Which of the following is true in relation to Principal Component Analysis (PCA)?

An n x n positive definite square matrix will have n eigenvectors, and not n - 1. Therefore statement I is incorrect.

A correlation and covariance matrix are related to each other through the matrix of standard deviations. If the covariance matrix is represented by V, the correlation matrix by C and D is the diagonal matrix of standard deviations, then V = DCD. However, there is no simple relationship between the eigenvalues of the two matrices, and it is not possible to derive the eigenvalues for one given the eigenvalues for the other. Therefore statement II is false.

Principal components are uncorrelated to each other. That is correct, and in fact PCA is useful because of this being so. Statement III is therefore true.

PCA does not explain 100% of the variation in a system with just three components - statement IV is false. (Remember though that most (though not 100%) of the variation in a system of term structures is explained by the first three components - trend, tilt and curvature).

Thus Choice 'd' is the correct answer.