𝐖𝐡𝐚𝐭 𝐚𝐫𝐞 𝐞𝐢𝐠𝐞𝐧𝐯𝐚𝐥𝐮𝐞𝐬 𝐚𝐧𝐝 𝐞𝐢𝐠𝐞𝐧𝐯𝐞𝐜𝐭𝐨𝐫𝐬?

They sound terrifying, I will explain them in the plain-English version (and with a simple diagram I made):

When you apply a matrix transformation to most vectors, they get rotated and scaled, so they change both direction and length.

But for some special vectors called eigenvectors, the transformation only scales them. It doesn't rotate them at all.

The amount of scaling is the eigenvalue.

That's the whole idea.

𝐖𝐡𝐲 𝐭𝐡𝐞𝐲 𝐦𝐚𝐭𝐭𝐞𝐫 𝐟𝐨𝐫 𝐀𝐈:

→ PCA (dimensionality reduction): eigenvectors of the covariance matrix point in the directions of maximum variance. You keep the ones with the largest eigenvalues, drop the rest.

→ Google's PageRank: the dominant eigenvector of the web's link matrix tells you which pages are most central and authoritative.

→ Stability in neural networks: the largest eigenvalue of a weight matrix (its spectral radius) tells you whether gradients will explode or vanish during training.

→ Recommender systems: SVD is the generalization of eigendecomposition, it decomposes user-item interactions into their underlying "directions of preference."

Every time you see "top components", "latent factors", or "principal dimensions" in data science, someone ran an eigenvalue decomposition (or its close cousin, SVD).

You can use sklearn's PCA without knowing any of this. But you'll never know why 3 components capture 95% of your variance.