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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
Attached Files
File Type: zip IIT Kanpur Linear Algebra Notes.zip (13.9 KB, 24 views)


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