This is a theory about how human artistic expression styles can be modelled by the concepts of linear transformation and eigenvectors from linear algebra.

Background

Much work is done in mathematics to create methods that uncover useful characteristics about the things we want to analyze. These things could be data, objects, functions, etc. Applying these methods that expose unique qualities enhances our analytical prospects at better understanding these things. For example, in statistics, properties like median and variance provide information about a set of data. We can then use them to describe the data, make inferences, deduce some meaning, or as variables in another more complex function.

The most useful characteristics expose a non-apparent, complex intrinsic property of the thing at hand, and are succinct and easily digestible to analyze further. They help us realize intrinsic qualities and analyze the inner workings of the thing.

In the field of linear algebra, linear transformations have an extraordinarily useful characteristic called the “eigenvector”. Along with “eigenvalues”, these provide information regarding the intrinsic characteristics of the transform itself. Fully understanding eigenvectors is not a prerequisite to understanding the basics of this theory, but it certainly would help.

Eigenvectors and Eigenvalues

A basic example.

Say I had a stretchable piece of paper lying flat on a table. Say that I owned a machine that has one function: it stretches a piece of stretchy paper to have double (2x) its length, but the width stays the same (and of course, the whole paper stays flat on the table). This machine is the “linear transform”. This transform has these two characteristics: it has an eigenvector in the direction of the length, and an eigenvalue of 2. The eigenvector describes the direction in which the transform (the machine) has stretched something, but not rotated it. The eigenvalue is how much stretching occurs - in this case, the length is doubled, so the eigenvalue is 2.

These two values describe the characteristics of the transform. I could say “my linear transformation here has an eigenvector in the y-direction, and an eigenvalue of 2, and those two pieces of information alone would be enough to essentially understand the transformation. Anything that is transformed by this machine, gets stretched in the y-direction, at a magnitude that doubles its initial measurement in that direction.

(Not to get too confusing, but this example also has an eigenvector along the width axis too, and the eigenvalue is 1, meaning that the width doesn’t change.)

Self-Expression Eigenvector Theory

I would argue that the main idea behind eigenvectors is found in everyday life, particularly in forms of self-expression through writing, art, music, etc. Art not only expresses the artist’s view of particular properties of the real world, but also their personal inner workings, biases, worldview, etc. It is no surprise the level of analysis that is done on artwork, but it is not only about what it is representing, but also the artist themself and how they are expressing the characteristics of their inner world to the outside.

Taking my example above, consider the artist’s work as the work of a transformation machine - but one that is mental, and not physical. They are taking concepts, experiences, objects, etc. from the real environment, and transforming it into artwork. Their style holds across different pieces of art, and in that style lies intrinsic characteristics about themself.

Main idea:

The action of taking a real-world experience and converting it into art is an action of transformation
This transformation process includes biases, predispositions, and inner workings of the artist’s brain
By knowing the details of the real-world experience, and the end artistic result, we can deduce the characteristics of the artist’s inner workings
Eigenvectors and eigenvalues can be thought of as intrinsic psychological qualities of the artist (eigenvectors), and how much the artist magnified these traits (eigenvalues)

Here I am considering the idea that the eigenvectors (the artist’s style based on their brain) will not necessarily change after each piece of art created, however, the eigenvalues (their magnification of certain style components) may vary. This makes sense per the mathematics side of things too, as the eigenvectors are really what determine how the transformation works (what it actually does to the original matrix), whereas the eigenvalues are a measure of how much the eigenvectors will transform the original matrix from its original state.

Understanding the Formula

Mathematically, linear transforms operate on matrices. A matrix is just a raw tabular data format without column or row names. The formula of transformation is literally just:

[MATRIX] x [TRANSFORM] = [TRANSFORMED MATRIX]

In my example, this would be [paper size] x [transformation machine] = [new paper size]

If we know the original MATRIX, and the TRANSFORMED MATRIX, we can solve for what the transform did to the original. In my example, we could deduce that the paper length was doubled by the machine, even if we had no access to the machine, and only saw the paper before and after it was transformed.

It’s quite literally just asking “what changed between the original and the new”, the same way we can ask “what is different between the events in the real world, and the events as discussed in a piece of writing”.

On a deeper level, this deduction of characteristics of the transformation machine (the artist) can expose the realm of an artist’s state of mind, intrinsic qualities, values, biases, predispositions etc. This is all part of art analysis, regardless of the medium or form. It’s a process of figuring out “who the artist is”, beyond the artwork itself.

Consider:

A person lives through some situation in the real world (modelled by a huge MATRIX of details)
This person transforms their experience of the situation (TRANSFORM) into some artistic medium - a TRANSFORMED MATRIX with many details
This person’s inner workings can be unveiled because we can “solve” for the characteristics of their transform by analyzing the difference between the matrices. We can tell “who they are” by the way they express themselves in the artwork. We can deduce the characteristics of the “machine” (the artist) that did the transforming, just like how we could deduce what the machine did (double the length of the paper).

Naturally, some artwork is much more static than representing an entire experience. This still fits within the formula, as the person may simply observe something in the real world, and transform it into art. Any art, regardless how simple, shows the artist’s style and characteristics.

Abstracting the Formula

Now, we can abstract the matrix transformation formula into the realm of human activities.

[A detailed place/event/experience/thing]
x
[An artist's transformation]
=
[The artist's artwork]

Here, “An artist’s transformation” has characteristics similar to eigenvectors and eigenvalues. The eigenvectors represent the artist’s innate mental qualities that are not going to change across various works of art, and the eigenvalues represent how much they have magnified these qualities in a piece of artwork. An artist’s style, of course, could change slowly over time, but it will still show the qualities of the artist at the time of the work.

The formula may be relatively concrete, but the theory is far from complete. The right words for “innate mental qualities” and “magnification” within the realm of artistry and psychology are going to be tough to find and apply to this situation.

Regardless, some forms of art are easier to pick up on the details of the artist. In the case of writing as artwork, bias is fairly easy to spot if the facts are understood by the reader. With paintings or music, it is most definitely more nuanced to understand the painter’s mentality, and requires a much more experienced viewer to pick up on it.

Wrapping Up

An artist’s articulation of the real world into an artistic medium (transformation) follows the axes of that artist’s mentality, opinions, worldview, biases, etc. (eigenvectors) at varying intensities (eigenvalues).

Quite literally, people take a situation, and in the process of sharing it, transform it, while simultaneously magnifying certain components, which reflects their brain’s inner workings. No matter what the artist does, or what medium they use, their intrinsic qualities will always be present. Just like how no matter what is being transformed, a transformation matrix maintains the same core characteristics - eigenvectors.

I have no idea if someone else has thought this before and quite frankly don’t even know what to search to try to figure that out. My explanations here need refinement to be clearer; this post will probably be updated.

Daniel