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September 23, 2023

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Description

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 Curriculum

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

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