Sean \(m+1\) variables, con \(n\) observaciones cada una:
\[ \begin{matrix} y & x_{1} & x_{2} & \cdots & x_{m} \\ \hline y_1 & x_{11} & x_{21} & \cdots & x_{m1} \\ y_2 & x_{12} & x_{22} & \cdots & x_{m2} \\ \cdots &\cdots & \cdots & \cdots & \cdots \\ y_n & x_{1n} & x_{2n} & \cdots & x_{mn} \\ \end{matrix} \]
se define la covarianza entre las variables \(x_j\) e \(y\) como:
\[ cov(x_{j},y) = S_{jy} = \frac{\sum_{i=1}^{n}(x_{ji} - \bar x_j)(y_i - \bar y)}{n-1}, \quad j \in [1,m], \quad i \in [1,n] \]
y la covarianza entre las variables \(x_j\) e \(x_k\) como:
\[ cov(x_{j},x_{k}) = S_{jk} = \frac{\sum_{i=1}^{n}(x_{ji} - \bar x_j)(x_{ki} - \bar x_{k})}{n-1}, \quad j,k \in [1,m], \quad i \in [1,n] \]
Se define la matriz de covarianzas para \(x_j\) y \(x_k\) como:
\[ S_{xx} = \begin{bmatrix} S_{11} & S_{12} & \cdots & S_{1m} \\ S_{21} & S_{22} & \cdots & S_{2m} \\ \cdots & \cdots & \cdots & \cdots \\ S_{m1} & S_{m2} & \cdots & S_{mm} \\ \end{bmatrix} \]
Se define la matriz de covarianzas entre \(x_j\) e \(y\) como:
\[ S_{xy} = \begin{bmatrix} S_{1y} \\ S_{2y} \\ \cdots \\ S_{my} \\ \end{bmatrix} \]