Random musings about academic nightmares and the joys of LLMs
Information overload
I encountered this fellow named Eigen for the first time in a class on Matrix/Matrices. For a square matrix A, a scalar λ is an eigenvalue if there exists a non-zero vector v such that : Av = λv, Av — λv = 0, (A — λI)v = 0 and det(A — λI) = 0 (for a non-trivial solution (v ≠ 0)), blah blah blah…
Then a slew of problems and some twists and turns involving Determinants, Characteristic equations, Linear transformations etc. were enough to zone me out. Unsurprisingly, I never bothered to register anything except for the name “Eigenvalue”. It had a nice ring to it. I remember someone sitting next to me gloating that “Eigen” is a german word that means “proper”. That’s all that I remembered.
A few years later, Professor Abbas Rangwala was teaching us Quantum Mechanics in the lecture halls of Bombay (Mumbai) University. He started off with the regular history of QM, Blackbody radiation, Planck’s law and lo and behold, after a few days what did I year — Eigenvalues are the results of applying a quantum operator (representing a physical observable) to an eigenfunction (or something to that effect).
So now I was being stalked in pairs. The “proper” german had a friend function too! Did I bother to understand what they meant? I tried correlating my earlier definitions in the matrix class to this new mathematical jugglery (I still maintain that’s what it is). No more matrix representations. We were dealing with “simplified” Dirac notation now (Bra-Ket operators) :
Operator|Eigenfunction> = Eigenvalue |Eigenfunction>
There was now an army of stalkers — EigenValue, EigenVector/EigenFunction, Eigenket
In this new notation, the eigenvalue equation for an operator  acting on an eigenket |ψ⟩ is written as Â|ψ⟩ = λ|ψ⟩, where λ is the eigenvalue. The eigenket |ψ⟩ represents the eigenvector (or eigenfunction) of the operator, and λ is the corresponding eigenvalue, a scalar value.
Sorry Dirac, in hindsight while I now recognize why this was considered simplified notation; during my learning years in college, I felt matrices were simpler than terminologies like “Hermitian conjugate” for something called “bra”!! 😂 More than 5 decades of Physics research work was being crammed into our heads. I was reading books by Feynman that had quotes like “nobody understands quantum mechanics”. Honestly, this was very gratifying to know.
With tools like Gemini, ChatGPT, LLaMA at my disposal, I’ve shed all my inhibitions to understand the relevance of “Eigen”. I wish we were not made to drink through a hose pipe in our college years. Saying that it was overwhelming is a gross understatement.
If only someone had then told me in clearer terms (supported by layman’s analogies and contextual deep dive information) that…
in Linear Transformations,
while applying it (like a rotation + stretch) to a vector space, most vectors change direction and length. But eigenvectors are special — they only get stretched or shrunk, not rotated. The eigenvalue tells you by how much.”
“Eigenvalues reveal the invariant properties of a matrix. They tell you what directions the transformation acts upon most strongly or weakly.”
in Stability Analysis
In differential equations and systems dynamics:
- If all eigenvalues of a system matrix have negative real parts, the system is stable.
- If any eigenvalue has a positive real part, the system is unstable.
in Principal Component Analysis (PCA)
eigenvalues of the covariance matrix tell you how much variance each principal component (direction) captures.
in Quantum Mechanics
eigenvalues of an operator (like the Hamiltonian) correspond to measurable quantities like energy levels.
in Graph Theory
eigenvalues of the adjacency matrix or Laplacian matrix of a graph give insights into connectivity, clustering, and dynamics on the graph.
…I would’ve befriended Mr. Eigen then and there. Well, better late than never :)
Leveraging LLMs to get back to those questions that you always wanted to ask (but shied away from asking) is a pure joy that kids can experience today. This is even more enjoyable for independent/self-paced learners. It truly has the power to put the joy back into learning. Someone with no ego patiently answering all your queries in science! Just Wow! \[That’s ok. Humans hallucinate in worse ways possible\] Unfortunately everything that’s feature rich is taken for granted and easy to abuse. The rising costs of education and the consequential conversion of education institutions into shameless business centers is inevitable. So LLMs had to be invented to make education inclusive. I won’t be surprised if educational institutions get shut down (over a few decades) — they will be better off as social centers for discussions, apprenticeship, sponsored experimentation and similar activities. Greedy businessmen cannot predict their next course. Only a visionary can.
Ende gut, alles gut, was sagt Herr Eigen?
#physics #quantummechanics #llm #genai #learning #eigenvalues #eigenvectors #eigenfunctions #eigenkets #practicallearning