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
- 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.
- Singular values are one generalization of eigenvalues. My question relates to another generalization.
- Since I stated the problem so as not to be limited to finite-dimensional vector spaces, one such application area is Fourier analysis.