**Introduction to Properties of Vectors, Norms, Positive Semi-definite Matrices** |

Lecture 01 - Vectors and Matrices - Linear Independence and Rank |

Lecture 02 - Eigenvectors and Eigenvalues of Matrices and their Properties |

Lecture 03 - Positive Semidefinite Matrices and Positive Definite Matrices |

Lecture 04 - Inner Product Space and its Properties: Linearity, Symmetry and Positive Semidefinite |

Lecture 05 - Inner Product Space and its Properties: Cauchy Schwarz Inequality |

Lecture 06 - Properties of Norm, Gaussian Elimination and Echelon Form of Matrix |

Lecture 07 - Gram-Schmidt Orthogonalization Procedure |

Lecture 08 - Null Space and Trace of Matrices |

Lecture 09 - Eigenvalue Decomposition of Hermitian Matrices and Properties |

Lecture 10 - Matrix Inversion Lemma (Woodbury Identity) |

Lecture 11 - Introduction to Convex Sets and Properties |

Lecture 12 - Affine Set Examples and Application |

**Beaming Forming in Wireless Systems, Multi-user Wireless, Cognitive Radio Systems** |

Lecture 13 - Norm Ball and its Practical Applications |

Lecture 14 - Ellipsoid and its Practical Applications |

Lecture 15 - Norm Cone, Polyhedron and its Applications |

Lecture 16 - Applications: Cooperative Cellular Transmission |

Lecture 17 - Positive Semidefinite Cone and Positive Semidefinite Matrices |

Lecture 18 - Introduction to Affine Functions and Examples |

**Convex Optimization Problems, Linear Program** |

Lecture 19 - Norm Balls and Matrix Properties: Trace, Determinant |

Lecture 20 - Inverse of a Positive Definite Matrix |

Lecture 21 - Example Problems: Property of Norms, Problems on Convex Sets |

Lecture 22 - Problems on Convex Sets (cont.) |

Lecture 23 - Introduction to Convex and Concave Functions |

Lecture 24 - Properties of Convex Functions with Examples |

Lecture 25 - Test for Convexity: Positive Semidefinite Hessian Matrix |

Lecture 26 - Application: MIMO Receiver Design as a Least Squares Problem |

**QCQP, SOCP Problems, Applications** |

Lecture 27 - Jensen's Inequality and Practical Application |

Lecture 28 - Jensen's Inequality Application |

Lecture 29 - Properties of Convex Functions |

Lecture 30 - Conjugate Function and Examples to Prove Convexity of Various Functions |

Lecture 31 - Example Problems: Operations Preserving Convexity and Quasi Convexity |

Lecture 32 - Example Problems: Verify Convexity, Quasi Convexity and Quasi Concavity of Functions |

Lecture 33 - Example Problems: Perspective Function, Product of Convex Functions |

**Duality Principle and KKT Framework for Optimization, Application** |

Lecture 34 - Practical Application: Beamforming in Multi-antenna Wireless Communication |

Lecture 35 - Practical Application: Maximal Ratio Combiner for Wireless Systems |

Lecture 36 - Practical Application: Multi-antenna Beamforming with Interfering User |

Lecture 37 - Practical Application: Zero-Forcing Beamforming with Interfering User |

Lecture 38 - Practical Application: Robust Beamforming with Channel Uncertainty for Wireless Systems |

Lecture 39 - Practical Application: Robust Beamformer Design for Wireless Systems |

Lecture 40 - Practical Application: Detailed Solution for Robust Beamformer Computation |

**Optimization for Signal Estimation, LS, WLS, Regularization, Application** |

Lecture 41 - Linear Modeling and Approximation Problems: Least Squares |

Lecture 42 - Geometric Intuition for Least Squares |

Lecture 43 - Practical Application: Multi-antenna Channel Estimation |

Lecture 44 - Practical Application: Image Deblurring |

Lecture 45 - Least Norm Signal Estimation |

Lecture 46 - Regularization: Least Squares + Least Norm |

Lecture 47 - Convex Optimization Problem Representation: Canonical Form, Epigraph Form |

**Application: Convex Optimization for Machine Learning, Principal Component Analysis, Support Vector Machines** |

Lecture 48 - Linear Program Practical Application: Base Station Cooperation |

Lecture 49 - Stochastic Linear Program, Gaussian Uncertainty |

Lecture 50 - Practical Application: Multiple Input Multiple Output Beamforming |

Lecture 51 - Practical Application: Multiple Input Multiple Output Beamformer Design |

Lecture 52 - Practical Application: Cooperative Communication, Overview and Various Protocols Used |

Lecture 53 - Practical Application: Probability of Error Computation for Cooperative Communication |

Lecture 54 - Practical Application: Optimal Power Allocation Factor Determination for Cooperative Communication |

**Application: Compressive Sensing** |

Lecture 55 - Practical Application: Compressive Sensing |

Lecture 56 - Practical Application: Compressive Sensing (cont.) |

Lecture 57 - Practical Application: Orthogonal Matching Pursuit Algorithm for Compressive Sensing |

Lecture 58 - Example Problem: Orthogonal Matching Pursuit Algorithm |

Lecture 59 - Practical Application: L1 Norm Minimization and Regularization Approach for Compressive Sensing Optimization Problem |

Lecture 60 - Practical Application of Machine Learning and Artificial Intelligence: Linear Classification |

Lecture 61 - Practical Application: Linear Classifier (Support Vector Machine) Design |

**Practical Application: Approximate Classifier** |

Lecture 62 - Practical Application: Approximate Classifier Design |

Lecture 63 - Concept of Duality |

Lecture 64 - Relation between Optimal Value of Primal and Dual Problems, Concepts of Duality Gap and Strong Duality |

Lecture 65 - Example Problem on Strong Duality |

Lecture 66 - Karush-Kuhn-Tucker (KKT) Conditions |

Lecture 67 - Application of KKT Condition: Optimal MIMO Power Allocation (Waterfilling) |

**Application: Optimal MIMO Power Allocation** |

Lecture 68 - Application: Optimal MIMO Power Allocation (Waterfilling) (cont.) |

Lecture 69 - Example Problem on Optimal MIMO Power Allocation (Waterfilling) |

Lecture 70 - Linear Objective with Box Constraints, Linear Programming |

Lecture 71 - Example Problems on Convex Optimization |

Lecture 72 - Examples on Quadratic Optimization |

Lecture 73 - Examples on Duality: Dual Norm, Dual of Linear Program |

**Application: Convex Optimization for Big Data Analytics** |

Lecture 74 - Examples on Duality: Min-Max Problem, Analytic Centering |

Lecture 75 - Semidefinite Program and its Application: MIMO Symbol Vector Decoding |

Lecture 76 - Application: SDP for MIMO Maximum Likelihood Detection |

Lecture 77 - Introduction to Big Data: Online Recommender System (Netflix) |

Lecture 78 - Matrix Completion Problem in Big Data: Netflix |

Lecture 79 - Matrix Completion Problem in Big Data: Netflix (cont.) |