Advanced Matrix Theory

IISc Bangalore, , Prof. Vittal Rao

Updated On 02 Feb, 19


Introduction : First Basic Problem Systems of Linear equations - Matrix Notation The various questions that arise with a system of linear eqautions - Second Basic Problem Diagonalization of a square matrix The various questions that arise with diagonalization.

Vector Spaces : Vector spaces - Subspaces - Linear combinations and subspaces spanned by a set of vectors - Linear dependence and Linear independence - Spanning Set and Basis - Finite dimensional spaces - Dimension

Solutions of Linear Systems : Simple systems - Homogeneous and Nonhomogeneous systems - Gaussian elimination - Null Space and Range - Rank and nullity - Consistency conditions in terms of rank - General Solution of a linear system - Elementary Row and Column operations - Row Reduced Form - Triangular Matrix Factorization

Important Subspaces associsted with a matrix : Range and Null space - Rank and Nullity - Rank Nullity theorem - Four Fundamental subspaces - Orientation of the four subspaces

Orthogonality : Inner product - Inner product Spaces - Cauchy Schwarz inequality - Norm - Orthogonality - Gram Schmidt orthonormalization - Orthonormal basis - Expansion in terms of orthonormal basis Fourier series
Orthogonal complement - Decomposition of a vector with respect to a subspace and its orthogonal - complement Pythagorus Theorem

Eigenvalues and Eigenvectors : What are the ingredients required for diagonalization? - Eigenvalue Eigenvector pairs - Where do we look for eigenvalues? characteristic equation - Algebraic multiplicity - Eigenvectors, Eigenspaces and geometric multiplicity

Diagonalizable Matrices : Diagonalization criterion - The diagonalizing matrix - Cayley-Hamilton theorem, Annihilating polynomials, Minimal Polynomial - Diagonalizability and Minimal polynomial - Projections - Decomposition of the matrix in terms of projections

Hermitian Matrices : Real symmetric and Hermitian Matrices - Properties of eigenvalues and eigenvectors - Unitary/Orthoginal Diagonalizbility of Complex Hermitian/Real Symmetric - matrices - Spectral Theorem - Positive and Negative Definite and Semi definite matrices

General Matrices : The matrices AAT and ATA - Rank, Nullity, Range and Null Space of AAT and ATA - Strategy for choosing the basis for the four fundamental subspaces - Singular Values - Singular Value Decomposition - Pseudoinverse and Optimal solution of a linear system of equations - The Geometry of Pseudoinverse

Jordan Cnonical form* : Primary Decomposition Theorem - Nilpotent matrices - Canonical form for a nilpotent matrix - Jordan Canonical Form - Functions of a matrix.

Selected Topics in Applications* : Optimization and Linear Programming - Network models - Game Theory - Control Theory - Image Compression


Lecture 1: Prologue Part 1

4.1 ( 11 )

Lecture Details

Advanced Matrix Theory and Linear Algebra for Engineers by Prof. Vittal Rao ,Centre For Electronics Design and Technology, IISC Bangalore. For more details on NPTEL visit httpnptel.iitm.ac.in



39 Ratings
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Excellent course helped me understand topic that i couldn't while attendinfg my college.

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Great course. Thank you very much.