# Fast matrix – vector multiplication

Hi all. This is the first part of my notes on ideas for fast multiplication of matrices with vectors. It’s also the first time that I use the LatexToWordpress tool (mentioned a few posts back). More to follow soon.

Low – rank approximation ideas

2.1. Low – rank approximation

Let’s suppose that the matrix M has rank r. Then, if ${r\leq N}$ we can use the singular value decomposition, writing M as follows,

$\displaystyle M = RSV \ \ \ \ \ (1)$

where R, V are complex and unitary matrices, ${RR^{T}=I=VV^{T}}$, and S is a diagonal matrix with non-zero elements, ${\sigma_{1}\geq \sigma_{2}\geq \sigma_{3}\ldots\sigma_{N}\geq 0}$.

This way we can carry out the multiplication at ${O(N\cdot r)}$ by doing the product ${RSVv}$. Since many matrices have full rank, this technique is of little applicability but produces exact results (up to the floating point error).

2.2. Effective low – rank approximation

In order to be able to use the above ideas in more general cases we do the following. Let the SVD of the matrix M be the same as before. Then we choose a real number ${\epsilon>0}$ and define the effective low – rank approximation of M, ${M_{\epsilon}}$, as the matrix ${RS_{\epsilon}V}$, where ${S_{epsilon}}$ is the diagonal matrix with all the singular values ${\sigma_{i}}$ such that

$\displaystyle \sigma_{i}\geq \epsilon\cdot \sigma_{1} \ \ \ \ \ (2)$

This new matrix ${M_{\epsilon}}$ has rank

$\displaystyle r_{\epsilon}=\#\left\{\sigma_{i}:\sigma_{i}\geq \epsilon\cdot \sigma_{1} \right\} \ \ \ \ \ (3)$

Using this idea, we can carry out the multiplication at ${O(N\cdot r_{\epsilon})}$ with an extra error, due to the approximation.