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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)

Lecture 75 - Proof of Singular Value Decomposition Theorem


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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