Computational aspects

In the validation phase $n$ linear systems of dimension $n-1$ need to be solved. The system matrices are obtained by dropping one row and one complumn of the complete kriging matrix. This can be efficiently accomplished by means of intersections of $n-1$-dimensional lines with apporpriate coordinate $n$-dimensional planes.

Note that the krigin matrix $C$ is symmetric, and thus its eigenvalues $\lambda_i$ are real. However, since

$$\sum_{i=1}^n\lambda_i =$$   Tr$$(C) = \sum_{i=1}^n c_{ii} = 0$$

where Tr$(C)$ is the trace of matrix $C$, it follows that some of the eigenvalues must be negative and thus $C$ is not positive definite. For this reason, the solution of the linear systems is usually obtained by means of direct methods, such as Gaussian elimination or Choleski decomposition. Full Pivoting is often necessary to maintain stability of the algorithm.