## Doctoral Theses

### Some Problems in Differential and Subdifferential Calculus of Matrices.

5-28-2014

5-28-2015

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Delhi)

#### Supervisor

Bhatia, Rajendra (TSMU-Delhi; ISI)

#### Abstract (Summary of the Work)

A central problem in many subjects like matrix analysis, perturbation theory, numerical analysis and physics is to study the effect of small changes in a matrix A on a function f(A). Among much studied functions on the space of matrices are trace, determinant, permanent, eigenvalues, norms. These are real or complex valued functions. In addition, there are some interesting functions that are matrix valued. For example, the (matrix) absolute value, tensor power, antisymmetric tensor power, symmetric tensor power.When a function is differentiable, one of the ways to study the above problem is by using the derivative of f at A, denoted by Df(A). In order to obtain first order perturbation bounds, it is helpful to have information about kDf(A)k. In general, finding the exact value of the norm of any operator is not an easy task. It might be easier and adequate to find good estimates on kDf(A)k. Higher order perturbation bounds can be obtained using the norms of the higher order derivatives.Some interesting functions like norms are not differentiable at some points. But they possess the useful property of being convex. In such a case, the notion of subderivative is used in place of the derivative.This thesis consists of two parts. In one of them, we study (higher order) derivatives of the maps that take a matrix to its kth tensor power, kth antisymmetric tensor power and kth symmetric tensor power. We obtain explicit formulas for these derivatives and compute their norms. We also obtain expressions for the map that takes a matrix to its permanent. In the other part, we study the subdifferentials of norm functions and use them to investigate Birkhoff-James orthogonality in the space of matrices. These results are then applied to obtain some distance formulas. Such formulas have been of interest to many mathematicians.Let M(n) denote the space of nÃ—n complex matrices. Let A(i|j) denote the (nâˆ’1)Ã—(nâˆ’1) submatrix obtained from A by deleting its ith row and jth column. Let det : M(n) â†’ C be the map that takes a matrix A to its determinant. This map is differentiable and the famous Jacobi formula gives its derivative as D det(A)(X) = tr(adj(A)X), (0.1)

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843053

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