Applications of Linear Algebra in Advanced Math: Chapter 6 Overview

[g.lemaitre]

this is the final chapter of my Linear Algebra book. It looks like they're just exploring how LA is used in other more advanced math. I have a feeling that it is not very useful for QM which is what I want LA for.

The purpose of this chapter is to give a taste of some of the diversity. Sec¬tion 6.1 introduces the linear algebra in function theory, the broad interaction between linear algebra and analysis, and some of the applications. Section 6.2 examines the singular value decomposition and generalized inverses, an area of pure linear algebra growing out of attempts to deal with numerical difficul¬ties Section 6.3 shows how the equation Ax = b can be solved by iterative methods, an approach often needed when A is very large and sparse. Section 6 4 introduces matrix norms and provides the theory needed for Section 6.3, for defining and understanding condition numbers, and for taking the first step into Banach spaces. Section 6.5 provides an introduction to abstract linear algebra over an arbitrary field. This is the proper context for abstract matrix theory, for the interaction between linear algebra and abstract algebra, and for understanding many applications such as coding theory

 

[chiro]

All the theoretical mathematics underlying quantum mechanics concerns not only the nature of infinite-dimensional vector spaces, but also the operators on them.

One of the first things in looking at these operators is to consider an inner product where <Ax,y> = <x,Ay> where instead of x,y being real vectors, they are now complex. Because of this, we need to look at how this impacts operators like A and the constraints imposed on them. The inner products have a new problem in that complex numbers have issues with regard to the inner product axioms holding (like say <v,v> >= 0 = ||v||^2 for some vector v) and as a result of this kind of scenario, we get results like the ones concerning operators in the situation like we have above with <x,Ay> and these kinds of operators are at the foundation of quantum mechanics at a theoretical and practical level.

Now you mentioned Banach Spaces in your quote: this study leads into operator algebras that are specifically studied and this kind of thing was motivated by Quantum Mechanics when John Von Neumann and his colleagues created the mathematically rigorous theory of QM.

Because of the nature of infinite-dimensional vector spaces, the analysis of the operators on those spaces is more complex due to the nature of requiring not only vectors that make sense (i.e. inner products converging), but also for the operators.

Just like you have eigen-analysis for finite-rank operators, you have a similar kind of thing for the infinite-dimensional operators as well but it's more abstract and you need to consider all these cases of the infinite-dimensional space that complicate things like the convergence properties, the continuity issues of infinite-dimensional spaces (i.e. how the sequences of infinite-dimensional spaces affect continuity, limits, that kind of thing), as well as whether an operator itself even makes sense in the context of using norms on operators.

The other thing is that the spectrum of an operator plays an important role in harmonic analysis and QM, and this is considered in infinite-dimensional theories of R^n and more importantly C^n (which is the Hilbert-Space stuff).

Also another thing that you might consider interesting, is that at a general level an operator (i.e. a general function) can be considered as a function of a linear operator and this is true even in the infinite-dimensional case. The results in the operator algebras establish how to represent a function of a linear operator like say e^x or SQRT(x) where x is a linear operator of full rank and then you go from the idea that a general operator applied to some vector is just a matrix, you end up with a new way of thinking about general functions and what this means geometrically.

You can also consider the space of projections and also the subject of projective geometry (which again has a relation to QM).

The reason why projections (and in particular sets of orthogonal projection operators to form a basis) are important is because these provide a way to decompose a vector (especially infinite-dimensional) in general, and this kind of thing is at the heart of integral transforms at the abstract level, and it's used in application in harmonic and Fourier analysis especially for wavelets.

Understanding projections on the infinite-dimensional spaces of C^infinity (infinite-dimensional complex number geometries) helps understand how you can decompose the space, which tells us a lot about the space and we can analyze it (remember: analysis is to break down and to break down is to decompose).

 

[g.lemaitre]

Thank for your in depth reply.