Higher order derivatives and norms of certain matrix functions

Tese de doutoramento, Matemática (Álgebra Lógica e Fundamentos), Universidade de Lisboa, Faculdade de Ciências, 2014

In this thesis we obtain formulas for higher order directional derivatives for the immanant map, which is a generalization of the determinant and the permanent maps. We also obtain formulas for the k-th derivative of the m-th Х-symmetric tensor power of an operator or a matrix, where Х is an irreducible character of the permutation group Sm. Moreover, we calculate the operator norm of these derivatives. We start by presenting some general concepts of multilinear algebra, representation theory and matrix analysis, in particular some results about characters of Sm and differential calculus applied to matrix functions, which will be useful throughout this work. We also present some well known results about the immanant map, as well as other results such as the generalized Laplace expansion for immanants. The starting point of this kind of problems is the famous Jacobi formula obtained in the 19th century by Carl Jacobi. This formula gives us the first order derivative of the determinant function. In recent work, R. Bhatia, T. Jain and P. Grover [9], [8] presented us formulas for higher order derivatives of the determinant and the permanent maps and also the expressions for the derivatives of the symmetric and antisymmetric tensor powers. These maps are all particular cases of the immanant and the Х symmetric tensor power, when Х is a linear character of Sm, namely the principal and alternating characters. The general case has much more complicated features and needs some new concepts and notations, which play a very important role throughout our work. We also study the norm of these higher order derivatives. This problem was first addressed by R. Bhatia and S. Friedland in [7] where they proved a formula for D ^m (A). This result has been extended in two different directions. In [9], R. Bhatia and T. Jain study the case for higher order derivatives and they obtain a formula for the norm of Dk ^m (A), whereas in [6], R. Bhatia and J. A. Dias da Silva demonstrate a formula for the norm of the first derivative for all symmetry classes. In our work we obtain a result that subsumes all the previous expressions