IISc Bangalore Course , Prof. Vittal Rao

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IISc Bangalore Course , Prof. Vittal Rao

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

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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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

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- 1.Prologue Part 1
- 2.Prologue Part 2
- 3.Prologue Part 3
- 4.Linear Systems Part 1
- 5.Linear Systems Part 2
- 6.Linear Systems Part 3
- 7.Linear Systems Part 4
- 8.Vector Spaces Part 1
- 9.Vector Spaces Part 2
- 10.Linear Independence and Subspaces Part 1
- 11.Linear Independence and Subspaces Part 2
- 12.Linear Independence and Subspaces Part 3
- 13.Linear Independence and Subspaces Part 4
- 14.Basis Part 1
- 15.Basis Part 2
- 16.Basis Part 3
- 17.Linear Transformations Part 1
- 18.Linear Transformations Part 2
- 19.Linear Transformations Part 3
- 20.Linear Transformations Part 4
- 21.Linear Transformations Part 5
- 22.Inner Product and Orthogonality Part 1
- 23.Inner Product and Orthogonality Part 2
- 24.Inner Product and Orthogonality Part 3
- 25.Inner Product and Orthogonality Part 4
- 26.Inner Product and Orthogonality Part 5
- 27.Inner Product and Orthogonality Part 6
- 28.Diagonalization Part 1
- 29.Diagonalization Part 2
- 30.Diagonalization Part 3
- 31.Diagonalization Part 4
- 32.Hermitian and Symmetric matrices Part 1
- 33.Hermitian and Symmetric matrices Part 2
- 34.Hermitian and Symmetric matrices Part 3
- 35.Hermitian and Symmetric matrices Part 4
- 36.Singular Value Decomposition (SVD) Part 1
- 37.Singular Value Decomposition (SVD) Part 2
- 38.Back To Linear Systems Part 1
- 39.Back To Linear Systems Part 2
- 40.Epilogue

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