2 Transformations of Data Matrices

Matrix algebra provides a natural notation for transformations. If we premultiply the matrix $_{i}{\bf B}_{j}$ by another, say $_{k}{\bf T}_{i}$, then the rows of ${\bf T}$ describe linear combinations of the rows of ${\bf B}$. The resulting matrix will therefore consist of $k$ rows corresponding to the linear transformations of the rows of $\bf B$ described by the rows of ${\bf T}$. A very simple example of this is premultiplication by the identity matrix, I, which, as noted earlier, merely has 1's on the leading diagonal and zeroes everywhere else. Thus, the transformation described by the first row may be written as `multiply the first row by 1 and add zero times the other rows.' In the second row, we have `multiply the second row by 1 and add zero times the other rows,' and so the identity matrix transforms the matrix B into the same matrix. For a less trivial example, let our data matrix be ${\bf X}$, then

$${\bf X}^{\prime} = \left( \begin{array}{rrrrr} -2 & -1 & 0 & 1 & 2 \\ -4 & 2 & 0 & -2 & 4\ \end{array} \right)$$

and let

$${\bf T} = \left( \begin{array}{rr} 1 & 1\ 1 & -1 \end{array} \right)$$

then

$$\begin{eqnarray*} {\bf Y^{\prime}} &=& {\bf TX}^{\prime}\\ &=& \left( \begin{... ... 1 & 0 & -1 & 6 \\ 2 & -3 & 0 & 3 & -2\ \end{array} \right). \end{eqnarray*}$$

In this case, the transformation matrix specifies two transformations of the data: the first row defines the sum of the two variates, and the second row defines the difference (row 1 $-$ row 2). In the above, we have applied the transformation to the raw data, but for these linear transformations it is easy to apply the transformation to the covariance matrix. The covariance matrix of the transformed variates is

$$\begin{eqnarray*} \frac{1}{N-1} {\bf Y^{\prime}Y} &=& \frac{1}{N-1} {\bf (TX^{\... ...ime}XT^{\prime}\\ &=& {\bf T}({\bf V_{x}}){\bf T}^{\prime}\\ \end{eqnarray*}$$

which is a useful result, meaning that linear transformations may be applied directly to the covariance matrix, instead of going to the trouble of transforming all the raw data and recalculating the covariance matrix.