I am not a mathematician by any means, so please be clement if my question happens to be naïve.

In many textbooks, matrices make their first appearance in the context of a discussion of linear transformations of vectors. Then the author usually proceeds to consider changes of basis, deduce formulae like $T^{-1}AT$, and so on. So far, so good, except for the fact the term «matrix» actually disguises the tensor nature of linear transformation.

Matrix is just a representation of tensor in this context. But what is important here is that despite their somewhat abstract nature, tensors correspond to physical values. However, the notion of covariance has no meaning when one talks about matrices. That makes it possible to consider matrices as in some sense as more general objects than tensors (speaking simplistically «every tensor is a matrix, but not every matrix is a tensor»; yes, tensors can be of any rank, but let's talk about 2D case only).

Many mathematical objects are intimately connected to the properties of physical objects. Of course, by definition, they are generalizations and abstractions, but this intimate touch with reality persists nonetheless. Here are a few examples of such connections: continuity — topology; counting — ordinary numbers and fractions; rotation — complex numbers and quaternions; the rate of change — derivative; linearity — tensors (vectors) and so on. I would not go too far and say that every mathematical object should have some physical meaning, even in principle. It is maybe that matrices are an example of such «pure mathematical machinery.» But, if so, how can it be that they appear in many physical equations (well, they pervade all mathematical physics!)? It looks like, for example, Pauli of Dirac matrices should have some meaning. And this is my question: can matrices (not the objects they represent) be associated with any property of the world.

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    $\begingroup$ I don't think matrices have a physical meaning, they are just tables of numbers. Linear transformations and quadratic forms have a physical meaning, as you point out. Matrices are just a convenient device to represent them in a chosen basis. Compare the distinction between natural numbers (meaningful) and sequences of digits (not meaningful, merely representations of natural numbers in a chosen base). $\endgroup$ – user856 Oct 24 '19 at 16:36
  • $\begingroup$ Matrices appear ubiquitously in physical equations because linear transformations and quadratic forms are ubiquitous in physics, and matrices are the standard way to represent them. One may well ask why sequences of decimal digits appear so often in records of scientific observations. $\endgroup$ – user856 Oct 24 '19 at 16:39
  • $\begingroup$ What about the inertia matrix? Moments of inertia are as physical as it gets. $\endgroup$ – Rodrigo de Azevedo Oct 24 '19 at 16:57
  • $\begingroup$ @Rahul I guess I got the idea behind your parallel between matrices and decimal digits. But I would suggest to draw another parallel, namely, one between matrices and complex number. One meets complex numbers when learning about quadratic equations. And at that moment the imaginary unit is just a new symbol. One learns its unusual properties. However, it is not until later one realises that $i$ is connected very tightly to everything involving rotations and waves. It can even be considered as an elementary bivector. I thought matrices might fall into this category as well. $\endgroup$ – S. N. Oct 25 '19 at 15:50
  • $\begingroup$ @RodrigodeAzevedo AFAIK there is no such thing as inertia matrix. And, strictly speaking, there can't be. There is matrix representation of the inertia tensor. And this distinction is at the heart of my original question. $\endgroup$ – S. N. Oct 25 '19 at 15:52

There are essentially two ways matrices arise from physics.

On one hand, physics is generally described through differential equations, and quantities are often defined through derivatives of other quantities. Now a derivative in essence means locally approximating a function by a linear function. And linear functions are represented by matrices.

Also the second derivative of scalar quantities is often used, which means local approximation by a quadratic form. And again, quadratic forms are described by matrices.

On the other hand, in modern physics many of the fundamental laws are derived from symmetries. Those symmetries form a group, and representation theory tells us that any group is isomorphic to a subgroup of the general linear group of an appropriate vector space. The general linear group consists of all invertible linear functions of that vector space to itself. Again, when the vector space is finite, the linear functions can be represented as matrices.

  • $\begingroup$ Thank you for this answer. If I got your idea straight, any matrix is a purely mathematical object that has no place in physics until it represents some physical object or process. If not, they are no more than machinery. $\endgroup$ – S. N. Oct 25 '19 at 15:56
  • $\begingroup$ @S.N.: Yes, ultimately in physics all mathematical objects either represent some physical object/process, or are mere machinery. Indeed, one of the disputes in the fundamentals of quantum mechanics is exactly in which of the two categories the quantum wave function belongs: Does it represent something physical (realist interpretations) or is it just a tool to calculate probabilities (instrumentalist interpretations)? $\endgroup$ – celtschk Oct 27 '19 at 13:38

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