Linear systems and their solutions -- when does a solution exist, how accurate is the solution, how to solve the equations while minimising the error, how to estimate whether the solution obtained is close to the "real" solution. In the process, I would touch on condition number of a matrix, Givens rotations, Householder transforms, LU decomposition.

We will begin with a quick review of basics of systems of linear equations and the notions of eigenvalues, eigenvectors and diagonalization of matrices. Then we consider systems of the first order linear differential equations and outline how the methods of linear algebra can be applied to the problem of determining their solutions. We shall also see how this relates to solving homogeneous linear differential equations of nth order. Applications to electrical networks may also be indicated.

The singular value decomposition is closely associated with the eigenvalue-eigenvector decomposition of symmetric matrices. It applies to any rectangular matrices. It has numerous applications. We shall discuss applications to the least squares problem and polar decomposition of matrices.

Introducing Transforms: Fourier Transform, Wavelet Transform, Radon Transform and generalising to multiple dimensions. Connecting transforms to linear algebra principles and illustration with examples. Mathematical principles of filtering: examples of signal and image filters.

Solving Large and Spare Linear Least Squares Problem with Applications to Power System State Estimation. We will explore the following. Overview of Overdetermined Full Rank Linear Least Squares problem. Solving it using normal equations approach and QR decomposition. Sparsity issues and reduction of fill ins during LDU & QR factorisation. Formulation of power system state estimation problem as LSE problem.

Module 1

(Applications of Singular Value Decomposition for Big-Data Analysis) Fault (abnormal behaviour) detection and diagnosis using Principle component analysis (PCA), Fault diagnosis using Fisher Discriminant Analysis, Soft sensor development using Partial Least Squares (PLS), time series modelling using Dynamic PCA.

Module 2

(Applications of Eigenvalue based Analysis for Design and Control) Analysis of local stability of engineered systems using local linearisation and eigenvalue based analysis, linear feedback controller synthesis using eigenvalue assignment, convergence of numerical schemes for solving linear algebraic equations using eigenvalue based analysis