Eigenvalues

Extracted from a Pynchon-list thread:

Eigenvectors, eigenfunctions, and eigenvalues are just basic terms out of "matrix" theory (matrix in this sense being the rectangular or n x m arrays of values, a mathematical term--and matrices can be in n dimensions, lest the quibblers correct my n x m example!).

The use in differential equations, quantum mechanics, chaos theory, etc., etc., comes because matrices are the basic way "transformations" (aka operators) are used. Cf. any encyclopedia article on matrices, eigenvectors, etc.

I don't recall Pynchon's use...perhaps I saw it considered it unremarkable. Pynchon, having worked for Boeing and obviously being familiar with physics, was probably able to use it transparently.

In many transformations, such as distortions (squeezings, compressions), one vector (direction) may get mapped into the same direction, albeit of different length. (I could show this more easily at a blackboard.) These vectors that maintain their direction are called eigenvectors. (A German term, originally.) The coefficient for their length is the eigenvalue.

About 5 minutes of explanation with pictures will burn this idea into nearly anyone's mind forever.

Though this may not have been Pynchon's use, one could somewhat obscurely use this as an allusion to the changes someone or some group felt. Perhaps, "The Industrial Revolution changed many things, but the aristocracy was an eigenvector of this transformation, maintaining its position, though increased by a large eigenvalue."

(Well, I said it would be an obscure reach. Presumably Pynchon's use was more subtle, though maybe even more obscure if eigenvectors and eigenfunctions are an unknown concept.)

Tim May

The "eigen-" of "eigenvector" normally is translated as "characteristic", and not as the other translations that I have seen here. The idea is that the characteristic vectors and the associated values will in certain circumstances entirely characterize the linear transformation (for instance over the complex numbers).

I always thought that the name was just a silly pun... "Dudley" meaning "zero", and having zero eigenvalue, the translation is not invertible. How this relates to the character is beyond me.

David Milne

I came in on the tail end of this eigenvalue discussion, but I don't think it's surprising that we're seeing two wildly (and widely) different opinions on the significance of Eigenvalue. That is, there have been a couple of people to wrestle with the meaning of eigenvectors, both mathematically and in terms of literary characterizations, and likewise another group of people who want to say, wait a minute, Eigenvalue is really not all that complicated. . . it's just a little joke for engineers in the audience.

I'd call this the Pynchon dilemma, and one that's central to all his work. Not that he made it up by any means, but over and over this is a dilemma that he suggests is built-in to any interpretation.

In Gravity's Rainbow, it's posited as zeros and ones, and in various other forms. Is it paranoia or is it a conspiracy to be legitimately concerned about? Is the baby Jesus happy or has he just farted? I could go on, and others of you could supply your own. The text is filled with this interpretive binaries. And yes, these are binaries that he seriously muddies: As he puts it, just because you're paranoid doesn't mean they're not following you. That's probably the clearest example I can think of where he merges the two interpretative options.

In the Crying of Lot 49, Oedipa Maas's quest poses similar questions of interpretation. Is the Trystero all an imagined construction joined in tinker-toy fashion by her own imagined connections or is it "real". Is it just Pierce's idea of a joke or an entire underground society?

To bring this back to eigenvalues and eigenvectors, I'd say Pynchon's texts anticipate these problems that we have when we read them. That, for me, has always been what most impresses me most about his novels: they are in some ways their own most rigorous critics; they are there working away, taking apart and putting together, long before we even get there.

Paul Maliszewski

An eigenvector is a fixed point of a linear map in a vector space - actually the vector's direction is fixed, it's length is multiplied by a scalar otherwise known as the `eigenvalue' associated with the eigenvector. For example, the transformation which rotates 3-d vectors 45 degrees about the Z axis has vector (0 0 1) as its only eigenvector and 1 as the corresponding eigenvalue. Another example would be the transformation which scales up by a factor of 2 along each axis. In this case every vector is an eigenvector with eigenvalue 2.

Presumably where the vectors are functions the eigenvector is called an eigenfunction. Function spaces occur in quantum mechanics where the probability functions for quantum level entities are elements of a vector space of functions. Eigenvectors are associated with observable variables.

P.S. In German `eigen' means `self' or `own' (it functions as an adjective).

Andrew Dinn

About the eigenfunction

The eigenfunction is a tool used to solve two-point boundary value problems, which are a class of differential equations. My exposure to eigenfunctions have been in the solution of linear differential equations but there is likely research, beyond my experience, which applies the concept to the nonlinear differential equations used to describe chaotic systems.

A typical two-point boundary value problem can be stated:
solve the equation dy(x)^2/dx^2 + lambda* y(x) = 0 with the boundary conditions ay(0) + by'(0) = 0 and cy(l) + dy'(l) = 0.

The functions y(x) which solve the above system of equations are called the Eigenfunctions. The values of lambda which solve the equations are called the eigenvalues. For practical engineering problems the solution is often in the form of an infinite series. The series is composed of harmonic functions such as sine, cosine, hyperbolic sine and cosine and other creatures such as bessel functions or Chebychev polynomials.

Please note that in many applications neat (although infinite series) analytic functions which solve the equations do not exist and we must resort to brute calculation. To perform the calculation we must integrate twice. Pynchon referred to this technique in "Gravity's Rainbow". These are the references to the double integral. The double S's of the SS. The double lightning bolts which the underground rocket factory Nordhausen mimics when viewed from above.

In fact, much of the rocket's dynamics were characterized by these two-point boundary value equations. The heat transfer equations, the yaw control equations, and the inertial guidance system used to calculate Brennschluss.

I hope this helps. I can provide more specific examples if you like and if you are mathematically inclined you can refer to any advanced text on differential equations. I referred to "Differential equations and Their Applications", M. Braun, Springer-Verlag, 1986.

Stephen Roe

Of course it all came flooding back as soon as I posted (I did this 16 years ago in school maths). An eigenfunction is an `annihilator function' for a linear mapping constructed using the eigenvalues. i.e. if mapping M has eigenvalues l1, ..., ln and associated eigenvectors e1, ..., en then the eigenvectors satisfy the equations

M ei = l1 * ei for each i = 1, ..., n

or equivalently

M ei - li * ei = 0

or factoring the multiplication

(M - li) ei = 0

So, for any given eigenvector ei the mapping (M - li) `annihilates' the vector ei. If you multiply (i.e. apply in sequence) the mappings

(M - l1)(M - l2)...(M - ln)

then this product `annihilates' all the eigenvectors i.e. it maps all eigenvectors to zero. Given suitable conditions it actually maps any vector to zero - in other words the product above is equivalent to a null or zero mapping.

The eigenfunction is the algebraic polynomial

(x - l1)(x -l2) ... (x -ln)

which expands out to something of the form

x^n + a1 * x ^ n-1 + ... + an-1 * x + an

Substitute the mapping M for x and you get the zero mapping. Substitute an eigenvalue for x and you get the value 0. So each eigenvalue is a root of the eigenfunction (equivalently, each eigenvalue is a zero of the annihilator functions).

n.b. sometimes the eigenvalue associated with an eigenvector is 0. In such a case the `dud eigenvalue' does not contribute to the annihilator function since it would merely add a redundant power of x to each term.

> P.S. In German `eigen' means `self' or `own' (it functions as an > adjective).

Someone mentioned that `eigen' was usually `translated' as `characteristic'. Actually, the `eigenxxx' terms do not need translating as they are not German in origin. Germans (so a colleague assures me) find it very funny when they see this prefix in maths classes. `Characteristic xxx' is a recent rewording.