Matrix Computation and its Applications

This course deals with applications of matrices to a wide range of areas of engineering and science. Initial some basics of linear algebra is discussed followed by matrix norms and sensitivity and condition number of the matrices. In this course , we will also discuss psudo-inverse of a matrix of any dimension (m x n) using Moore- Penrose theorem. Singular value decomposition of a general matrix is also discussed along with applications. Householder transformations, QR factorization will also be covered.INTENDED AUDIENCE :UG/PGPRE-REQUISITES : Some knowledge of matrix theoryINDUSTRY SUPPORT : Any software/financial industry will be interested.

Week-1:Some basics of Linear Algebra :- Vector spaces, Linear transformations, eigen values and eigen vectors.
Week-2:Matrix norm, Sensitivity analysis and condition numbers, Linear systems, Jacobi, Gauss-Seidel and successive over relaxation methods, LU decompositions, Gaussian elimination with partial pivoting, Banded systems, positive definite systems, Cholesky decomposition – sensitivity analysis, Gram- Schmidt orthonormal process, Householder transformation, QR factorization, stability of QR factorization. Week-3:Solution of linear least squares problems, normal equations, singular value decomposition (SVD), Moore-Penrose inverse, Rank deficient least squares problems, Sensitivity analysis of least-squares problems, Sensitivity of eigenvalues and eigenvectors.
Week-4: Reduction to Hessenberg and tridiagonal forms; Power, inverse power and Rayleigh quotient iterations, Explicit and implicit QR algorithms for symmetric and non-symmetric matrices, Reduction to bi diagonal form, Sensitivity analysis of singular values and singular vectors, conjugate gradient method.