Above, we have generated confidence intervals from the covariance matrix cov( ?) = ? ²( X ^T X) ^{? 1}. While the value of ? ² may be determined by fluctuations in experimental conditions that are beyond our control, we can control, through our choice of experimental design, the design matrix X. We would like to design our experiments so that they provide enough information to estimate the parameters to sufficient accuracy. We now consider the application of eigenvalue analysis of X ^T X to experimental design.

We diagonalize the positive-semidefinite matrix X ^T X = V ? V ^T, where ? = diag( ? ₁, ? ₂, , ? _P), and P = dim( ?). The matrix V is an orthogonal P P matrix, V ^?1 = V ^T, whose columns contain the normalized eigenvectors of X ^T X. The covariance matrix of ? is

where . We see that the small eigenvalues of X ^T X, with the corresponding largest diagonal elements in ? ^{? 1}, contribute the most to the uncertainty.

When is an eigenvalue ? _j satisfying ( X ^T X) v ^{[ j ]} = ? _j v ^{[ j ]} small? Writing

we see that we obtain a small eigenvalue whenever no row in the design matrix has a significant...