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27th November 2015, 10:09 AM
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IIT Kanpur Linear Algebra Notes
Cab you provide me the topics for Lecture Notes in Mathematics - Linear Algebra and Applied Matrix Theory of I.I.T. (Indian Institutes of Technology) Kanpur as I need it for preparation of exams?
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#2
27th November 2015, 10:46 AM
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Re: IIT Kanpur Linear Algebra Notes
The topics for Lecture Notes in Mathematics - Linear Algebra and Applied Matrix Theory of I.I.T. (Indian Institutes of Technology) Kanpur as you need it for preparation of exams are as follows: Contents Chapter 1 Fields 1. Groups PROBLEMS 2. Permutation Groups PROBLEMS 3. Rings PROBLEMS 4. Fields PROBLEMS 5. Vector Spaces and Algebras over a Field PROBLEMS 6. Polynomials over a Field PROBLEMS 7. The Division Algorithm PROBLEMS 8. HCF of Polynomials PROBLEMS 9. Prime Factorization of Polynomials PROBLEMS 10. Extension of Fields PROBLEMS Chapter 2 Matrix Theory 1. Matrices PROBLEMS 2. Matrix Addition PROBLEMS 3. Matrix Multiplication PROBLEMS 4. Scalar Multiplication PROBLEMS 5. Block-Partitioned Matrices and Block Operations PROBLEMS 6. Vector Spaces of Matrices PROBLEMS 7. The Standard Inner Product in R n and C n PROBLEMS 8. Linear Independence of Matrices PROBLEMS 9. Some Standard Matrices PROBLEMS 10. Elementary Row and Column Operations PROBLEMS 11. Row-Reduced Echelon Form PROBLEMS 12. Determinant of a Square Matrix PROBLEMS 13. Properties of the Determinant Function PROBLEMS 14. Cofactor Expansion PROBLEMS 15. Rank of a Matrix PROBLEMS 16. The System of Linear Equations: Ax = b PROBLEMS 17. Eigenvalues and Eigenvectors 18. Companion Matrices and Characteristic Polynomial 19. Method of Danilevsky for Characteristic Polynomial PROBLEMS 20. Matrices with a Full-Set of Eigenvectors PROBLEMS 21. The Cayley-Hamilton Theorem PROBLEMS 22. Triangulization and Unitary Diagonalization of a Matrix 23. Schurs Lemma and the Spectral Theorem PROBLEMS 24. The Equation: AX-XB = C PROBLEMS 25. Circulants and the Discrete Fourier Transform (DFT) PROBLEMS 26. DFT of a Vector Signal and Solution of a Circulant System PROBLEMS 27 l -Matrices and Similarity PROBLEMS 28. Elementary Transformations on l -Matrices PROBLEMS Chapter 3 Canonical Factorizations 1. Row-Reduced Echelon Form 2. Hermite Canonical Form 3. Rank Factorization 4. Rank Factorization (Rectangular Form) 5. Rank Factorization (Left Invertible and Right Unitary) 6. Triangular Reduction 7. Conjugate Diagonal Reduction of Hermitian Matrices 8. Triangular Reduction by a Unitary Matrix 9. QR-Decomposition 10. Gram-Schmidt Triangular Reduction 11. Simultaneous Reduction of Positive Definite A and Hermitian B 12. Non-Singular LU-Decomposition PROBLEMS 13. Cholesky LL* -Decompostion PROBLEMS 14. Singular Value Decomposition PROBLEMS 15. Polar Decomposition PROBLEMS 16. When do the Polar Factors Commute? 17. Unicity of Polar Decomposition 18. Equivalence of Polar Decomposition and SVD 19. To Rotate a Vector in the Direction of another Vector 20. Tridiagonalizing a Hermitian Matrix by Householder Reflections (in n-2 steps) 21. QR-Algorithm for Hessenberg Matrices Chapter 4 Vector Spaces 1. Vector Space over a Field PROBLEMS 2. Linear Independence of Vectors PROBLEMS 3. Bases in a Vector Space PROBLEMS 4. Dimension of a Vector Space PROBLEMS 5. Direct Sum Decomposition of a Vector Space PROBLEMS 6. Linear Transformations (Operators) PROBLEMS 7. Change of Bases PROBLEMS 8. Jordan Canonical Form PROBLEMS 9. Rank of a Linear Transformation PROBLEMS 10. Linear Functionals PROBLEMS 11. The Transpose of a Linear Transformation PROBLEMS 12. Invariant Subspaces and Direct Sum of Operators PROBLEMS 13. Norms on Finite Dimensional Real or Complex Vector Spaces PROBLEMS 14. The Hölder and Minkowskis Inequalities 15. Equivalence of Norms on Finite Dimensional Spaces 16. Matrix Norms PROBLEMS 17. Inner Product Spaces PROBLEMS 18. Norm Induced by an Inner Product PROBLEMS 19. Orthogonality in Inner Product Spaces PROBLEMS 20. The Angle Between Two Vectors PROBLEMS 21. Gram-Schmidt Orthonormalization Procedure PROBLEMS 22. Projections and Orthogonal Projections PROBLEMS 23. The Kaczmarz Method 24. The Method of Residual Projections Chapter 5 Second Order Forms Links 1 1. Bilinear and Multilinear Expressions on Product Spaces PROBLEMS 2. Bilinear Forms on a Vector Space PROBLEMS 3. Symmetric and Skew-Symmetric Bilinear Forms PROBLEMS 4. Quadratic Forms PROBLEMS 5. Sesqi-Linear and Hermitian Forms PROBLEMS 6. Characterization of a Positive Definite Matrix PROBLEMS Chapter 6 Simultaneous Triangulization and Diagonalization Links 1 1. Characteristic and Minimal Polynomials PROBLEMS 2. T-Invariant Subspaces and T-Conductors PROBLEMS 3. Triangulization and Diagonalization PROBLEMS 4. Simultaneous Triangulization and Diagonalization PROBLEMS 5. Simultaneous Diagonalization PROBLEMS Chapter 7 The Primary and The Cyclic Decomposition Theorems Links 1 1. The Primary Decomposition Theorem PROBLEMS 2. T-Cyclic Subspaces and Vectors PROBLEMS 3. The Cyclic Decomposition Theorem PROBLEMS 4. The Rational Form PROBLEMS 5. The Invariant Factors PROBLEMS 6. The Jordan Canonical Form PROBLEMS 7. A Rank Based Determination of the Jordan Canonical Form PROBLEMS Chapter 8 Generalized Inverses Links 1 1. Definition of a g-Inverse PROBLEMS 2. Reflexive g-Inverse A-r PROBLEMS 3. Least-Squares g-Inverse A-r PROBLEMS 4. Minimum Norm g-Inverse A-m PROBLEMS 5. Moore-Penrose Inverse A+ PROBLEMS Chapter 9 Courant-Fischer Min-Max Theorems Links 1 1. A Variational Characterization of Eigen Values 2. Givens Method for a Hermitian Matrix 3. Eigenvalues of a Tridiagonal Hermitian Matrix Chapter 10 Inequalities in Matrix Theory Links 1 1. The Volume of an m-Dimensional Parallelepiped 2. Change of Variables in Multiple Integrals 3. Surface and Volume of a Hyper-Sphere 4. Volume of an n-Dimensional Ellipsoid 5. Principal Semi-Axes of a Section of an Ellipsoid 6. Application of Poincaré Separation Theorem to Ellipsoids Chapter 11 Perron-Frobenius Theory of Non-Negative Matrices Links 1 1. Non-Negative Matrices and Vectors PROBLEMS 2. Perrons Theorem on Positive Matrices PROBLEMS 3. Irreducible Matrices and Directed Graphs PROBLEMS 4. The Perron-Frobenius Theorem PROBLEMS 5. The Structure of Cyclic Matrices PROBLEMS Chapter 12 Stability of a Matrix Links 1 PROBLEMS 1. Lyapunovs Stability Theorem PROBLEMS 2. Equivalent Formulations of Sylvesters Inertia Theorem PROBLEMS 3. Some Equivalent Criteria of Stability |