Data

Matrices as Datasets

Another useful way to think of matrices is as observations of Random Variables. For example, if we have two random variables

x

and

y

, with three observations each, we can tabulate our data as

\vec{X} = (\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix})

or even put both variables together as

D = [\begin{matrix} x_{1} & y_{1} \\ x_{2} & y_{2} \\ x_{3} & y_{3} \end{matrix}]

. It is common and convenient to 'center' these data matrices by substracting off the mean for each variable (the vector of means is easily added back in when appropriate. With data arranged this way, our familiar matrix operations become very useful; Our inner product yeilds the variance (a fact used to great effect in the geometry of statistics):

X' \otimes X = (\begin{matrix} (x_{1} - μ_{x}) & (x_{2} - μ_{x}) & (x_{3} - μ_{x}) \end{matrix}) \otimes (\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}) = \sum_{i} {(x_{i} - μ_{i})}^{2} = cov (x)

Extending this to multiple variables brings us into the land of covariance matrices, the fundamental object in applied statistics:

D' \otimes D = (\begin{matrix} x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \end{matrix}) \otimes [\begin{matrix} x_{1} & y_{1} \\ x_{2} & y_{2} \\ x_{3} & y_{3} \end{matrix}] = [\begin{matrix} var (x) & cov (x, y) \\ cov (y, x) & var (y) \end{matrix}]

Note that these covariance matrices are always real, symmetric, and positive semi-definate.