We often spend a lot of time calculating, and performing operations with, matrix determinants. I have never quite understood what the concept is actually meant to represent however.
Suppose we have the following matrix with α,δ being integers:
If you multiply a 2D vector (x1,x2) by A, you stretch it horizontally by α and vertically by δ. This linear transformation is equivalent to stretching a quadrilateral corresponding to such vector (with 0, (αx1,0), (δx2,0), and (αx1,δx2) being its vertices). Also det(A)=σδ, which is the scaling of the area of such a polygon, so α and δ are also its eigenvalues.
Here, we have an algebraic structure in algebraic space producing a topology. The algebra we consider as general elements of vectors and matrices, and the topology is the points or lines and so on corresponding to such. With R on the horizontal axis, and i on the vertical axis, we get complex space and consider such space an algebra. With x on the horizontal axis and y on the vertical axis, we get Euclidean space (the classic Cartesian coordinates, where we can get to any point within via some polynomial ring of x and y1).
Topological space is not a sufficient condition for algebraic space, yet algebraic space tends to produce a topology. However, we tend not to describe the algebra as itself topological ontologically. It corresponds with and represents transformations associated with topological objects, yet does not represent such objects (set elements) per se. However, such objects can exist independently of, and without any, algebraic structure. Euclidean (and non-Euclidean) geometries fuse the two concepts, yet we can see ontologically they seem distinct. Also symplectic manifolds have a topology and algebraic structure, yet are not metric spaces, so this adds a further complication to our notion of geometry defined in the usual Euclidean or “smooth” and “curved” manner.
Now consider another matrix:
Det(B)=-1, yet quite clearly we cannot have a negative area. Here, you multiply a vector with B, and you get a reflection along the vertical axis.
Therefore the sign of the determinant contains information on orientation and rotation. Shape rotating (reflecting?) is typically regarded as geometry, yet the operation associated with this is fundamentally algebraic. To see why, I will now introduce the modular group.
Γ, the modular group
Let SL_2(Z) be a set of linear 2×2 matrices, with integer elements. They are special, so their determinants are always 1. In other words, unit matrices. From this, we can derive the following statements:
Let A,B be matrices in SL_2(Z), so det(A)=det(B)=1. Then det(AB)=det(A)det(B)=1 (given linearity, so homomorphic), so we have closure.
Linear matrix multiplication is associative by definition.
Can we find a linear isomorphism with these rules? Certainly! The identity matrix gives det(I)=1×1=1, so we have a linear operation in the matrix elements where f:X→X (a permutation where domain = codomain). Therefore we have an identity. Applying I^(-1) using the usual linear transformations the set is defined by gives det(I^(-1))=1 also, so a linear isomorphism exists. I'm off on a tangent however, and we only need to find an identity which we've got!
If det(A)=1 for elements α,β,γ,δ then αδ-βγ=1. det(A^(-1))=δα-(-β)(-γ)=δα-βγ. This is a linear operation of the matrix elements in R, which by definition is commutative. So δα-βγ = αδ-βγ=1. Therefore there exists an inverse operation.
As such, SL_2(R) forms group operations when its matrices represent linear transformations. However, what are the actual elements forming the group itself?
One could make the argument that algebraic space merely correspond and describe how different abstract elements relate to each other (imposing structure via operation), and the elements the algebra describes form the geometry. In this sense, geometry is a group where there exists a mapping from its associated group algebra to the group elements. So we can link Cartesian coordinates in Euclidean R^2 to the complex set. The former describes the algebra where the points can be described via a polynomial ring, and the latter provides the elements itself.
This only works for R^2 and C however. Locally, an object can be Euclidean and globally not, which gives rise to manifolds in higher dimensions (so generalising this concept to R^n for any real number n), and by relaxing smoothness we get other higher-dimensional polygons. This is where we get from algebra to geometry, and vice-versa, yet I have only a vague sense of this, without formal definition.
Hilbert, in his fourth problem, provides the best formal definition, yet the mapping from algebra to topology is the weak link. In R^2, we can get an isomorphism between C and the Cartesian coordinates. If we can find such in higher dimensions, then of course my working definition of geometry extends to R^n.
Haussdorf measures are one important example of such, given the constant scaling between cardinality measures in one dimension vs another. Yet some objects these describe, such as Kakeya sets, give rise to lines with zero Lebesque measures (removed from the “shapes” or planes with metric structure we typically associate with geometry), which are certainly topologies yet why do we need a distinct concept of geometry to describe such?
In my last post, I built up the intuition that the strong nullstellensatz tells us something about the correspondance between “algebra” and “geometry”, yet this intuition needs sharpening and refining (as ChatGPT would say). As you can probably tell, I'm still learning these concepts myself. Nonetheless I think there exists a fascinating philosophical (metaphysical) question here that mathematicians tend to gloss over. This series is dedicated to providing some possible (directions to) answers for this.