Necessarily, $p \leq \min(m,n)$.

In much of this lecture, we'll think of $X$ as a matrix of **data** in which

In much of this lecture, we'll think of $X$ as a matrix of data in which

* each column is an **individual** -- a time period or person, depending on the application

In the $ m < < n$ case in which there are many more individuals $n$ than attributes $m$, we can calculate sample moments of a joint distribution by taking averages across observations of functions of the observations.

In this $ m < < n$ case, we'll look for **patterns** by using a **singular value decomposition** to do a **principal components analysis** (PCA).

In this $ m < < n$ case, we'll look for patterns by using a singular value decomposition to do a principal components analysis (PCA).

In the $m > > n$ case in which there are many more attributes $m$ than individuals $n$ and when we are in a time-series setting in which $n$ equals the number of time periods covered in the data set $X$, we'll proceed in a different way.

We'll again use a **singular value decomposition**, but now to construct a **dynamic mode decomposition** (DMD)

We'll again use a singular value decomposition, but now to construct a **dynamic mode decomposition** (DMD)

## Singular Value Decomposition