What are eigen values? I accessed Eric Weisstein’s excellent site for this, of course, first, only to find out that because of a copyright lawsuit they had to remove the whole MathWorld site!!! This is a blow to mathematics information, and another triumph for those people of whom it is said,“What do you call a boatload of lawyers at the bottom of the sea?” Ans: “A good start.” My real question is when will society do something to eliminate lawyers and accountants from taking over everything? That Weisstein site was wonderfully organized for easy access even to someone ignorant of mathematics and computers such as I. No, my real question is what are eigen values, without mentioning matrices as assumed knowledge. Just curious, as I keep hearing this term, eigen values. (Anybody remember Jack Eigen?)
From a non-math guy, who’s a long way from his college math classes:
Eigenvalues (it’s usually spelled as one word) are the values that, when entered into a set of equations (usually represented as a matrix) produce a particular, desired result. I’ve forgotten the mathematical details of the desired result, but typically what they produce is a physically possible system.
“Eigen” is a German word, usually translated as “characteristic”. So eigenvalues are characteristic values – that is, values that the system is characterized by.
Think of a guitar string – when you pluck the string it vibrates with a certain frequency, its “characteristic” frequency.
The most commonly encountered use of the term is in quantum mechanics. The solution of Schrodinger’s wave equation describes the state of a system (usually, for explanatory purposes, an isolated hydrogen atom). The boundary conditions determine which states are physically possible, and a natural result of calculating these states is a set of eigenvalues which characterize the system. Rather than describing all the details of the state of the system, it is sufficient just to list the eigenvalues, since the state is determined by them. So they become a kind of shorthand for physicists. A useful tool.
Typically it’s one word: eigenvalue. “Eigen” is, IIRC, German for “characteristic”, so it’s a composite word from two languages.
See Eigenvectors, Eigenvectors and Matrices, Eigenvectors and Eigenvalues, and Eigenvalues.
An eigenvalue is a number. An eigenvector is a mathematical object that can be be operated on by an operator. Some definitions:
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operator: anything you can define that transforms one mathematical object into a simliar mathematical object. Similar means it’s in the same space as the original object.
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space: basically a set of these operated-on objects. (The set must have a few specific properties, though.) An example of a space is the space of all polynomial functions f(x). Some sample members of this space (or, set, if you’d like) are 2x, 3x[sup]4[/sup]-2x+8, e[sup]x[/sup] (which can be written as a polynomial), etc.
To take this function space example further, let’s define some operators. We are free to define any operator we want so long as it always gives us a result that is also in our function space. Some possibilities:
- The “multiply by two” operator. (Take the function, multiply it by two.) If you give this operator an f(x) from the space, you will get another valid f(x) back.
- The “take the derivative” operator. Simply differentiate the function. Again, you are guaranteed a polynomial back if you start with a polynomial, so this operator is valid.
- The “take the derivative and then multiply by x” operator.
Instead of using words to describe these, it might be useful to write them mathematically:
- 2
- d/dx
- x*d/dx
Now, the good stuff. Pick a function from the space, say x[sup]3[/sup]. Let’s operator on it three different times with our three operators:
- 2(x[sup]3[/sup]) = 2x[sup]3[/sup]
- d/dx(x[sup]3[/sup]) = 3x[sup]2[/sup]
- xd/dx(x[sup]3[/sup]) = 3x*[sup]3[/sup]
Notice that all three returned functions that are in our original space, like we wanted. But also notice that operators (1) and (3) gave us the same function we started with, only it has a number in front of it. For operator (1) we started with x[sup]3[/sup] and ended up with 2x[sup]3[/sup]. We say, then, that x[sup]3[/sup] is an eigenvector of operator (1) with eigenvalue 2 (since 2 was the number we gained in front). For operator (3), we say that x[sup]3[/sup] is an eigenvector of operator (3) with eigenvalue 3. The function x[sup]3[/sup] is apparently not an eigenvector of operator (2).
(Aside: Often one would say “eigenfunctions” instead of “eigenvectors” in this example to emphasize what type of objects we have. “Eigenvector” is used more often when the object is actually a finite-dimensional vector. You can say eigenanything, though. Common in quantum mechanics, for example, is “eigenstate”.)
Other things to notice about our example operators:
- Every function in our space is an eigenvector of operator (1) with eigenvalue 2.
- Eigenvectors of operator (2) include only exponentials (e[sup]x[/sup] (eigenvalue 1), 3e[sup]-2x[/sup] (eigenvalue -2), …)
- Only single-term polynomials (i.e., monomials) are eigenvectors of operator (3). That is, x[sup]6[/sup] is an eigenvector (with eigenvalue 6), but x[sup]3[/sup]-2x[sup]2[/sup] is not (as it gives us, after operation, 3x[sup]3[/sup]-4x[sup]2[/sup], which is not just a multiple of what we started with.)
So, anything mathematical system of the form Av=cv (where A is an operator, v is a vector or something, and c is a number) can be thought of in terms of eigenvectors and eigenvalues. v is an eigenvector of operator A with eigenvalue c. Solving the above equation for a certain system is what a lot of physics entails.
Also, matrices and eigenstuffs are often mentioned in the same breath because knowing the eigenvalues of a matrix is important and because many operators (certainly most useful ones) can be expressed in matrix form.
Pasta beat me to the punch - I was going to post an explanation, but s/he beat me to it. Only one nit to pick
In order to find eigenvectors, you have to be talking about a linear operator - it can’t be anything.
So, let me jump on the other comment in the OP. I have also found Weisstein’s site to be of great value. So much so that I bought his book and CD (well, actually, convinced my office mates that it was valuable enough to add to our library). The book/CD is a snapshot of the website at a particluar point in time (and amounts to a tome 3 inches thick). So your comment about him being forced to take it down due to a lawsuit is interesting. One usually hears of these copyright cases as an author/artist/etc. being ripped off by something like Napster. In this case, however, Mr. Weisstein wants to put up the website, but his publisher is the one preventing it. Now, one wonders why he entered a publishing deal for this material when he intended to “give it away” online?
Kinda turns the usual situation inside out.
Pasta: Thank you. One of the best answers I’ve ever seen on the SDMB.
BTW, regarding the Weisstein mess, check here and here for some interesting background information.
I also think Pasta’s explanation was excellent, but at the same time I find I have a lingering feeling of - so what.
I guess I’m wondering what this stuff could be used for. It also seems that Pasta’s first example is trivial and even his last example could be duplicated in form by any number of strung together operators.
Please do not think that I am making pejorative comments here, I’m actually just trying to get some idea of why this methodology was developed, and how it could be used.
A valid question. I suppose there are some mathematicians who would tell you that mathematical concept don’t have to have something so crass as physical usefulness to be interesting. However, luckily, eigenvalues and eigenvectors can be used to represent a slew of physical concepts.
One example that springs to mind from my own background is the vibration of physical systems (springs and masses, or vibrating beams or rods, or pretty much anything else). You can use an equation (or set of equations) to model a vibrating system. The eigenvalues of the equation(s) correspond to the natural frequencies of the system; the eigenvectors correspond to the mode shapes (the shape of the system when it vibrates at the natural frequency).
As a matter of fact, the centerpiece of my doctoral dissertation was finding the natural frequencies and mode shapes of a fairly complex (and general) vibrating system. This entailed finding the eigenvalues and eigenvectors associated with a fairly nasty linked set of partial differential equations–pretty tedious for an engineer.
You might argue that this particular use of eigenvalues is pretty esoteric, and in some sense you’d be right, but my point is that the technique is applicable to a wide range of vibrating systems; from a simple spring-and-mass to a highly complex system. I sure would have been stuck without the math.
I’m sure other posters could contribute examples from other fields.
Ring, eigenvalues and eigenvectors are probably kind of like cell phones: you don’t see a use for them until you start using them, and then they’re indispensible.
zut has beaten me to the punch as far as posting a good example goes, but let me toss out another. I study hyperbolic geometry, and any hyperbolic isometry (that is, any rigid motion in hyperbolic space) can be described by a matrix. The most common hyperbolic isometry in a certain sense is a loxodromic or “screw” motion: a translation combined with a rotation, like a plane in the middle of a barrel roll. It turns out that the axis of a loxodromic motion, along with the length of the translation and the angle of rotation, are all determined in a straightforward way by the eigenvectors and eigenvalues of the corresponding matrix.
As for your other question of why all this eigenstuff was developed: AFAIK, the original reason was to better understand matrices (and later linear operators) in general. The eigenvectors and eigenvalues of a matrix determine the matrix completely, so they’re important to a theoretical understanding of matrices in general. (I’m fudging a bit: to determine a matrix completely you sometimes need something called the “generalized eigenvectors” as well. But I digress.) Since then of course practical applicaitons of eigenstuff have appeared.
While I understand the sentiment, “so what?” is a question that tends to lead to big embarrassment when asked about high-powered mathematics. Number theory, for example, was long thought of as the purest (i.e. least useful) of all mathematics, and now advanced cryptography is essentially nothing but number theory.
I agree with everyone about pasta’s great definition of eigenvalues and vectors: let me try one that’s more concrete, and possibly more intuitive.
Say you’re moving around in the middle of some flat place like Kansas. You could describe any motion you made, if you wanted, as a combination of “north” and “east.” Three paces east and one pace north, or negative five paces north and two paces east.
You could do the very same thing, if you wanted, with the directions north-north-west and south-east, say. It’d be a little weird, but any spot you chose could be described as a combination of those directions: 500 yards north-north-west, and 100 yards south-east. Because you CAN do that, north-north-west and south-east count as eigenvectors. Not every combination would work: north and south, for example, would just move you in a straight line, and not give you much flexibility.
Sure, you might wonder why you would ever CHOOSE north-north-west and south-east over simple north and east. Well, if your compass needle was bent and only indicated north-north-west, this new method might be pretty handy. Mathematicians seem to find this way of doing things useful.
Does this concept have anything to do with operating on state vectors in order to determine various observables in QM? If so how would that fit with operating on Schrodinger’s Wave equation as described by <b>Pluto</b>?
Also does operating on an <i>eigenvector</i> produce another eigenvector times an eigenvalue or does operating on a <i>vector</i> produce (Jesus I’m getting tired of writing eigen) an eigenvector times an e…value?
Please pardon my ignorance. If these are stupid questions maybe someone could answer what I actually meant to ask.
I’m not a mathematician, so take this for what it’s worth. 
To my way of thinking, the eigenvectors (eigenstuffs) seem more…I don’t know…essential to the whole thing than the eigenvalues.
As Pasta said, you have some linear operator A. Could be a matrix, could be a differential operator, could be pretty much anything. Feed it an input v, and you’ll get something out. Fine.
An eigenvector is an input that, when fed into the operator, comes out unchanged except for being rescaled.
Av = cv. This is what makes the eigenvectors special; feed in just any old input u and, in general, you will not get a multiple of u out.
The eigenvalue is simply the scaling factor associated with a particular eigenvector. When dealing with matrices, it’s usually easier to determine the eigenvalues first, and then find their associated eigenvectors - this sequencing muddled my understanding for quite a while.
To answer your question, Ring; operating on an eigenvector produces that eigenvector times its associated eigenvalue.
Say A has two eigenvectors, v[sub]1[/sub] and v[sub]2[/sub]. Each eigenvector will have an associated eigenvalue c[sub]1[/sub] and c[sub]2[/sub], such that Av[sub]1[/sub] = c[sub]1[/sub]v[sub]1[/sub], and Av[sub]2[/sub] = c[sub]2[/sub]v[sub]2[/sub].
To hint at how this could be made useful, think of an input u that is not an eigenvector of A, but that is a linear combination of v[sub]1[/sub] and v[sub]2[/sub]: u = k[sub]1[/sub]v[sub]1[/sub] + k[sub]2[/sub]v[sub]2[/sub]. Then we can know right away that Au = c[sub]1[/sub]k[sub]1[/sub]v[sub]1[/sub] + c[sub]2[/sub]k[sub]2[/sub]v[sub]2[/sub]. IOW, linear combinations of eigenvectors remain linear combinations of only those same eigenvectors when operated on. Depending on what you’re doing, this can be very useful knowledge.
Second question first:
Your first suggested use is closer to correct than your second use. The idea is that you have a vector and you can operate on it to produce another vector. If the vector you started with and the vector you ended up with are the same vector except for a multiplier, then that vector is called an eigenvector. It’s still okay to call it a plain old vector because it is just a vector. But for this particular operator the vector has this special property, so you can add “eigen” to the front to remind yourself of that.
I would phrase your statement as, “Operating on an eigenvector produces the (self-same) eigenvector times its eigenvalue.” (Both your sentences are technically correct, though; they just suggest a slight misunderstanding.)
First question second:
Absolutely. In fact, that’s one of the postulates of QM: For every observable there is an associated operator that operates on the state space (called the Hilbert space). The only possible values one can measure for the observable are the eigenvalues of its associated operator. (There are other parts of this postulate that are unimportant here (like the fact that the operator must be linear and Hermitian.))
The Schroedinger equation is an excellent example. The equation (the time-independent version at least) looks like this: Hs=Es. H is an operator called the Hamiltonian. It is the operator associated with the familiar observable called energy. s is the state vector. s can be a function of spatial position (x, y, z), of internal parameters (like the purely QM quality of particle spin), of anything else the state of the system might depend on. E represents an energy eigenvalue. Once you know what the Hamiltonian looks like and what your Hilbert space looks like (i.e., what variables s depends on), you can begin your task of finding those state vectors that satisfy the equation – finding the eigenvectors (or, as they are usually called, the eigenstates.)
Once you know the eigenstates, what then? It turns out that knowing the eigenstates of the Hamiltonian tells you how a known state of a sample system will evolve in time. That is, if you have a system in the lab, and you know its state right now, you need to know the eigenstates of the Hamiltonian to be able to say what your state will be some time later. Knowing the eigenstates of other QM operators is equally important, though.
Slightly tangential (and sure to make you all retract your earlier compliments (BTW - thanks!)): I said that the only values that you can measure for any observable (like energy) are the eigenvalues of its (the observable’s) operator. One might say something like, “I have a system here in my lab. It is in state s[sub]1[/sub]. It has energy E[sub]1[/sub] because s[sub]1[/sub] is an eigenstate of my system’s energy operator with eigenvalue E[sub]1[/sub].” But, what if my system is in a state that is not an eigenstate of the Hamiltonian? What is its energy then? It so happens that you can always write the state of the system as a sum of eigenstates of the Hamiltonian. For instance, say you have a system with only three eigenstates, s[sub]1[/sub], s[sub]2[/sub], and s[sub]3[/sub]. They have associated energy eigenvalues (or “eigenenergies”) E[sub]1[/sub], E[sub]2[/sub], and E[sub]3[/sub], respectively. If your system is in state s= s[sub]3[/sub], say, then you are all set – its energy is E[sub]3[/sub]. But if your system is not in an eigenstate, you can always write it as a sum of the three eigenstates: s=as[sub]1[/sub]+bs[sub]2[/sub]+cs[sub]3[/sub], where a, b, and c are just numbers specifying how much of each eigenstate there is in my state s. If you measure the energy of this state, you still can only get E[sub]1[/sub], E[sub]2[/sub], or E[sub]3[/sub] (because the postulate says so). s is indeed not an eigenstate of our Hamiltonian because we can just check: Hs = H(as[sub]1[/sub] + bs[sub]2[/sub] + cs[sub]3[/sub]) = aHs**[sub]1[/sub] + bHs**[sub]2[/sub] + cH**s**[sub]3[/sub] = a**E[sub]1[/sub]s[sub]1[/sub] + b**E[sub]2[/sub]s[sub]2[/sub] + c**E[sub]3[/sub]s[sub]3[/sub], so our result is not just a multiple of our original s (unless E[sub]1[/sub] = E[sub]2[/sub] = E[sub]3[/sub], in which case we can factor it out the E’s, but that’s another story… Also, the fact that H is a linear operator lets me operate on each term individually.) Anyway, so s is not an eigenvector, and it turns out that you have a certain probability of getting each of these three possible values for energy. Namely, you have a probability of a[sup]2[/sup] of getting E[sub]1[/sub], probability b[sup]2[/sup] of getting E[sub]2[/sub], and probability c[sup]2[/sup] of getting E[sub]3[/sub] is you measure the energy of a system in state s! You cannot say for certain beforehand which energy you will measure. (Aside: the numbers a, b, and c are in general complex numbers, so you actually take the modulus squared to get the probabilities.)
Enough rambling for one post, I’d say.
Okay, one more reason for eigenstuff, one that is probably closer to folks in everyday life than hyperbolic geometry: computer games!
To get all the pretty graphics to fit in the memory of folks’ graphics cards better, there are new compressed texture formats that allow the hardware to quickly access and decode the data needed. Basically, it breaks up the image into little chunks, and in each chunk generates a small set of useful colors, much less than the number of pixels in each chunk. To calculate which colors are useful, it’s very useful to use eigenanalysis. Otherwise, you wind up with a set of bad colors, or you take way too long to calculate the set.