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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


Go to the Course Home or watch other lectures:

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