### Lecture 1: Prologue Part 1

##### 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 http://nptel.iitm.ac.in

##### Course Description :

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 : 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

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