# Subdiferencijal svojstvene vrijednosti simetrične matrice

Markek, Goran (2015) Subdiferencijal svojstvene vrijednosti simetrične matrice. Diploma thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

In this master thesis we study the extended notion of diferentiability of real locally Lipschitz functions defined on space $\mathbb{R}^{n}$. In the first part we study the well known directional derivative of convex functions and its properties, some of which are, for instance, boundness, locally Lipschitz continuity and sublinearity. After that, the subdifferential of convex function in $x \in \mathbb{R}^n$ is defined, and we show that it is a nonempty, convex and compact set, which can be empty if the function f is not convex. Next, we study the Clarke’s directional derivative and Clarke’s subdifferential. We list some of their properties and compare them to the usual directional derivative. Similarly to classical analysis, there are calculus rules in nonsmooth analysis for Clarke’s subdifferentials, such as for linear combination of functions, product, quotient and chain rule, but in the case of Clarke subdifferential, only one inclusion can be showed. We also generalize some important results, such as mean value theorem and max-function theorem. In the last section we calculate the subdifferential and directional derivative of the largest eigenvalue of the real symmetric matrix with the help of the results from the previous chapters. Another generalized derivative is defined, called the Michel-Penot’s directional derivative. After defining the Michel-Penot’s subdifferential we show that Clarke’s and Michel-Penot’s subdifferentials coincide for mth largest eigenvalue of a real symmetric matrix.