Blog post originally published in French in 2022.
Sur l’autre corps bidimensionnalisé, on pouvait distinguer les os et les vaisseaux sanguins, et il était aisé de reconnaître chaque partie de son anatomie. Durant le processus de bidimensionnalisation, chaque objet tridimensionnel était projeté selon des principes géométriques précis sur la surface bididimensionnelle…
— Liu Cixin, La Mort immortelle, Actes Sud, 2018
A few months ago, I was reading the excellent Three-Body Problem trilogy by Liu Cixin, in which – without wanting to spoil too much but I won’t be able to avoid it, and this long sentence gives you the chance to stop reading if that bothers you, aliens play with the laws of physics. Some of them, in particular, enjoy sending terrifying objects through space that collapse dimensions whenever something displeases them. Like humanity, for instance.
The other thing is that I was recently looking for an activity to rest my brain between chess games and game analysis, because reasoning over 64 squares is quite exhausting. So I decided to take up another hobby in parallel: refreshing my linear algebra knowledge. Turns out, I had forgotten nearly everything.
However, yesterday, while I was lost in a chapter on eigenvectors (let’s be honest, the English word eigenvector just sounds way cooler than the French “vecteurs propres”), I suddenly had a revelation about The Three-Body Problem.
These kinds of impromptu revelations aren’t rare throughout human history. For instance, Poincaré, while about to board a bus, discovered Fuchsian functions. Here’s a funny anecdote:
[Translated from the original]
When we arrived in Coutance, we got on an omnibus for some sort of outing; as I was stepping onto the platform, the idea suddenly came to me – without anything in my prior thoughts seeming to have led me there – that the transformations I had used to define Fuchsian functions were the same as those of non-Euclidean geometry.
— Henri Poincaré, Science et Méthode, Flammarion, 1908.
Which, I guess, has happened to anybody taking public transport – sorry for my American friends who cannot relate. Now, to be fair, pleasant revelations don’t always occur during transit. Even for scientists. Just to give one example, Pierre Curie would have loved if horse-drawn carts didn’t exist.
Anyway, my humble contribution to humanity is the realization that the objects the aliens send in Liu Cixin’s work are matrices. Their weapon is matrix multiplication. So, whether or not these weapons are terrifying depends largely on your fear of linear algebra, but their effectiveness depend on their theoretical ground. Now, as a side note, let’s admit that trying to annihilate a civilization with matrices is particularly cool – much less common and more elegant than doing it with the differential equations traditionally used for missiles.
So, now that we know the weapon, the question becomes whether it would be mathematically possible to undo the damage and bring humanity back to life. But first, why do I say these are matrices? Let me explain – you’ll see, it’s quite simple.
If we define a vector \(\boldsymbol{v} = \begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}\) as a two-dimensional vector in a coordinate system \((O, \vec{i}, \vec{j})\) then by definition we have \(\boldsymbol{v} = v_{1} \times \vec{i} + v_{2} \times \vec{j}\).
Now, let \(\boldsymbol{A}\) be a matrix of size \((m,2\)). If we define
$$ \boldsymbol{A} \times \boldsymbol{x}=\begin{bmatrix} a_{1,1} & a_{1,2}\\ \vdots & \vdots\\ a_{m,1} & a_{m,2} \end{bmatrix} \times \begin{bmatrix} v_{1} \\ v_{2} \end{bmatrix} = v_{1} \times \begin{bmatrix} a_{1,1} \\ \vdots\\ a_{m,1} \end{bmatrix} + v_{2} \times \begin{bmatrix} a_{1,2} \\ \vdots\\ a_{m,2} \end{bmatrix} $$
We can observe that the two column vectors forming the matrix \(\boldsymbol{A}\) can be seen as the vectors of a new coordinate system \((O, \vec{a_{*,1}}, \vec{a_{*,2}})\) in which we represent the vector \(\boldsymbol{v}\) after it has been transformed by \(\boldsymbol{A}\). In other words, the vector \(\vec{a_{*,1}}\) is what \(\boldsymbol{A}\) does to the standard basis vector \(\vec{i}\), and \(\vec{a_{*,2}}\) is what \(\boldsymbol{A}\) does to \(\vec{j}\).
What’s amusing — and what led to my revelation — is that if, for example, we take \(m = 3\) and assume the two column vectors of \(\boldsymbol{A}\) are linearly independent, then what we intuitively understand is that multiplying \(\boldsymbol{A}\) by \(\boldsymbol{v}\) gives the projection of \(\boldsymbol{v}\) onto a plane in 3-dimensional space. In the same way — and here lies the tragedy — if we set \(m = 1\), we end up projecting our vector onto a single dimension, a line — just like, and this is no coincidence, the dot product.
Thus, in The Three-Body Problem, the aliens use matrices and matrix multiplication as a weapon. So whether humanity can be saved comes down to whether we can find an inverse matrix \(\boldsymbol{A^{-1}}\) (if it exists), which would cancel out the transformation and restore the universe to its glorious dimensions. For example, if \(\boldsymbol{A}\) applied a 45° rotation, then the matrix \(\boldsymbol{A^{-1}}\) would logically be a rotation of -45°.
However, in this case, we lose a dimension. And by losing a dimension, we instinctively understand that we get superpositions that cannot be told apart. You just cannot project the universe onto a piece of paper and preserve its richness, its vibrancy – no offense to writers. More formally, just by saying that \(\boldsymbol{A}\) is a rectangular matrix of size \((1, 2)\), we were already doomed — because an invertible matrix is by definition square, and satisfies \(\boldsymbol{AB} = \boldsymbol{BA} = \boldsymbol{I_{n}}\). Here, the inverse matrix \(\boldsymbol{A^{-1}}\) simply does not exist.
So, can humanity be saved? From my reading and reflection, I’ve concluded that humanity is lost – and linear algebra is to blame. How’s your day?