Joint Spectral Radius : theory and approximations

Acknowledgements I first would like to thank my promotor Vincent Blondel for accepting me as his first Ph.D student, and providing me with a challenging research subject. His constructive comments, his pragmatism and his initiative were essential in the realization of this thesis. Several researchers contributed to this thesis. I would like to especially thank Alexander Vladimirov and Yurii Nesterov for their important collaborations. I am also very grateful to Fabian Wirth whose knowledge of the subject was of invaluable help. I learned a lot working with him. I am also grateful to Paul Van Dooren for clarifying matrix theoretical problems on several occasions. And I thank Vladimir Protasov for his numerous relevant comments on this work. I am also indebted to the members of my jury for their careful analysis of my work, allowing for a rich feedback on this thesis, which translated into an exciting private defense. I wish every Ph.D. student to get such a feedback on his work. Fund (FNRS) for granting me a fellowship. Two members of the CESAME successively had to stand my sense of humour all day long : Amaury Lendasse and Antoine Vandendorpe. My stay in the CESAME would not have been as pleasant without them. I also want to thank Vincent Canterini for helping me at the beginning of this thesis, and for his friendship. I also want to thank the other members of the CESAME for making it such a nice place (among others, Adrien Leygue, Agnès Provost, Luc Moens, Jonathan de Halleux and Jean-Charles Delvenne). I also thank the secretaries team consisting in Isabelle, Michou, Dominique and Lydia, for making everything so easy to organize. Last but not least, I thank Marie for standing by me throughout these years. Basics N set of nonnegative integer numbers Z set of integer numbers R field of real numbers C field of complex numbers j complex unit, j = √ −1 Re(z) real part of z Im(z) complex part of z ¯ z complex conjugate of z |z| modulus of z x ceiling of x : the smallest integer ≥ x ⇐⇒ if and only if := equal by definition to δ ij Kronecker delta: equals 1 if i = j and zero otherwise · p p-norm (1 ≤ p ≤ +∞) · 2 Euclidean norm (vectors) / spectral norm · 1 1-norm (vectors and induced matrix norm) · 1 ∞-norm (vectors …