# InfoCoBuild

## Matrix Analysis with Applications

Matrix Analysis with Applications. Instructors: Dr. S. K. Gupta and Dr. Sanjeev Kumar, Department of Mathematics, IIT Roorkee. This course contains the concepts related to matrix theory and their applications in various disciplines. It covers a depth understanding of matrix computations involving rank, eigenvalues, eigenvectors, linear transformation, similarity transformations, (diagonalisation, Jordan canonical form, etc). It also involves various iterative methods, including Krylov subspace methods. Finally, topics like positive matrices, non-negative matrices and polar decomposition are discussed in detail with their applications. (from nptel.ac.in)

 Introduction

 Lecture 01 - Elementary Row Operations Lecture 02 - Echelon Form of a Matrix Lecture 03 - Rank of a Matrix Lecture 04 - System of Linear Equations Lecture 05 - System of Linear Equations (cont.) Lecture 06 - Introduction to Vector Spaces Lecture 07 - Subspaces Lecture 08 - Basis and Dimension Lecture 09 - Linear Transformations Lecture 10 - Rank and Nullity Lecture 11 - Inverse of a Linear Transformation Lecture 12 - Matrix Associated with a LT Lecture 13 - Eigenvalues and Eigenvectors Lecture 14 - Cayley-Hamilton Theorem and Minimal Polynomials Lecture 15 - Diagonalization Lecture 16 - Special Matrices Lecture 17 - More on Special Matrices and Gerschgorin Theorem Lecture 18 - Inner Product Spaces Lecture 19 - Vector and Matrix Norms Lecture 20 - Gram Schmidt Process Lecture 21 - Normal Matrices Lecture 22 - Positive Definite Matrices Lecture 23 - Positive Definite and Quadratic Forms Lecture 24 - Gram Matrix and Minimization of Quadratic Forms Lecture 25 - Generalized Eigenvectors and Jordan Canonical Form Lecture 26 - Evaluation of Matrix Functions Lecture 27 - Least Square Approximation Lecture 28 - Singular Value Decomposition Lecture 29 - Pseudo-Inverse and Singular Value Decomposition Lecture 30 - Introduction to Ill-conditioned Systems Lecture 31 - Regularization of Ill-conditioned Systems Lecture 32 - Linear Systems: Iterative Methods I Lecture 33 - Linear Systems: Iterative Methods II Lecture 34 - Non-stationary Iterative Methods: Steepest Descent I Lecture 35 - Non-stationary Iterative Methods: Steepest Descent II Lecture 36 - Krylov Subspace Iterative Methods (Conjugate Gradient Method) Lecture 37 - Krylov Subspace Iterative Methods (CG and Preconditioning) Lecture 38 - Introduction to Positive Matrices Lecture 39 - Non-negativity and Irreducible Matrices Lecture 40 - Polar Decomposition

 References Matrix Analysis with Applications Instructors: Dr. S. K. Gupta and Dr. Sanjeev Kumar, Department of Mathematics, IIT Roorkee. This course contains the concepts related to matrix theory and their applications in various disciplines.