This is the unit's data-science payoff topic — the point where the fully-proved SVD (Topic 6) is put to work. Three results, each building on the last: the Eckart-Young theorem, which says truncating the SVD gives the best possible low-rank approximation to a matrix in a precise sense; the Moore-Penrose pseudoinverse, which generalises matrix inversion to rectangular or singular matrices via the SVD; and the full PCA derivation, closing a loop opened all the way back in LA-I's PCA-first-look note, which explicitly deferred the complete eigenportfolio construction to this unit.
By the end of this topic you should be able to:
- state and prove the Eckart-Young theorem;
- construct the Moore-Penrose pseudoinverse and use it to solve rank-deficient least-squares problems;
- derive PCA's variance-maximisation property from the SVD, choosing components;
- describe image compression and latent semantic analysis as direct Eckart-Young applications.
In this lesson
- Contents
- The big picture
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