Course Details

Review of Schur’s unitary triangularization theorem, its consequences: Denseness of diagonalizable matrices, simultaneous triangularization, continuity of eigenvalues, eigenvalues of rank one perturbation, Various characterizations of normal matrices.
Singular value decomposition and its applications including Moore-Penrose inverse, Polar decomposition.
Hermitian matrices and congruences: Characterizations, Variational principles of eigenvalues, Courant Fischer minimax theorem, Weyl’s inequality and Cauchy’s interlacing theorem.
Algebraic, Analytic and geometric properties of vector norms, various matrix norms, Gershgorin’s theorem, spectral radius and Gelfand’s formula.
Positive definite matrices: Various characterizations, Schur product, Hadamard and Fischer inequalities.
Non-negative matrices: Perron-Frobenius theory, Birkhoff theorem for doubly stochastic matrices.

Course References:

Text Book:
1. R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge university press, 2013.
1. F. Zhang, Matrix Theory: Basic Results and Techniques, 2nd Ed., Springer, 2011.
2. M. Fiedler, Special Matrices and Their Applications in Numerical Mathematics, Dover Publications, 2008.