2011-04-18

(2) Max determinant problem

I would like to talk about why mathematicians are interested in max determinant problem. This is just my personal theory and I could not find an article that say this directly. So, I warn you that I might be completely wrong.

The max determinant problem is mentioned in a context of partial differential equation.  Is a partial differential equation interesting? I safely say, yes. This includes heat and wave problem. We can design buildings, computers, cars, ships, airplanes, ... and so on. There are so many applications of this in our world.

Hadamard is one of the mathematicians who contributed the max determinant problem. His one of the interests was partial differential equation. A basic partial differential equation, for instance, a wave equation is like this.
This can be re-written as
(By the way, if we wrote it as above, an operator d^2/d x^2 looks like to have an eigenvalue λ. Like Mu = -λu. This is a clue of relationship between integral equation and linear algebra.)

Fredholm wrote this kind of integral equation in a finite sum form. This is a matrix form. He had an idea to solve this equation by taking the limit of it, e.g., the dimension of matrix goes to infinite.

If we can write an integral equation in his form, each finite sum equation can be solved by solving linear equations. At that time, people solved the linear equations by Cramer's rule. Cramer's rule has 1/det(A) form. If a matrix A's size becomes larger, the solution can converge or not is depends on the (absolute) max determinant value, whether it is less than 1 or not. I imagine that this is the reason of mathematicians are interested in the max determinant problem. Though, I could not find a direct article about the motivation of why they are interested in the max determinant problem.

(Note: I explained only a little that how an integral equation has a matrix form. Because this is quite detail, I will add appendix in case one is interested in this.)

Although Fredholm didn't use Cramer's rule directly, the proof of convergence needs maximal value of determinant (see. 30 lectures of Eigenproblem, Shiga Kouji, p.121. (in Japanese)).

Hilbert removed the determinant from this problem and established eigenvalue based solution --- Hilbert space. He climbed up the view one level. I think determinant is still an important subject, though, eigenanalysis is much interesting. This is also just my impression, but, when Hilbert established the Hilbert space, people's interest moved to eigenvalues from the determinant. I just imagine this.

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