MIT讲师用线性代数揭示股票隐藏的独立因子

Dr. Choongbum Lee, MIT lecturer:

take just 5 stocks and record their prices for 365 days. that is 1,825 numbers staring back at you, but linear algebra asks a much more dangerous question: how many genuinely independent things are actually happening underneath them? because if those stocks move together, hundreds of columns of data may be repeating the same few hidden directions.

this is what eigenvectors and SVD are built to expose. in Lee's hand-worked example, a matrix produces eigenvalues of 25, 9 and 0, which become singular values of 5, 3 and 0. the zero direction carries no information and can be deleted completely; six visible numbers are being reorganized into the few directions where the data actually has movement.

scale that idea to the 5-stock example and the difference becomes absurd. a full decomposition can introduce a 365 × 365 matrix 133,225 entries - even though the original data can have rank no greater than 5. the reduced SVD throws away the dimensions where nothing independent is happening and keeps only the structure capable of explaining the data.

and this is where linear algebra stops being classroom mathematics. build a matrix from stock data, multiply it into the right form, and the eigenvectors begin grouping companies by how they move together. instead of watching 100 charts independently, you start asking whether those 100 assets are really just different combinations of 3, 5 or 10 underlying forces.

the lecture is free. 1,825 price observations can hide only a handful of meaningful directions, a 365 × 365 block can become irrelevant, and a matrix that looks like a table of historical prices can secretly contain the correlation structure of an entire market. the edge is not collecting more data; it is finding how little of that data is actually independent.