# Intro to Singular Value Decomposition

### Singular Value Decomposition (SVD)

SVD is a foundational technique in Machine Learning. This series approaches the SVD from a few view points to build more useful intuition what is actually happening behind the equations.

The equation behind SVD is a very elegant mathematical statement, but also a very “thick” one as well.

The SVD of a matrix is defined as

$X = U S V^T \\ (n \times m) = (n \times k) (k \times k) (k \times m) \\ n: \text{ number of rows/data points} \\ m: \text{ number of columns/features} \\ k: \text{ number of singular vectors}$

In data problems, $$X$$ is typically the dataset, and the SVD helps create a more “condensed” representation. Although a simple statement to state, it is not so simple to understand what this statement gives us.

As a data scientist, I prefer much more the following way to write the SVD:

$X V_{\text{weighted}} = U \\ \text{where } V_{\text{weighted}} = V W \\ \text{and } W \text{ is a matrix that weights the columns of V}$