Matrix & operator summary

[ognik]

Hi, I feel thoroughly muddled like I am drowning in a soup of terminology and notation, and I have assignment deadlines.

So I have tried to compile a rough table that will give me a consistent base which I can use now - and add to going forward; trying also to stick with the usage in my textbook. It is deliberately shallow which I ask you to bear with within reason, I just want to sort out what generally goes with what, and where there are multiple 'options' for a table cell, to choose a preference for myself.

I have been building other related and detailed notes (on Hilbert spaces for example), so am aware my table will probably make you cringe a little, but it will help me move on - and then doing problems will help me grow a better understanding; at the moment I feel I have no 'structure' to relate the detail to.

For example I have separated Matrices and operators, but I know that operators are often Matrices. But, for example, the ODE operator is not a matrix (afaik), so to speak of operators that are not matrices as being symmetric makes less sense to me than referring to them as self-adjoint; maybe later through experience I will revise that. I have also noted that there might be differences between finite and infinite spaces, but that is not important at this point in the course, we will cover that later.

I would appreciate it therefore if you could offer corrections with the above in mind, even if only for limited aspects at a time.
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[jvicens]

Hello,

I can understand your frustration with the terminology and notation used in mathematics. It can be overwhelming at times, especially when trying to compile a consistent base for your studies. Your approach of creating a table to organize your understanding is a good one.

I would like to offer some clarifications and corrections to your table:

1. Matrices and operators are closely related, but they are not the same thing. Matrices are arrays of numbers, while operators are functions that operate on vectors or functions. So, a matrix can be seen as a representation of an operator, but not all operators can be represented by matrices.

2. It is true that not all operators are symmetric, but not all self-adjoint operators are symmetric either. A self-adjoint operator is one that satisfies a specific condition, while a symmetric operator is one that satisfies a different condition. It is important to understand the difference between these terms and how they relate to operators.

3. The ODE operator is not a matrix, as you mentioned, but it can be represented by a matrix in certain cases. However, it is more common to refer to it as a differential operator, as it operates on functions to produce their derivatives.

4. When it comes to finite and infinite spaces, there are some key differences that you should be aware of. For example, in finite-dimensional spaces, all operators can be represented by matrices, while this is not always the case in infinite-dimensional spaces. Additionally, there are different types of operators that are specific to infinite-dimensional spaces, such as compact operators.

I hope these clarifications and corrections help you in your understanding. As you continue to work through problems and gain experience, you will develop a better understanding of the relationships between different concepts in mathematics. Keep up the good work and don't hesitate to ask for help when needed. Good luck with your assignments!