Some important differentials

This chapter discusses the differentials of some scalar functions of X (eigenvalue, determinant), a vector function of X (eigenvector), and some matrix functions of X (inverse, Moore-Penrose (MP) inverse, adjoint matrix). There are two problems involved in differentiating eigenvalues and eigenvectors. The first problem is that the eigenvalues of a real matrix A need not, in general, be real numbers — they may be complex. The second problem is the possible occurrence of multiple eigenvalues. When employing arguments that require limits such as continuity or consistency, some care is required when dealing with eigenvectors and associated concepts. Because of the symmetry of A , all its eigenvalues are real and they are uniquely determined. However, eigenvectors are not uniquely determined, not even when the eigenvalue is simple. Also, while the eigenvalues are typically continuous functions of the elements of the matrix, this is not necessarily so for the eigenvectors.