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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)

Lecture 23 - Positive Definite and Quadratic Forms

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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