x} in the form of a harmonic mode: \mathbf{x} = \mathbf{v}\exp(-i\omega t) for a frequency ω.  (Of course, the physical solution is the real part of this.)  Then one obtains the eigenequation:
:\hat{A} \mathbf{v} = \omega^2 \mathbf{v}
and so one obtains the frequency from the eigenvalue.  Since \hat{A} is Hermitian and positive semi-definite, the frequencies are real, corresponding to oscillating modes (as opposed to decaying/growing modes for complex ω).  Furthermore, because \hat{A} is Hermitian the eigenvectors are orthogonal (hence, the normal modes), and form a complete basis (at least, for a reasonable physical \hat{A}).   Other nice properties follow, too (e.g. the eigenvalues are discrete in an infinite-dimensional problem with compact support.)
If losses (damping) are included, then \hat{A} becomes non-Hermitian, leading to complex ω that give exponential decay.
Notice that the amplitude per se is totally arbitrary, as usual for eigenvalue problems.  One can scale \mathbf{v} by any constant and still have the same normal mode.
User:Jitse Niesen claimed at Wikipedia:Featured article review/Eigenvalue, eigenvector and eigenspace that the article was talking about the eigenvalues of the "time-evolution operator" \exp(-\hat{A}t).  The first problem with this is that \exp(-\hat{A}t) is not the time-evolution operator, because this is a second-order problem and is not determined by the initial value \mathbf{x}(0).  You could convert it to a set of first-order equations of twice the size, but then your eigenvector is not the mode shape any more, it is the mode shape along with the velocity profile.  Even then, the eigenvalue is only a factor multiplying the amplitude, since the absolute amplitude is still arbitrary.  Anyway, it's a perverse approach; the whole point of working with harmonic modes is to eliminate t in favor of ω.
Note that the discussion above is totally general and applies to all sorts of oscillation problems, from discrete oscillators (e.g. coupled pendulums) to vibrating strings, to acoustic waves in solids, to optical resonators.
As normal modes of oscillating systems, from jumpropes to drumheads, are probably the most familiar example of eigenproblems to most people, and in particular illustrate the important case of Hermitian eigenproblems, this subject deserves to be treated properly. (I'm not saying that the initial example needs the level of detail above; it can just be a brief summary, with more detail at the end, perhaps for a specific case.  But it should summarize the right approach in any case.) 
—Steven G. Johnson 15:29, 24 August 2006 (UTC)
replies: text: I readily agree that my comment at FAR was sloppy, and I'm glad you worked out what I had in mind. Originally, I thought that the "perverse" approach was not a bad idea to explain the concept of eigenvalues/functions, but I now think that it's too confusing for those that have already seen the standard approach. -- Jitse Niesen (talk) 12:14, 25 August 2006 (UTC)
replies: text: Thanks for your thoughtful response.  Let me point out another problem with saying that the "amplitude" is the eigenvalue.  Knowing the amplitude at any given time is not enough to know the behavior or frequency.  You need to know the amplitude for at least two times.   By simply calling the eigenvalue "the" amplitude, you've underspecified the result.  —Steven G. Johnson 15:23, 25 August 2006 (UTC)
thread_title:  Vibrational modes - erroneous treatment in the article 

message: The Standing wave example of eigen values isn't very clear. It is just stated that the standing waves are the eigen values. Why is this the case? How do they fit the definition / satisfy the criterion for being an eigen value? —The preceding unsigned comment was added by 67.80.149.169 (talk • contribs) .
:Actually, the wave is the eigenfunction, not the eigenvalue. Did you not see this part?:
The standing waves correspond to particular oscillations of the rope such that the shape of the rope is scaled by a factor (the eigenvalue) as time passes. Each component of the vector associated with the rope is multiplied by this time-dependent factor. This factor, the eigenvalue, oscillates as times goes by.
:I don't see any way to improve that. —Keenan Pepper 03:10, 7 September 2006 (UTC)
replies: text: Except that calling the amplitude the "time-dependent eigenvalue" is horribly misleading and bears little relation to how this problem is actually studied, as I explained above.  Sigh.  —Steven G. Johnson 20:54, 7 September 2006 (UTC)

text: As I explained above, the shape of the rope isn't the eigenvector of this operator.  Because it is second order in time, you would need the shape of the rope plus the velocity profile.  And there are other problems as well, as I explained above.  —Steven G. Johnson 12:48, 8 September 2006 (UTC)
replies: text: Sorry for the delay. Perhaps then the statment "the standing was is the egienfunction" could be elobrated on a little more. I'm having trouble visualising what that means, and how the notition of a vector whos direction reamains unchanged by the transformation applies to this example. My apologies for the confusion. --165.230.132.126 23:35, 11 October 2006 (UTC)
thread_title:  Request for Clarificaion of the Standing wave example for eigen values 

message: When are eigenvectors orthogonal? Symmetric matrix says "Another way of stating the spectral theorem is that the eigenvectors of a symmetric matrix are orthogonal." So A is symmetric => A has orthogonal eigenvectors, but does that relation go both ways? If not, is there some non-trivial property of A such that A has orthogonal eigenvectors iff ___? —Ben FrantzDale 20:36, 7 September 2006 (UTC)
replies: text: eigenvectors corresponding to distinct eigenvalues are orthogonal iff the matrix is normal. in general, eigenvectors corresponding to distinct eigevalues are linear independent. Mct mht 22:31, 7 September 2006 (UTC)

text: btw, "symmetric matrix" in the above quote should mean symmetric matrix with real entries. Mct mht 22:34, 7 September 2006 (UTC)
replies: text:  Thanks. Normal matrix does say this; I'm updating other places which should refer to it such as symmetric matrix, spectral theorem, and eigenvector. —Ben FrantzDale 13:30, 8 September 2006 (UTC)
thread_title:  Orthogonality 

message: i think the geographic pole is different from the magnetic pole...-- Boggie 09:57, 21 November 2006 (UTC)
replies: text: You are correct that the geographic south pole is different from the magnetic south pole. However, it is the geographic pole that we need here, isn't it? -- Jitse Niesen (talk) 11:33, 21 November 2006 (UTC)
replies: text: yes, u r right, i just mixed them up;-) -- Boggie 16:51, 22 November 2006 (UTC)
thread_title:  An arrow from the center of the Earth to the Geographic South Pole would be an eigenvector of this transformation 

message: Some confusion about the existence of eigenvectors, eigenvalues and eigenspaces for transformations can possibly arise when reading this article (as it did when I asked a question at the mathematics reference desk, after reading the article half-asleep - my bad!).
My question here would be, what could be done to correct it. The existence of eigenvectors is a property of the transformation not a property of the eigenvector itself, making it unclear which of the pages needs revision. Does anybody have any ideas?
Also this article seems a bit long, I suggest placing the applications in a separate article (as are some of the theorems in the earlier sections).
**Putting the application elsewhere would be very damaging.  Math without applications is useless, i.e. not interesting!
:: I disagree.  All truly interesting math is useless.  ;)  —Preceding unsigned comment added by Crito2161 (talk • contribs) 03:02, 8 March 2008 (UTC)
thread_title:  Existence of Eigenvectors, eigenvalues and eigenspaces 

message: Figure 3 in the article has a nice picture of the absorption spectrum of chlorine. The caption says that the sharp lines correspond to the discrete spectrum and that the rest is due to the continuous spectrum. Could someone explain some things?:
* What operator it is that this is the spectrum of?
** The Hamiltonian of the Chlorine atom
* If the discrete spectrum corresponds to eigenvalues, are those eigenvalues shown on the x or y axis of the graph?
** The x-values, i.e. the energies of the tomic eigenstates.
* What would the corresponding eigenfunctions be for elements of the spectrum?
** Those are multielectronic wavefunctions which are often approximated (better expanded in a basis of) Slater determinants.
Thanks. —Ben FrantzDale 15:35, 20 December 2006 (UTC)
thread_title:  Continuous spectrum 

message: I am so sad this article is not featured anymore.  I did the job to get it featured.  I have no time to improve it now.  I wish someone could do the job.  The article is not paedagogic anymore.  Such a pain!  Vb 09:38, 28 December 2006 (UTC)
thread_title:  Going back to featured status! 

message: Excerpt from the text:
"Having found an eigenvalue, we can solve for the space of eigenvectors by finding the nullspace of A − (1)I = 0."
Shouldn't this be ...finding the nullspace of A - λI = 0?
216.64.121.34 04:06, 8 February 2007 (UTC)
thread_title:  minor typo? 

message: It says in the article: A vector function A is linear if it has the following two properties:
    Additivity \ A(\mathbf{x}+\mathbf{y})=A(\mathbf{x})+A(\mathbf{y})
    Homogeneity \ A(\alpha \mathbf{x})=\alpha A(\mathbf{x})
where x and y are any two vectors of the vector space L and α is any real number.[10] 
doesn't alpha not necessarily have to be a real number, just a scalar from the base field of the vector space? LkNsngth (talk) 23:43, 11 April 2008 (UTC)
replies: text: Thanks for the good and concrete question. I am checking it and will try to answer ASAP. --Lantonov (talk) 06:14, 14 April 2008 (UTC)

text: Seems that you are right. Korn and Korn (section 14.3-1) say that α is a scalar. I could not find said there that α is necessarily real. I will look through some more books for this definition and change it accorsingly. --Lantonov (talk) 10:24, 14 April 2008 (UTC)

text: On the other hand, Akivis, where I took this definition from, explicitly states that α is a real number. --Lantonov (talk) 15:11, 14 April 2008 (UTC)

text: Mathworld says that α is any scalar. --Lantonov (talk) 15:14, 14 April 2008 (UTC)

text: Shilov - any α --Lantonov (talk) 15:17, 14 April 2008 (UTC)

text: Strang - all numbers ... --Lantonov (talk) 15:24, 14 April 2008 (UTC)

text: Linear transformation - any scalar α --Lantonov (talk) 15:28, 14 April 2008 (UTC)

text: Kuttler - a and b scalars --Lantonov (talk) 15:34, 14 April 2008 (UTC)

text: Beezer - \alpha \in \mathbb{C} --Lantonov (talk) 15:37, 14 April 2008 (UTC)

text: Now the last one really convinced me. Changing ... --Lantonov (talk) 15:37, 14 April 2008 (UTC)
thread_title:  small technical question 

message: Where to put this in the article? Something to the effect: A right eigenvector v corresponding to eigenvalue λ satisfies the equation Av = \lambda v for matrix A. Contrast this with the concept of left eigenvector u which satisfies the equation u^T A = \lambda u^T. ?? --HappyCamper 04:39, 15 March 2007 (UTC)
replies: text: It's briefly mentioned in the section "Entries from a ring". Are you saying that you want to have it somewhere else? -- Jitse Niesen (talk) 06:16, 15 March 2007 (UTC)
replies: text: No no...just that I didn't see the note on the first reading. I think the little note is plenty for the article already. --HappyCamper 23:36, 16 March 2007 (UTC)
thread_title:  Left and right eigenvectors 

message: As far as mathematics articles goes, this is certainly one of the best.  To the general reader, however, it remains quite arcane, but I'm not too sure how it could be improved, or whether a little dumbing down would be either possible or desirable.
To begin with the earliest eigenvalue problem, I understand, was that of the frequency of a stretched string.  Pythagoras didn't solve it, but observed the simple relationship between string lengths, which formed harmonies, and also that the frequency depended on the parameters (string tension and length) and not on the excitation.
Using the examples of the Earth's axis and Schrodinger's equation, endows the article with a certain ivory tower feel, as if this subject were only relevant to the 'difficult' problems which 'clever' people preoccupy themselves with.  Maybe the intention is to impress, rather than inform, but I doubt it, the attempt to communicate with joe public appears sincere.
However, eigenvalue problems are to be found everywhere, they are not obscure mathematical concepts applicable to arcane problems, they are to be found in stock market crashes and population dynamics.  Even the 12inch ruler on the desk, with its buckling load, illustrates an eigenvalue problem.  They are to be found in every musical instrument from tin whistle to church organ.  The swinging of a pendulum, the stability of a bullet, the automatic pilot of an aircraft. Such questions as why does a slender ship turn turtle?  Why does a train sway at high speed?  Why is the boy racer's dream car a 'whiteknuckles special'?  Why do arrows have flights?  Why doesn't a spinning top fall over? All of these are much more immediate and relevant to the lay person, who is surely the one we seek to educate.
Some articles do indeed fall into the category of 'ignotum per ignotius'; an explanation which is more obscure that the thing it purports to explain.  This is not one of them.Gordon Vigurs 07:45, 10 April 2007 (UTC)
replies: text: The introduction seems particularly opaque.  I'm going to try to say something that a non-mathematician could understand. Rick Norwood 14:29, 27 April 2007 (UTC)
replies: text: Finally, some Wikipedia mathematicians that are actually concerned about the general reading public. :) 70.187.40.16 (talk) 04:54, 29 June 2008 (UTC)
thread_title:  Comment on level of presentation 

message: The picture of the Mona Lisa appears only have been turned a little, and it is not clear that the blue arrow has changed direction.  A more extreme distortion, where the blue arrow was clearly pointing in a different direction, would make the point better.  If the length of the red arrow were also changed, that would also be good. Rick Norwood 15:02, 27 April 2007 (UTC)
replies: text: You misunderstand.  It hasn't been turned at all.  It's been sheared.  There's a link in the caption.  The change of the arrow direction is quite clear, and the length of the red arrow does not change in a shear. -- GWO
I understand that it has been sheared.  That's why I said "appears" instead of "is".  But the average viewer is going to assume that it has been turned, because that's what it "looks like".  Also, the change in direction of the blue arrow is not clear, it is a few degrees at most.  Which is why a more extreme distortion would better convey the idea that the red arrow keeps its direction while the blue arrow changes direction. Rick Norwood 12:42, 28 April 2007 (UTC)
thread_title:  picture 

message: I don't think the current Mona Lisa illustration is as helpful as it could be. In very many applications, the operator is symmetric and so there is a set of n real orthogonal eigenvectors. In the case of an image transformation, that would be an orthotropic scaling, not necessarily axis-aligned. The current image shows shearing which has only one eigenvector and so isn't representative of most "typical" eigenvector problems. I propose replacing the sheared Mona Lisa with an orthotropically stretched Mona Lisa with two eigenvectors. —Ben FrantzDale 07:14, 29 April 2007 (UTC)
replies: text: Let me amend that. I think the example would do better with a Cartesian grid rather than a picture since a Cartesian grid has a clear origin and so is easier for novice to connect to linear algebra. Those new to linear algebra won't necessarily connect image transformation with matrix operations. —Ben FrantzDale 12:01, 25 May 2007 (UTC)
replies: text:  Perhaps a grid could be superimposed on the image? I think it is not such a bad idea to illustrate an "atypical" problem (e.g. to combat the misperception that there is always a basis of eigenvectors). On the other hand, I don't think it is a good lead image, and the text there should be used to annotate the copy of the image in the "Mona Lisa" example (which should be renamed to avoid OR). Something like the electron eigenstates image seems like a more appealing lead image to me. Geometry guy 16:54, 9 June 2007 (UTC)
thread_title:  Illustration 

message: The lead has been very much improved recently.  The definition of linear transformation is now very clear! Thanks to the editors! Vb 12:50, 30 April 2007 (UTC)
thread_title:  Great contribution 

message: The article summary repeatedly uses the serial comma for lists — it seems to me that the title is the only place without it.  Admittedly, I'd like to rename the article just for my love of commas; but, as a legitimate reason, shouldn't we rename it for consistency? ~ Booya Bazooka 15:43, 12 June 2007 (UTC)
replies: text: Support. —Ben FrantzDale 16:59, 4 November 2007 (UTC)

text: Wikipedia on the whole uses the serial comma reasonably consistently.  For this reason, and because the serial comma is used throughout the article, and is often needed for clarity, the title of the page, and any remaining outliers should be fixed, for consistency above all.--169.237.30.164 (talk) 11:05, 13 December 2007 (UTC)
thread_title:  Return the serial comma? 

message: The colors used to highlight the text in the first picture is rather misleading - blue is used for links usually while red is usually for broken links. I think just bolding them would be sufficient, the colors are perhaps unnecessary.--Freiddie 18:55, 4 August 2007 (UTC)
thread_title:  Misleading colors 

message: Just a quick note of thanks especially for the practical examples of eigenvectors/values such as the Mona Lisa picture and the description of the vectors of the earth and the rubber sheet stretching.  I find that having a number of different descriptions helps me generalise the concept, much more so than the mathematical treatment.
I certainly don't imply that the rigorous mathematical treatment should be removed, as this is also important, however I do feel that often mathematical symbology (while very concise and important) hides the meaning of what can actually be simple concepts (at least to the beginner).  Thanks to all for providing both practical and theoretical aspects in this article.
As a suggestion, it would also be very helpful to have some simple examples of why eigenvalues/eigenvectors are useful in some different problem domains.  For example, why would I want to find the eigenvectors in the first place?  How do they help us?
131.181.33.112 07:12, 4 September 2007 (UTC)
thread_title:  Note of thanks and a brief suggestion regarding need for eigenvalues/eigenvectors 

message: The example of how to find an eigenvalue value and eigenvector is far too simplistic. Steps are skipped or poorly explained so often that if the reader doesn't already have a good understanding of the material he will not be able to follow along. I think the whole thing needs to be redone starting with a more typical problem. One with two distinct eigenvalues and two distinct eigenvectors. Because the example has only one eigen vector it is misleading about what to expect when working with eigenvalues.Marcusyoder 05:35, 17 October 2007 (UTC)
thread_title:  Poor Example 

message: I removed the following paragraph from the Infinite-dimensional spaces section.
(I have also made visible a hidden note in the paragraph.)  I am not sure this fits with the rest of the section, and I have a hard time seeing what it is driving at.  Thoughts? --TeaDrinker 23:27, 10 November 2007 (UTC)
replies: text: I reintroduced the paragraph and tried and respond to your comments. Vb 15:24, 16 January 2008 (UTC)
thread_title:  Infinite dimensional spaces 

message: Redirects here, but was not mentioned in the text, so I put in a short definition.
The formal definition is:
an eigenvalue λ of A has algebraic multiplicity
a = a(\lambda) = a_A(\lambda) \in N, if \pi_A(z) = (z - \lambda)^a p(z),
where \pi_A(z) = \det(A - z I) is the characteristic polynomial and p(\lambda) \neq 0.
Thomas Nygreen (talk) 00:10, 6 December 2007 (UTC)
thread_title:  Algebraic multiplicity 

message: Factor_analysis,Principal_components_analysis,Eigenfaces,Tensor of inertia,Stress_tensor.
To me at least the first three are actually the exact same application, just with different types of data. Each of those three have a spread of data, looked at from it's center of mass, in the case of the tensor of inertia the data points are each differential element of mass. Then to find the principle axes, if you didn't know about the eigenvalue/eigenvector shortcut, you'd ask what direction of a best fit line would give you the least squared error/moment of inertia. The derivative of error with respect to direction will be zero at the local minima & maxima.
Once you have the components into the covariance matrix/tensor of inertia, you notice that stresses and shears behave exactly the same way as the variance and covariance components. I mean I think it's no accident that people use sigma as the symbol for stresses. In fact I've found it helpful to use Mohr's circle in those earlier cases of statistics and inertia.
This is the long way of saying that I don't think we should let terminology get in the way of the explanation. There seems to be a few people very interested in this article, so I won't just redo that section without your input.Sukisuki (talk) 14:58, 29 December 2007 (UTC)
replies: text: I agree with these comments.  I however think this section should not appear as one more mathematical concept but as example of practical application in utterly distinct domains.  The stress should be on the applications not on the theory behind.  This section must be appealing to the user of math not to the mathematicians! Vb 15:04, 16 January 2008 (UTC)
thread_title:  Applications 

message: Is it right, that P is never to be normed, but it's gonna be easier to calculate P-1, since then, if P=orthogonally => P^{-1}=P^T ?
On my calculator (TI-89), however, he norms P always, when searching the EigenVectors ... --Saippuakauppias ⇄ 09:14, 16 January 2008 (UTC)
thread_title:  When is P, the matrix of the Eigenvectors, to be normed? 

message: I feel that this article should reflect the importance of the eigenvalue problem to solving the frequency and damping ratios of reciprocating systems. I think this should be in the 'applications' section.
130.88.114.226 (talk) 13:03, 23 January 2008 (UTC)
thread_title:  Vibrating System 

message: I think the definite point of turnoff are the definitions. Do we have to make so much definitions on such small space, even of things we haven't mentioned so far? MathStyle says that it is good to have 2 definitions for a thing: 1 formal, and 1 informal, and many explanations around each definition. Changing the structure of the article to introduce everything one in a time will make it more accessible. Lead section and history are tolerable. My suggestion for structure after them is:
#Introduce eigenvector by easy examples
#Make a formal definition
#Make informal definition or, alternatively
#Explain the formal definition with more examples
#Do 1-4 for eigenvalue
#Do 1-4 for eigenspace
#Introdice and/or define the characteristic equation
#Introduce and define eigenmatrix
#Finding eigens for the 2-case
#Finding eigens for the 3-case
#Introduce complications: zero eigenvectors and equal eigenvalues
#Calculations in the complicated cases: complex matrices
#Linking those concepts with matrix orthogonalization as it is where most applications come from
#Applications from simple to complex, starting with geometry on a plane, space, n-dim, then electrodynamics, relativity, etc.
Although I like the image of Mona Lisa, having it twice is too much. Also, a heading "Mona Lisa" on a section, which is an example of finding specific eigenvalues is just a little bit misleading. --Lantonov (talk) 13:33, 11 February 2008 (UTC)
thread_title:  Structure 

message: At the top of this article there is a notice saying that "This article or section is in need of attention from an expert on the subject." In view of the very thorough revision by Lantonov over the last four weeks, I think that this notice is no longer necessary. Also this talk page does not indicate what is supposedly wrong with the article. Therefore I have decided to remove the notice, with thanks to Lantonov for much hard work well done. If anyone else wants to replace the notice, please indicate what you think should still be changed.   Dirac66 (talk) 17:06, 11 March 2008 (UTC)
replies: text: Thanks, Dirac. I am not yet half done. In the moment I am making figures with Inkscape to illustrate shear and rotation. Good program but not easy to work with. --Lantonov (talk) 17:14, 11 March 2008 (UTC)
thread_title:  Request for attention from an expert no longer needed. 

message: These are point by point answers to comments above:
*The eigenvectors of rotation are \begin{pmatrix}1\\i\end{pmatrix} and \begin{pmatrix}1\\-i\end{pmatrix}. True.
*Now, I don't know how many times I need to keep  saying it but THESE VECTORS ARE NOT THE COMPLEX SCALARS 1+i and 1-i.  OK?  Can you agree with that? True. Agreed.
*They ARE NOT ELEMENTS OF THE SET OF COMPLEX NUMBERS in exactly the same way that the vector \begin{pmatrix}5\\7\end{pmatrix} is NOT AND ELEMENT OF THE SET OF REAL NUMBERS. True.
*A vector of real numbers is not itself a real number. True.
*It is not an elemnt of the real line R, but it is in R^2 (the Cartesian product of two copies of the real line). False. A vector (directed intercept) can be an element of the real line. Namely, if we take a point with coordinates (a, 0) where a is real number as point 1, and a point with coordinates (b, 0) where b is real number as point 2 and build a vector between point 1 and point 2, this vector  \begin{pmatrix}a+b\\0\end{pmatrix} will be an element of the real line because it lies in the real line R. This is so because in the underlying field of scalars of R^2 there are scalars (a,0). Vectors on the real line are subset of vectors in R^2. Furthermore, the real line can be a one-dimensional vector space because it is closed under addition and multiplication by real scalar. Check: \begin{pmatrix}a\\0\end{pmatrix} + \begin{pmatrix}b\\0\end{pmatrix}= \begin{pmatrix}a+b\\0\end{pmatrix} and a\begin{pmatrix}b\\0\end{pmatrix} =  \begin{pmatrix}ab\\0\end{pmatrix} and the resulting vectors are in R.
*A vector of complex numbers is not itself a complex number. True. However, we must be careful what we understand by a "vector of complex numbers". In the notation, \begin{pmatrix}z_1\\z_2\end{pmatrix}=z_1\times\begin{pmatrix}1\\0\end{pmatrix} + z_2\times\begin{pmatrix}0\\1\end{pmatrix} where z_1 and z_2 is a vector of complex numbers (scalars) and where \times is supposed to mean multiplication of vector by scalar and not vector product, a vector for which z_1 is real and z_2 is complex is also a complex vector. Note that the 2 eigenvectors of rotation are exactly of this type: z_1 is real, and z_2 is complex. More specifically, z_1 = 1 and z_2 = i or z_2 = -i. 
*It is not an element of the complex plane C, but it is in C^2 (the Cartesian product of two copies of the complex plane). False, if stated in this way. True, if it is supposed to mean that the complex plane is not a vector space of complex vectors. A complex vector can be an element of a complex plane C for a similar reason as the above. Such plane is defined if one takes 2 lines (one-dimensional spaces) such that one line contains the whole set of real numbers (line X) while the other contains the whole set of purely imaginary numbers (line Y). Then make the Cartesian product  X \times Y to obtain a set of ordered pairs (x, iy). If these elements are points, they will form a two dimensional space, in which each point has coordinates (x, iy) where x and y are  real. If we take in this plane a point with coordinates (x1, iy1) as point 1 and a point with coordinates (x2, iy2) as point 2 and direct an intercept between these two points in such a way that point 1 is origin, and point 2 is end, we will obtain (build) a vector in the complex plane which as a column matrix is \begin{pmatrix}x_1 + x_2\\iy_1 + iy_2\end{pmatrix}. This vector will be element of the complex plane because it is built in this plane as vector and lies in this plane. Whether such plane is a vector space is another matter. It is NOT a vector space because it is not closed under multiplication by a complex number and such numbers are in the set of the underlying field of scalars. 
*You cannot plot the vector \begin{pmatrix}1\\i\end{pmatrix} on the complex plane, because it is not a complex number.  If you try, and you have, you will fail, and you did. False. I can plot the vector \begin{pmatrix}1\\i\end{pmatrix} on the complex plane. For this I must use the standard textbook definition of complex plane: it is a plane of 2 real axes (rays, directed lines, composed of real points only) that are geometric sites of points corresponding to the real R(z) and imaginary Im(z) parts of the complex number z = x + iy. Axis X contains the points for x (all real numbers), and axis Y contains the points for y (all real numbers). X and Y are orthogonal. The Cartesian product X \times Y gives the set of ordered pairs (x,y). In order for this set to be a ring with a unit, we define two operations: addition (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2) and multiplication (x1, y1) . (x2, y2) = (x1 . x2 + y1 .  y2, x1 . y2 + x2 . y1). These operations satisfy the conditions of associativity, distributivity, and multiplicative closure. Also define the unit as (1,1) and zero as (0,0). We do not need to define inverse element for the present purpose (do not need a field) because a ring with unit is all that is needed to define a vector space. Those elements (x,y) defined in such way are the scalars. We define vectors as \begin{pmatrix}(x,0)\\(0,y)\end{pmatrix}. To define vector space over the ring of scalars, we define the following two operations: addition of vectors \begin{pmatrix}(x_1,0)\\(0,y_1)\end{pmatrix}+\begin{pmatrix}(x_2,0)\\(0,y_2)\end{pmatrix}=\begin{pmatrix}(x_1,0)+(x_2,0)\\(0,y_1)+(0,y_2)\end{pmatrix} and multiplication of vector by scalar (x_1,y_1)\cdot\begin{pmatrix}(x_2,0)\\(0,y_2)\end{pmatrix}=\begin{pmatrix}(x_1 \cdot x_2,0)\\(0,y_1 \cdot y_2)\end{pmatrix}. These two operations should satisfy all necessary conditions for vector space: commutative addition, closure upon multiplication by scalar, associativity for multiplication by scalar, distibutive laws and so on (too lazy to check this). To cut it short, now we have a vector space which is one dimensional relative to the scalars (x,y) but two dimensional relative to the real numbers x and y (which are scalars (x,0) and (0,y) on the X and Y axes). I need again to stress: this is a real vector space and the vectors and scalars in it are real. However, vectors \begin{pmatrix}(x,0)\\(0,y)\end{pmatrix} have all the properites of complex vectors \begin{pmatrix}x+0 \cdot iy\\0 \cdot x+iy\end{pmatrix} and scalars (x,y) can be added and multiplied as complex scalars x + iy. Other properties of complex numbers can be easily obtained if we suitably define a field of (x,y) to have division and other operations with complex numbers. Now we have all that is needed to plot the eigenvalues x + iy and eigenvectors \begin{pmatrix}x+0 \cdot iy\\0 \cdot x+iy\end{pmatrix} on the complex plane. I did it, and I succeeded. Note that we need only 2 real numbers x and y and this helps to plot eigenvalues and eigenvectors in two dimensions. 
I understand what all those comments are driving at: to represent the diagonalized rotation matrix \begin{pmatrix} e^{i \varphi} & 0 \\0 & e^{-i \varphi}\end{pmatrix} as a homothety in the complex C^2 space. This can also be drawn on two-dimensional sheet because we need only two real numbers. For it, X and Y can contain all complex numbers but this is not good for further theorizing. Therefore, it is preferrable that X contains only the complex numbers x + iy and Y contains their conjugates x - iy. In this way axis X can be represented by the upper half-(complex planes) + real axis X and Y can be represented by the lower half-(complex planes) + real axis X which, in fact, are the eigenspaces of the two eigenvectors of rotation. As a unit on X we will choose 1 + i, and as a unit on Y -- 1 - i. All vectors on such plane must be defined so that they form a vector space with vectors of the type \begin{pmatrix}x+ix\\y-iy\end{pmatrix} where x and y are real. We need only 2 real numbers so we can draw this on a sheet. Then, \begin{pmatrix} e^{i \varphi} & 0 \\0 & e^{-i \varphi}\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix}=\sin \varphi \begin{pmatrix}x + ix\\y -iy\end{pmatrix} is a homothety with eigenvalue sin φ. I might be mistaken somewhere in the last paragraph, especially in the construction of this supposed complex space, and welcome any good-faith corrections. Also, I would welcome any arbitration or third-editor opinion on this. --Lantonov (talk) 11:14, 21 May 2008 (UTC)
replies: text: You said "A vector (directed intercept) can be an element of the real line." That is true in a way: as you point out, some 2-D vectors lie on the x axis, so in as much as the x axis is isomorphic to the real number line, two-vectors of the form (x, 0) are isomorphic to the real numbers. However, arbitrary two-vectors cannot be considered elements of the real line. The thing is, vectors are first-order mathematical objects, even if we often write them in a coordinate system. Using vector operations alone, how can I write the two-vector v as an element on the real line? I can't: R2 is not isomorphic to R. —Ben FrantzDale (talk) 21:37, 23 May 2008 (UTC)
thread_title:  Complex vectors