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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 39 - Unitary Equivalence

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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