<filename>svd-intuit-fcomp.tex
\subsection{SVD as a function composition}

The first thing to remember, is that matrices are the operational
representation of an special type of function between vector
spaces\footnote{The reader is invited to review any Linear Algebra
  textbook, to recall the definition of a vector space}
called linear transformations (also called linear mappings or linear
  functions). We say is an operational representation, in the 
sense that they provide an explicit recipe to apply the
function. Furthermore, the functions they represent are
special, as they have the nice property of preserve algebraic
structure across domain and codomains. Such 
property can be summarized as: \\

\[
f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)
\]
\hfill

Where the addition and products mentioned above, are the vector
addition and multiplication by an scalar; defined for vector
spaces. For the specific case of this work, where we restrict
our attention to real matrices, we can tell that they do represent
functions $f: \R{n} \fromto \R{m}$. \\

In this context of linear functions, the matrix multiplication is the
operational representation of the composition of the associated
functions. A matrix factorization is in essence, a way to understand
what the underlying function does; the whole product can be seen as
serial algorithm, where each matrix represents on particular step or
transformation. Each of the four matrices that appear on the SVD
factorization, has its own function as follows:  

\begin{itemize}
\item $A$ is a function $\func{F_A}: \R{n} \fromto \R{m}$
\item $V$ is a function $\func{F_V}: \R{n} \fromto \R{n}$
\item Same goes for  \trans{V}, which is $\func{F_{\trans{V}}}: \R{n} \fromto \R{n}$
\item $\Sigma$ is a function $\func{F_\Sigma}: \R{n} \fromto \R{m}$
\item $U$ is a function $\func{F_U}: \R{m} \fromto \R{m}$
\end{itemize}
\hfill

Thus, in the context of function compositions, the SVD factorization
can be restated as: \\

\[
\func{F_A}(\vec{x}) = \func{F_U}(\func{F_\Sigma}(\func{F_{\trans{V}}}(\vec{x})))
\]
\hfill
