Wednesday, September 12, 2012

Eigenvectors Generalized?

I have occasionally mentioned some of the mathematics that many find useful in reading Sraffa1. Here I want to raise a question. Consider the following two problem statements:

Problem 1: Given a vector space V and a linear function A mapping that vector space into itself, find a vector v in V such that the image of v under A is merely the original vector lengthened or shortened. In other words:

A(v) = λ v.

Problem 2: Given a vector space V and two linear functions, A and B, mapping that vector space into itself, find a vector v in V such that the images of v under A and B are two vectors, where one such image is the other vector lengthened or shortened.In other words:

A(v) = λ B(v).

The second problem statement is, in some sense, a generalization2 of the first. I know of lots of theory for analyzing the first problem and many application areas3 unrelated to economic models of circulating capital. I do not know of any literature on the second problem outside of mathematical economics and the analysis of joint production. Likewise, I have a name, eigenvector, for a solution to the first problem. But I have no such name for a solution to the second. Where, if anywhere, can one find literature on the second problem and other application areas motivating its application?

Maybe this is a question for math overflow.

Footnotes
1. In such discussions, I usually do not worry about whether or not the mathematics on which I draw is constructive. Arguably, Sraffa insisted that his proofs be constructive. This topic should be of interest to Wittgenstein scholars.
2. Singular values are one generalization of eigenvalues. My question relates to another generalization.
3. Since I stated the problem so as not to be limited to finite-dimensional vector spaces, one such application area is Fourier analysis.

Calgacus said...

Your problem 2 is called a Generalized Eigenvalue Problem. Of course if B is invertible, you can just invert it & get to problem 1. The phrase "Generalized Eigenvalue" by itself, however, means that (A- λI)^n v = 0 for some n & v.

Robert Vienneau said...

Thanks for the suggestions. I think one can assume either A or B is generically invertible.

I see I was sufficiently obscure on the motivation for not reducing problem 2 to problem 1. In the applications I have in mind, including the circulating capital model, one would like the maximum eigenvalue to exceed unity and the corresponding eigenvector to contain all non-negative elements. The Perron-Frobenius theorems state sufficient conditions for A to satisfy such that these properties follow. And these conditions have a clear economic interpretation for the circulating capital case. (Caveat: Sraffa's "beans".)

If the second problem is reduced to the first, these conditions do not have a clear interpretation in the case of joint production. Furthermore, the desired properties, I think, do not always hold.

Maybe the joint production case should only be considered in the context of the analysis of the choice of techniques. A large literature draws on John Von Neummann's growth model. I guess one could say that mathematical programming is the generalization to look at here. And, of course, this literature is applied in many fields outside of economics.

Calgacus said...

Meant to link to wikipedia article on

Eigendecomposition_of_a_matrix#Generalized_eigenvalue_problem> somehow made a bad link. It says "However, in most situations it is preferable not to perform the inversion, but rather to solve the generalized eigenvalue problem as stated originally. This is especially important if A and B are Hermitian matrices, since in this case is not generally Hermitian and important properties of the solution are no longer apparent." Similar to what you say for your interests. Just meant to give you a name to start from.

pqnelson said...

Well, I think you are starting to derive the singular value decomposition for a matrix.

Basically, if B is invertible, this boils down to a normal eigenproblem.

So lets consider the situation when B is not invertible. We can still say it is similar to a diagonal matrix D where some of the diagonal components are zero.

This "more general" eigen-decomposition is precisely the singular value decomposition!

Robert Vienneau said...

I apologize for my slowness in replying.

Calgacus, thanks for the wikipedia link. It seems to address my question.

pqnelson, thanks for the suggestion. I have never been able to link up the little I know about matrix decompositions to my favorite work in economics.

Anonymous said...

The Hermitian property -- very important to understand actually. Major control theory applications.... and what are you trying to do with an economy, if not to control it?

--Nathanael

pqnelson said...

The infinite-dimensional generalization is interesting! That's more related to my field (mathematics relating to QFT), so let me think about that one in some more detail...

Feel free to email me with follow up queries related to this, too! :)