preface i a review of linear algebra i.1 vector spaces and inner product spaces i.2 linear operators and matrices i.3 direct sums i.4 tensor products i.5 symmetry classes i.6 problems i.7 notes and references ii majorisation and doubly stochastic matrices ii.1 basic notions ii.2 birkhoff's theorem ii.3 convex and monotone functions ii.4 binary algebraic operations and majorisation ii.5 problems ii.6 notes and references iii variational principles for eigenvalues ili.1 the minimax principle for eigenvalues iii.2 weyl's inequalities iii.3 wielandt's minimax principle iii.4 lidskii's theorems iii.5 eigenvalues of real parts and singular values iii.6 problems iii.7 notes and references iv symmetric norms iv. 1 norms on cn iv.2 unitarily invariant norms on operators on cn iv.3 lidskii's theorem (third proof) iv.4 weakly unitarily invariant norms iv.5 problems iv.6 notes and references v operator monotone and operator convex functions v.1 definitions and simple examples v.2 some characterisations v.3 smoothness properties v.4 loewner's theorems v.5 problems v.6 notes and references vi spectral variation of normal matrices vi.1 continuity of roots of polynomials vi.2 hermitian and skew-hermitian matrices vi.3 estimates in the operator norm vi.4 estimates in the probenius norm vi.5 geometry and spectral variation: the operator norm vi.6 geometry and spectral variation: wui norms vi.7 some inequalities for the determinant vi.8 problems vi.9 notes and references vii perturbation of spectral subspaces of normal matrices vii.1 pairs of subspaces vii.2 the equation ax - xb = y vii.3 perturbation of eigenspaces vii.4 a perturbation bound for eigenvalues vii.5 perturbation of the polar factors vii.6 appendix: evaluating the (fourier) constants vii.7 problems vii.8 notes and references viii spectral variation of nonnormal matrices viii.1 general spectral variation bounds viii.4 matrices with real eigenvalues viii.5 eigenvalues with symmetries viii.6 problems viii.7 notes and references ix a selection of matrix inequalities ix.1 some basic lemmas ix.2 products of positive matrices ix.3 inequalities for the exponential function ix.4 arithmetic-geometric mean inequalities ix.5 schwarz inequalities ix.6 the lieb concavity theorem ix.7 operator approximation ix.8 problems ix.9 notes and references x perturbation of matrix functions x.1 operator monotone functions x.2 the absolute value x.3 local perturbation bounds x.4 appendix: differential calculus x.5 problems x.6 notes and references references index