Advanced Matrix Theory

Eigenvalues, eigenvectors and similarity, Unitary equivalence and normal matrices, Schur’s theorem, Spectral theorems for normal and Hermitian matrices; Jordan canonical form, Application of Jordan canonical form, Minimal polynomial, Companion matrices, Functions of matrices; Variational characterizations of eigenvalues of Hermitian matrices, Rayleigh-Ritz theorem, Courant-Fischer theorem, Weyl theorem, Cauchy interlacing theorem, Inertia and congruence, Sylvester's law of inertia; Matrix norms, Location and perturbation of eigenvalues Gerschgorin disk theorem; Positive semidefiniteness, Singular value decomposition, Polar decomposition, Schur and Kronecker products; Positive and nonnegative matrices, Irreducible nonnegative matrices.