The second-order derivatives of matrices of eigenvalues and eigenvectors with an application to generalized F-statistic Original Research Article

In this paper we derive the second-order derivatives of an orthogonal matrix of eigenvectors and of a matrix of eigenvalues of a real symmetric matrix. Obtained expressions depend on the first-order derivatives of these matrices, which were presented in Linear Algebra Appl. 264 (1997) 489. These results we use to find the main term of the bias of a generalized F-statistic [Biometrical J. 38 (1996) 5] in the case of normal population. A simulation experiment is carried out, in which we compare the sample mean and unbiased estimator of F-statistic with its asymptotic mean, which was obtained in Linear Algebra Appl. 321 (2000) 27.