Tridiagonalizing random matrices

Tridiagonalizing Complex Symmetric Matrices in Waveguide Simulations

We discuss a method for solving complex symmetric (nonHermitian) eigenproblems Ax = λBx arising in an application from optoelectronics, where reduced accuracy requirements provide an opportunity for trading accuracy for performance. In this case, the objective is to exploit the structural symmetry. Consequently, our focus is on a nonHermitian tridiagonalization process. For solving the resultin...

On Tridiagonalizing and Diagonalizing Symmetric Matrices with Repeated Eigenvalues

We describe a divide-and-conquer tridiagonalizationapproach for matrices with repeated eigenvalues. Our algorithmhinges on the fact that, under easily constructivelyveriiable conditions,a symmetricmatrix with bandwidth b and k distinct eigenvalues must be block diagonal with diagonal blocks of size at most bk. A slight modiication of the usual orthogonal band-reduction algorithm allows us to re...

Random Matrices and Random Boxes

OF THE DISSERTATION Random matrices and random boxes by Linh V. Tran Dissertation Director: Prof. Van Vu This thesis concerns two questions on random structures: the semi-circular law for adjacency matrix of regular random graph and the piercing number for random boxes. • Random matrices We proved in full generality the semi-circular law for random d-regular graph model in the case d tends to i...

Random Growth and Random Matrices

We give a survey of some of the recent results on certain two-dimensional random growth models and their relation to random matrix theory, in particular to the Tracy-Widom distribution for the largest eigenvalue. The problems are related to that of finding the length of the longest increasing subsequence in a random permutation. We also give a new approach to certain results for the Schur measu...

An Introduction to Random Matrices An Introduction to Random Matrices