User:SirMeowMeow/sandbox/SVD

= Singular Value Decomposition = Any real or complex matrix of size $$c \times r$$ may be decomposed into the triple product $$\mathrm{U} \mathrm{\Sigma} \mathrm{V}^*$$, where $$\mathrm{U}$$ is $$c \times c$$ and orthonormal, $$\mathrm{\Sigma}$$ is $$r \times r$$ is a positive-definite real diagonal matrix, and $$\mathrm{V}^*$$ is $$r \times r$$ and orthonormal.

= Example = Let $$\Phi$$ be the matrix below. Let every row of the matrix correspond to an individual observation, and let every column in the matrix represent different "dimensions" of the observation. For example, let every row in this matrix correspond to a user of a music service, and let every column represent a different song on the service, and let the weight correspond to how much that particular user enjoyed a song.$$\begin{bmatrix} 10 & 8 & 0 & 0 & 0 & 0 \\ 7 & 6 & 8 & 0 & 0 & 0 \\ 9 & 8 & 7 & 4 & 3 & 0 \\ 7 & 6 & 8 & 0 & 0 & 0 \\ 0 & 0 & 0 & 8 & 9 & 8 \\ \end{bmatrix}$$It is conceivable that data representation for such services will always experience sparse data sets because the majority of their customers will only sample a very small minority of songs.

The SVD decomposition of this matrix will give us a triple product of matrices popularly named $$\mathrm{U} \mathrm{\Sigma} \mathrm{V}^*$$. For the orthonormal matrix $$\mathrm{U}$$,