AIS Matrix Analysis (2016)

Objectives:
1) Advanced Instructional Schools are organized for those students who have undergone training in the Annual Foundation Schools or who have already studied the topics taught in these schools.
2) An AIS is normally devoted to one subject or two inter-related subjects. An AIS is like a Ph.D. course and it assumes the contents covered in the three Annual Foundation Schools. 
3) The main objective is to help researchers and teachers and learn advanced mathematics in an enjoyable way. 

Course content:
The Advanced Instructional School on Matrix Analysis consisted of two streams. In one of them, there were lectures on analysis of matrices, in particular on matrix decompositions, eigenvalues, perturbations and inequalities. In the other stream, there were lectures on applications of matrix analysis. In the latter part, the lectures of the first two weeks were based on non-negative matrices and graph theory. In the third week, there were lectures on matrix computations.
The topics studied in Linear Algebra, Functional Analysis and related courses were aimed to be covered rapidly from advanced viewpoint. To prepare the participants beforehand, they were advised to revise these topics before coming to the school and lists of a few exercises was provided to help them brush up the concepts that were supposed to be routinely used in the course.

The exact portions covered by the respective speakers are given below:

• Prof. Rajendra Bhatia  (Indian Statistical Institute Delhi) :Spectral theorem, Schur triangular form,  QR Decomposition, Singular value decomposition, Polar decomposition,  Norms and approximations, Extremal principles for eigenvalues, Perturbation of eigenvalues of Hermitian matrices, Hoffman-Wielandt theorem, Sylvester equation.
• Prof. R. B. Bapat (Indian Statistical Institute Delhi):Nonnegative matrices, irreducible matrices, directed graphs, strongly connected graphs, graph associated with a matrix, Perron’s theorem for positive matrices, Perron-Frobenius theorem for irreducible matrices, Collatz-Wielandt function, Primitive matrices, index of cyclicity, Reducible matrices, Frobenius Normal form, M-matrices, matrices over max algebra, eigenproblem over max algebra, circuit geometric mean, Kronecker product, compound matrices, totally positive matrices, Inequalities for Perron root including Levinger and Kingman inequalities.
• Dr. Priyanka Grover (Shiv Nadar University) :Tensor products of vector spaces, Tensor products of linear operators, Antisymmetric tensor product, majorization and doubly stochastic matrices, Hall’s marriage problem, The König-Frobenius Theorem, Birkhoff’s theorem, Spectral variation of normal matrices, Positive definite matrices and block matrices.
• Dr. Krishnan Sivasubramanian (Indian Institute of Technology Bombay) :Geršgorin type theorems and bounds for eigenvalues, Applications of eigenvalues- Biclique decomposition and the squashed cube dimension theorem of Witsenhausen, theorems on set systems like all pairs intersection, odd-even club formation, application to Ramsey theory, Delsarte-Hoffman Theorem and algebraic approach to EKR (Erdös-Ko-Rado) type theorems.
• Dr. Tanvi Jain (Indian Statistical Institute Delhi):Differential calculus on Banach spaces with emphasis on matrix analysis, including Daleckii-Krein formula, an introduction to Loewner matrices and a brief introduction to operator monotone functions-the focus was on the following results (a) A function on an interval is operator monotone if and only if every Loewner matrix corresponding to f and every choice of numbers in the interval is positive semidefinite. (b) The power function t ↦ tr   is operator monotone if and only if 0 ≤ r ≤ 1, a short introduction to conditionally negative definite matrices, and sign regular matrices, Inertia of Hermitian matrices and Sylvester's law of inertia and its converse, the paper Inertia of Loewner matrices.
• Dr. Shreemayee Bora (Indian Institute of Technology Guwahati) :Rotators and Reflectors, QR decomposition by rotators and reflectors, Unitary similarity transformation to upper Hessenberg and tridiagonal forms, Power method and its variations for eigenvalue problems, QR algorithm for eigenvalue problems and its implementation.

Timetable:

 
LIST OF TUTORS:

1) Dr. Jagjit Singh Matharu, University Institute of Engineering and Technology, Panjab University.
2) Dr. Satyanarayana Reddy, Shiv Nadar University.
3) Dr. Mushtaq Ahmad Bhat, Indian Institute of Technology Bombay.
4) Mr. Mukesh Kumar Nagar, Indian Institute of Technology Bombay.
5) Mr. Swarup Kumar Panda, Indian Statistical Institute Delhi.
6) Ms. Nandita Roy, Indian Institute of Technology Guwahati.
 

 How to reach

Please visit the SNU website (contact us) for the address and for maps that give general directions from NOIDA as well as a detailed local schematic map.

By Air: The nearest airport is Indira Gandhi International Airport, New Delhi. From IGIA drive to Dhaula Kuan and take the Ring Road towards NOIDA. From NOIDA join the expressway to Greater NOIDA. From Pari Chowk in Greater NOIDA follow the instructions in the schematic map whose link is given above.

By Rail: From the Delhi Stations (Delhi/New Delhi/Nizamuddin) proceed to NOIDA and then follow the directions given above. Another option is to disembark at Ghaziabad (if you are on that line) and drive down on NH91 to Dadri. SNU is about 3 km after Dadri on the right hand side.