3.2 Data Matrix

The general \((n\times p)\) data matrix with \(n\) objects and \(p\) variables can be written as follows:

\[ \begin {bmatrix} x_{11} & \cdots& x_{1j} & \cdots& x_{1p}\\ x_{21} & \cdot \cdots & x_{2j} & \cdots& x_{2p}\\ \vdots & & \vdots & & \vdots \\ x_{i1} & \cdots& x_{ij} & \cdots& x_{ip}\\ \vdots & & \vdots & & \vdots \\ x_{n1} & \cdots& x_{nj} & \cdots& x_{np}\\ \end {bmatrix} \]

the general \(n\times p\) data matrix will be denoted by \(X\). The element in the \(i^{\text {th}}\) row and \(j^{\text {th}}\) column is \(x_{ij}\) this denotes the observation of the \(j^{\text {th}}\) variable on the \(i^{\text {th}}\) object. Using the matrix algebra notation, we write

1.

\(X=(x_{ij})\)

2.

\(\underline {X}'_1, \underline {X}'_2,\ldots ,\underline {X}'_n\) as the rows of \(X\) where \(\underline {X}'_i\) is the \(i^{\text {th}}\) row written as a column.

3.

\(\underline {X}_{(1)},\underline {X}_{(2)},\ldots ,\underline {X}_{(p)}\) as the columns of \(X\). \[X= \begin {bmatrix} \underline {X}'_1 \\ \underline {X}'_2 \\ \vdots \\ \vdots \\ \underline {X}'_n\\ \end {bmatrix} = \begin {bmatrix} \underline {X}_{(1)} & \underline {X}_{(2)}& \cdots& \underline {X}_{(p)}\\ \end {bmatrix} \]

where \(\underline {X}_i= \begin {pmatrix} x_{i1}\\ x_{i2}\\ \vdots \\ \vdots \\ x_{ip}\\ \end {pmatrix}\qquad i=1,2,\ldots n \)

\[\underline {X}_j= \begin {pmatrix} x_{1j}\\ x_{2j}\\ \vdots \\ \vdots \\ x_{nj}\\ \end {pmatrix} \qquad j=1,2,3,\ldots ,p \]

in multivariate analysis the rows \(\underline {X}'_1,\underline {X}'_2,\ldots ,\underline {X}'_n\) usually form a random sample whereas the columns do not.

There are various ways of summarising multivariate data.