# InfoCoBuild

## Matrix Theory

Matrix Theory. Instructor: Prof. Chandra R. Murthy, Department of Electrical Communication Engineering, IISc Bangalore. In this course, we will study the basics of matrix theory, with applications to engineering. The focus will be two-fold: on the beautiful mathematical theory of matrices, and their use in solving engineering problems. This course covers topics: Vector spaces, matrices, determinant, rank, etc; Norms, error analysis in linear systems; Eigenvalues and eigenvectors; Canonical, symmetric and Hermitian forms, matrix factorizations; Variational characterizations, the quadratic form; Location and perturbation of eigenvectors; Least-squares problems, generalized inverses; Miscellaneous topics. (from nptel.ac.in)

 Course Introduction

 Lecture 01 - Course Introduction and Properties of Matrices Lecture 02 - Vector Spaces Lecture 03 - Basis, Dimension Lecture 04 - Linear Transforms Lecture 05 - Fundamental Subspaces of a Matrix Lecture 06 - Fundamental Theorem of Linear Algebra Lecture 07 - Properties of Rank Lecture 08 - Inner Product Lecture 09 - Gram-Schmidt Algorithm Lecture 10 - Orthonormal Matrices Definition Lecture 11 - Determinant Lecture 12 - Properties of Determinants Lecture 13 - Introduction to Norms and Inner Products Lecture 14 - Vector Norms and their Properties Lecture 15 - Applications and Equivalence of Vector Norms Lecture 16 - Summary of Equivalence of Norms Lecture 17 - Dual Norms Lecture 18 - Properties and Examples of Dual Norms Lecture 19 - Matrix Norms Lecture 20 - Matrix Norms: Properties Lecture 21 - Induced Norms Lecture 22 - Induced Norms and Examples Lecture 23 - Spectral Radius Lecture 24 - Properties of Spectral Radius Lecture 25 - Convergent Matrices, Banach Lemma Lecture 26 - Recap of Matrix Norms and Levy-Desplanques Theorem Lecture 27 - Equivalence of Matrix Norms and Error in Inverse of Linear Systems Lecture 28 - Errors in Inverses of Matrices Lecture 29 - Errors in Solving Systems of Linear Equations Lecture 30 - Introduction to Eigenvalues and Eigenvectors Lecture 31 - The Characteristic Polynomial Lecture 32 - Solving Characteristic Polynomials, Eigenvector Properties Lecture 33 - Similarity Lecture 34 - Diagonalization Lecture 35 - Relationship between Eigenvalues of BA and AB Lecture 36 - Eigenvector and Principle of Biorthogonality Lecture 37 - Unitary Matrices Lecture 38 - Properties of Unitary Matrices Lecture 39 - Unitary Equivalence Lecture 40 - Schur's Triangularization Theorem Lecture 41 - Cayley-Hamilton Theorem Lecture 42 - Uses of Cayley-Hamilton Theorem and Diagonalizability Revisited Lecture 43 - Normal Matrices: Definition and Fundamental Properties Lecture 44 - Fundamental Properties of Normal Matrices Lecture 45 - QR Decomposition and Canonical Forms Lecture 46 - Jordan Canonical Form Lecture 47 - Determining the Jordan Form of a Matrix Lecture 48 - Properties of the Jordan Canonical Form, Part 1 Lecture 49 - Properties of the Jordan Canonical Form, Part 2 Lecture 50 - Properties of Convergent Matrices Lecture 51 - Polynomials and Matrices Lecture 52 - Gaussian Elimination and LU Factorization Lecture 53 - LU Decomposition Lecture 54 - LU Decomposition with Pivoting Lecture 55 - Solving Pivoted System and LDM Decomposition Lecture 56 - Cholesky Decomposition and Uses Lecture 57 - Hermitian and Symmetric Matrix Lecture 58 - Properties of Hermitian Matrices Lecture 59 - Variational Characterization of Eigenvalues: Rayleigh-Ritz Theorem Lecture 60 - Variational Characterization of Eigenvalues (cont.) Lecture 61 - Courant-Fischer Theorem Lecture 62 - Summary of Rayleigh-Ritz and Courant-Fischer Theorems Lecture 63 - Weyl's Theorem Lecture 64 - Positive Semi-definite Matrix, Monotonicity Theorem and Interlacing Theorems Lecture 65 - Interlacing Theorem I Lecture 66 - Interlacing Theorem II (Converse) Lecture 67 - Interlacing Theorem (cont.) Lecture 68 - Eigenvalues: Majorization Theorem and Proof Lecture 69 - Location and Perturbation of Eigenvalues: Dominant Diagonal Theorem Lecture 70 - Location and Perturbation of Eigenvalues: Gershgorin's Theorem Lecture 71 - Implications of Gershgorin Disc Theorem, Condition of Eigenvalues Lecture 72 - Condition of Eigenvalues for Diagonalizable Matrices Lecture 73 - Perturbation of Eigenvalues: Birkhoff's Theorem, Hoffman-Wielandt Theorem Lecture 74 - Singular Value Definition and Some Remarks Lecture 75 - Proof of Singular Value Decomposition Theorem Lecture 76 - Partitioning the SVD Lecture 77 - Properties of SVD Lecture 78 - Generalized Inverse of Matrices Lecture 79 - Least Squares Lecture 80 - Constrained Least Squares

 References Matrix Theory Instructor: Prof. Chandra R. Murthy, Department of Electrical Communication Engineering, IISc Bangalore. In this course, we will study the basics of matrix theory, with applications to engineering.