Hidden North Stars: On Eigenvalues & Eigenvectors

“Read between the lines.”

“In an ever changing world/landscape.”

“In a nutshell..”

Clichés.

Being exposed to a term or phrase so much it actually loses the original impact or meaning when it gets repeated or used in everyday writing or speech (or even in music and movies, really).

Calling something cliché has, perhaps, even reached the point that it’s now cliche itself (don’t think about that too long).

Whatever clichés you have in your own personal lexicon, you likely still use them from time to time even when you know they feel flat and other word choices could be more eloquently used in different contexts.

Boring, unoriginal – generic even – phrases that despite their blandness still hold an important place and are cemented in our minds.

So why do they persist?

There could be many reasons here, but largely they form a “north star” of sorts in our everyday language.

You know what the phrases mean or infer, along with everyone else due to exposure over time — so it’s a shared, quantized – if boring – meaning space for communicators, more or less.

While context can certainly change the meaning of some clichés, they slowly become rigid placeholders in a constantly evolving language set, making them reliable anchors or shortcuts in many situations.

They each have their own shared unique characteristic – an “eigen” in a sense in our language space.

“Eigen”, from the German language means “own” or “characteristic” or.. “proper” – some special property or meaning belonging to the thing you’re describing.

The first time I encountered the word “Eigen” was not in any German studies, but like much of this site’s theme, I encountered it in mathematics – specifically in linear algebra.

Eigenvalues and eigenvectors – which I’ll go into more details below – play a vital role in many areas of mathematics, from linear algebra to graph theory to extensions in topology and geometry.

They also serve as anchors across quantum physics, thermodynamics, electromagnetism and other physical sciences.

Much like clichés, eigenvalues and eigenvectors form a shared meaning space between fields – loosely speaking representing the same core idea across many different branches of study, even if the context around it changes or transforms it – it’s pointed in the same direction.

Eigenvalues and Eigenvectors Serve As North Stars

In all fields, studying how environments (and the objects in the environments) change over time is ubiquitous, so looking for eigenvalues and eigenvectors is a way to identify landmarks to use as reference points in “an ever changing landscape” of the environment, so to speak.

Another, perhaps, cliché phrase “north star” is another way to describe them.

Navigators use the north star, Polaris, as a fixed, stable star in the sky to help guide their routes, as other stars move as the earth turns.

A generic, static landmark that has a shared meaning across all cultures – unchanging anchors in dynamic environments.

A (Semi)Formal Look At Eigenvalues and Eigenvectors In Linear Algebra

The most natural place where one encounters eigenvalues and eigenvectors is in linear algebra.

If you take some vector “v” and apply some linear transformation to it (multiplying by the square n x n matrix “A”, the result is just a scaled version (stretching or shrinking) of “v”, with λ being the amount the original vector is scaled by:

Av = λv

v is called an eigenvector of A and the scaling factor λ (some number/scalar) is called the eigenvalue.

This essentially means that when v is multiplied by A, v maintains its direction – A only stretches or shrinks v (by the numerical factor λ).

Eigenvectors and associated eigenvalues of some matrix are not always unique – there can be many, in fact, but they all hold the same properties regardless: holding the direction characteristic constant, or unchanged through transformation.

They are special – even if boring under transformation – objects in the language of mathematics.

Eigenvalues and Eigenvectors In Graph Theory

Last week I went through a short brief on graph theory – concepts like vertices and the edges that connect them to form a graph and the trajectory of traversal through the graph with walks, paths and trails.

Graph theory (a hidden north star on its own in the world) also gives way to the concept of eigenvalues and eigenvectors.

Looking for the eigenvectors of a particular graph means looking for the most important vertices – or nodes – of the graph.

There’s much, much more to eigenvectors in graph theory, but finding them is important for generalizing the importance of influence of any particular vertex (or vertices) in the graph — traversing the graph (a walk) means ultimately ending up (or passing through) the vertex (node) associated to the eigenvector.

Final Takeaway For SEOs & Marketers

Building intuition.

With search & AI/LLM spaces, we need as many anchors or “cliché” terms to build up some core ways to interpret how they behave and evolve over time. Eigenvectors and associated eigenvalues play the role of “meta-anchors”, in a way – both defining the anchors we need and existing as anchors as we continue to find better ways of measuring these spaces.

“In a nutshell”, even in the most complex systems and networks we study we find eigens — when we “read between the lines”.

It might be a bit cliché to say, but there’s a beauty in the boring.