Matrices: Matrices and Linear Transformations, Rank, Determinant (as a measure of
volume of the space enclosed by the rows/columns of a matrix), trace of a matrix. Solving simultaneous equations using matrices: Gaussian Elimination. Overdetermined and underdetermine systems, Inverse, pseudo inverse. Condition number of a matrix, eigenvalues, eigenvectors, singular values, singular vectors. Computation of eigenvalues and eigen vectors (physical significance of eigen vectors and singular vectors – e.g image as a matrix). Eigen vectors of a symmetric positive definite matrix and its meaning. Diagonalization of a matrix and applications. Derivatives of scalars w.r.t a vector. Jacobian (4 weeks)
Short Exam on LA, Programming Assignment on Eigen/Singular values and vectors –
e.g. image analysis and reconstruction using eigen decomposition.
Orthogonality : Inner Product, Orthogonality, Gram-Schmidt Orthogonalization, Vector and Matrix Norms - Applications to optimization problems and graph theory, machine learning. (1 week)
Probability and Random Processes: Basic Topics (recap) – Sample points and Sample spaces: Events, algebra of events, partitions, Bayes theorem, probability axioms, joint and conditional probability. (1 week)
Short Exam and written assignment on basics. Introduction to random variables and random vectors: Discrete and continuous random variables, random vectors. Transformation of continuous random variables and vectors by deterministic functions. Density functions of transformed continuous random variables and vectors (Jacobian). (2 weeks).
Introduction to Random Processes: Statistical averages – ensemble and time averages. Random process (definition), Bernoulli random process, binomial process, sine wave process. Weak and strict sense stationarity of a random process. Ergodicity. Autocorrelation and Autocovariance functions of random processes, ACF and its relation to spectra (if time permits). Auto covariance matrix – properties of the autocovariance matrix. PCA/Eigen analysis of the ACF. Poisson process (application to arrival times, interarrival times etc.), Gaussian process, Martingale model and Markov Chains. Estimation of parameters from data: method of moments, method of maximum likelihood. Tests of fit: Chi-Squared, Student-t test, normality test. Cramer-Rao bound on estimators. Comparison of two different distributions of the same random variable/vector – Kullback Leibler divergence. (6 weeks)
Programming Assignments: (1) Eigen analysis of covariance matrix of sample data (2-D) generated from synthetic unimodal Gaussian distributions. (2) Generation of a random process from a given distribution, estimation of parameters, performing tests of fit.