Η κατά απομάκρυνση ενός τυχαίου τετραγωνικού πίνακα από την κανονικότητα - Κούτση, Φελήσια (Εθνικό Μετσόβιο Πολυτεχνείο. Σχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών. Τομέας Μαθηματικών, 2018)
In this thesis, we concentrate our study in the departure from normality of an arbitrary square matrix A of order n with complex elements as defined by Henrici, which is directly computable only for the Frobenius norm, and will show its significance in estimating other quantities. The departure from normality of matrix A is associated with the concept of the closest normal matrix to A and measures the distance of A from all the normal matrices which have exactly the same eigenvalues as A, considering their multiplicities. Firstly, we mention the Schur Theorem, which is a very helpful tool in the triangulation of matrices and which we will use in some proofs of theorems. Then, we give a bound derived by Henrici for the departure from normality, describe iterated matrices, their norms and some approximate solutions of linear systems. Later, we refer to spectral variation and eigenvalue variation of matrices, which are two concepts that combine the eigenvalues of two different matrices. Also, we describe the field of values of a matrix and give some theorems about the distance of the boundary of the field of values of a non-normal matrix from the convex hull of its eigenvalues. The departure from normality of a matrix is a real scalar that is impractical to compute if a matrix is large and its eigenvalues are unknown. A simple formula is presented for computing an upper bound for departure from normality in the Frobenius norm. This new upper bound is cheaper to compute than the upper bound derived by Henrici. Finally, the best available bounds for departure from normality are given. Moreover, these new lower and upper bounds are sharp for Hermitian matrices, skew-Hermitian matrices and, in general, any matrix with eigenvalues that are horizontally or vertically aligned in the complex plane.