15 - Norms

Abstract

This chapter discusses vector norms, matrix norms, spectral radius of a square matrix, inequalities involving eigenvalues of matrices, inequalities for the characteristic polynomial, named theorems on eigenvalues, and variational principles. Primarily, it describes the general properties and principal of norms. The first group of inequalities relating to the eigenvalues λ and satisfying P(λ)=0 are unnamed, whereas the second group of inequalities are named theorems that apply to the explicit form of the characteristic polynomial P(λ). The named inequalities that are discussed include Parodi's theorem, Corollary of Brauer's theorem, Ballieu's theorem, and Routh–Hurwitz theorem. In the theorems involving eigenvalue inequalities, the elements “aij” of matrix A enter directly, and not in the form of the coefficients of the characteristic polynomial. These include Schur's inequalities, Sturmian separation theorem, Poincare's separation theorem, Gerschgorin's theorem, Brauer's theorem, Perron's theorem, Frobenius theorem, Perron–Frobenius theorem, Wielandt's theorem, Ostrowski's theorem, and first and second theorem due to Lyapunov.