Eigenspaces, measurement and Physics
A chat I had with ChatGPT5 about eigenspaces measurement and physics. You can scan through it if you like: What can we do with eigenspaces? ChatGPT said: Eigenspaces may sound abstract, but they are work-horses of applied math, data science, and engineering. Given a linear transformation 𝑇 : 𝑉 → 𝑉 T:V→V (or a square matrix 𝐴 A), an eigenspace is the set of all vectors that are simply stretched or compressed (scaled) by 𝐴 A, not rotated into some new direction. Below are the main things we do with them—ranging from pure theory to very practical computations. 1. Decompose and Simplify Linear Systems Diagonalization / Spectral Decomposition If 𝐴 A has a basis of eigenvectors, 𝐴 = 𝑃 𝐷 𝑃 − 1 A=PDP −1 with 𝐷 D diagonal. → Solving 𝐴 𝑘 𝑣 A k v or 𝑒 𝐴 𝑡 e At becomes trivial (just scale each eigenvector). Differential equations Systems 𝑥 ˙ = 𝐴 𝑥 x ˙ =Ax have solutions 𝑥 ( 𝑡 ) = ∑ 𝑐 𝑖 𝑒 𝜆 𝑖 𝑡 𝑣 𝑖 x(t)=∑c i e λ i t v i . Each eigenspace de...