Beating the force of gravity, armed with the power of matrices. It sounds like complete nonsense, or is there something to it? We’ll look at existing applications of matrices in physics and then indulge in some Beating Gravity style creativity and speculation. Let’s briefly start with some basics though.
What’s a matrix anyway?
A simple definition of a matrix is “a set of numbers arranged in rows and columns so as to form a rectangular array”[1]. And just in case your math is rusty, it looks like this:
$$\begin{vmatrix}4 & 5 & 2\\9 & 3 & 5\\3 & 9 & 7\end{vmatrix}$$
And while this is a 3×3 matrix as it has 3 columns and 3 rows, a matrix can have any number of columns and rows. So, basically a matrix is a table with some numbers, that’s a pretty wide definition! And thus it’s not that surprising that we find applications of matrices in physics. That’s what we’ll focus on but the use of matrices is also common in many other fields of science, and in engineering.
Applications of matrices in physics
We’ll now discuss some of the main applications of matrices in physics. Please note that this list is not exhaustive though!
Use of matrices in quantum mechanics
Quantum Mechanics (QM) deals with the world of atoms and their even smaller constituents, subatomic particles. QM is a complicated theory and we are not going to try and explain it in this article. See the references for a somewhat casual introduction book though.[2] For now, if you are not familiar with QM, just accept the fact that “operators” and “states” are important concepts in QM, and that these can be represented using matrices. The “matrix mechanics” formulation of QM is a major application of matrices in physics. Matrix mechanics was invented about 100 years ago by Werner Heisenberg and other physicists[3]. And if physics was never your thing, you may still have heard the name Heisenberg in the epic TV series Breaking Bad. Walter White’s criminal alter ego, Heisenberg, refers to the very same Werner Heisenberg![4]
$$\begin{vmatrix}0 & 1\\1 & 0\end{vmatrix}\begin{vmatrix}0 & \text{-}i\\i & 0\end{vmatrix}\begin{vmatrix}1 & 0\\0 & \text{-}1\end{vmatrix}$$
And what’s this? The 3 matrices above are the famous Pauli matrices.[5] Don’t let the use of the imaginary number \(i\) [6] intimidate you, and appreciate how simple these matrices look. How wonderful that something so simple can play an important role in something so complicated like quantum mechanics!
Einstein’s special and general relativity
If you are not familiar with Einstein’s theories and have no time or desire to study these, you can still appreciate the use of matrices in these theories. Special relativity involves something called Lorentz transformations, and we can express these using matrices. Without explaining how it works, here is a matrix that represents a “Lorentz boost in the X-direction”:[7]
$$\begin{vmatrix}\gamma &\text{-}\gamma \beta &0&0\\\text{-}\gamma \beta &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{vmatrix}$$
General relativity heavily uses “tensors”, and certain tensors are easily presented in matrix format. If you are not familiar with such tensors, please trust us that the following is a good example. This is the “Schwarzschild metric” which describes the gravitational field in certain situations:[8]
$$\begin{vmatrix}\scriptsize{{\frac {2GM}{rc^{2}}}\text{-}1}&0&0&0\\0&\frac{1}{1\text{-}{\frac {2GM}{rc^{2}}}}&0&0\\0&0&r^{2}&0\\0&0&0&\scriptsize{r^{2}\sin ^{2}\theta} \end{vmatrix}$$
The above examples from Einstein’s theories are merely to illustrate some of the applications of matrices in physics. We fully appreciate that these will not help you to understand the theories if you don’t have any prior knowledge. If you want to properly familiarize yourself with Einstein’s theories, there are no shortcuts, you’ll need to dive into some physics books![9][10]
Matrix theory / M-theory / superstring theory
Another application of matrices in physics is matrix theory.[11][12][13] Not surprisingly, matrix theory involves using matrices. This theory proposes that it is equivalent to M-theory, which in turn is a unification of different variants of superstring theory. M-theory involves complicated math and is one of the leading candidates for a so-called theory of everything.[14] At Beating Gravity, we share the healthy skepticism that some have expressed regarding this theory[15], but let’s not get into that now.🙃
Why do matrices apply to so many physics theories?
That’s something you might wonder about after reading the previous section. The main reason is that physics theories often deal with systems of equations. And the use of matrices happens to be a good method to represent and work with systems of equations.[16][17] Therefore, we should probably see matrices as a tool rather than a magical ingredient for physics theories. We can likely find applications of matrices for almost any physics theory, in one way or another.
What’s so good about using matrices?
While they are simple tables with numbers (or variables representing numbers), matrices come in many types, and have lots of cool properties. If you are familiar with matrices, then you’ll recognize terms like: unit matrix, determinant, trace, inverse, eigenvalue, eigenvector, adjoint, cofactor, minor, symmetric, antisymmetric, orthogonal, and so on. And you might be aware that we can do lots of cool things with matrices. We can multiply, add, subtract, diagonalize, transpose, invert, square, etc. If this is all new to you, there are plenty of math websites[18][19] and books[20] that explain this stuff. And if you don’t want to do any matrix calculations manually, there are plenty of tools that can come to the rescue. Physicists, and other scientists, might use Mathematica, R, Python, and so on. But Excel or some simple online tools[21] also work.
Ok, but what’s now actually so great about the use of matrices? As mentioned in the previous section, matrices are often a good way to represent systems of equations. They can help to summarize equations for the sake of overview. Moreover, matrices can help us find consistent solutions for a system of equations. We can benefit from the well-established mathematics concerning matrices, and use all of the wonderful matrix properties and tricks to solve a problem once we present it in matrix format. Furthermore, thinking about a problem in terms of matrices can result in creative new ideas that otherwise might not have come up. So, new applications of matrices in physics can result in useful ideas. That’s the thought behind this article anyway!
Matrices are cool
Seriously, matrices are cool? They absolutely are! You just got to love those elegant arrangements of numbers in tables, and the beautiful mathematical tricks they come with. So much clarity, so much joy, matrices just feel right. It’s impossible not to like a matrix! Ok, maybe that’s an overstatement but at Beating Gravity we really like matrices and strongly believe they have so much more to give. And if that sounds a little crazy to you, that’s fine with us. Call it a gut feeling but we believe in the power of using matrices.
$$\begin{vmatrix}\text{M}&\text{A}&\text{T}&\text{R}\\\text{I}&\text{C}&\text{E}&\text{S}\\\text{A}&\text{R}&\text{E}&\\\text{C}&\text{O}&\text{O}&\text{L}\end{vmatrix}$$
Do we live in a matrix?
Most probably, you’ve heard of The Matrix[22] movies. Whatever you might think of these movies, they do make an interesting point. Is it possible that we live in a giant simulation? In fact, simulation is probably a more accurate term for what these movies seem to refer to as “The Matrix”. We suppose we can’t really blame the makers though, as “The Simulation” would have been a boring name. But anyway, insofar the matrix refers to a giant simulation in these movies, that’s something completely different from the good old matrix used in physics and mathematics.
Sure, the movie repeatedly shows a bunch of green characters, which is supposedly the program code behind the simulation. However, that has nothing to do with applications of matrices in physics as discussed in this article. But if we stick with the terminology of the movie, and refer to a giant simulation as a matrix, then we might actually live in one! At least, that’s what some people seem to think.[23][24] However, this paragraph is already too lengthy and grossly off-topic. So please decide for yourself if you are a mere digital character in a virtual world. In the meantime, let’s assume we are real and get back on topic in the next section.
Creative new applications of matrices in physics
We already mentioned that we think matrices are cool and that we believe in the power of matrices. But what exactly do we think humanity can achieve further with the help of these beautiful arrangements of numbers? Well, we think matrices might just help us to beat the force of gravity. And it’s not that we’ll “jump on a matrix” (whatever that means) and just take off. We mean using matrices as a way of thinking about a problem. Call it a hunch, a gut feeling, or even a sick form of worshiping, but we believe matrices have a role to play in humanity overcoming gravity. Perhaps we can find completely new applications of matrices in physics, or just use matrices as a tool to come up with new ideas. In the next section, we’ll reserve space for wild ideas like that. In matrices we trust, let’s take off!