MA20107 (Sec 1) Matrix Algebra

Syllabus:
Rank-properties, row space, column space, row-reduced echelon form, vector space, subspace, basis, dimension, linear transforamtion, null space, rank-nullity theorem, trace, Eigenvalues, eigenvectors, properties, symmetric, skew-symmetric matrices, Hermitian, skew-Hermitian, orthogonal, unitary matrices and their eigenvalues, Cayley-Hamilton theorem, Characteristic polynomial, minimal polynomial, diagonalization, generalized eigenvectors, Jordan canonical form, Matrix functions: p(A), exp(A), sin(A), cos(A), arctan(A), exp(At), Inner product spaces-properties, orthogonal, orthonormal sets, Gram-Schmidt orthonormalization process, Schur's Triangularization Theorem, Real quadratic forms (QF): positive definite, negative definite, positive semi-definite, negative semi-definite, indefinite, various characterizations of real quadratic forms: in terms of minors, in terms of eigenvalues, rank, index, signature of QF, diagonal form/principal axis theorem of QF, Lagrange's reduction to diagonal form, Bilinear forms-properties, Norms , matirx norm, induced norms-properties, examples of various norms including p-norm, equivalence of norms, Cauchy-Schwartz inequality, Minkowski inequality, Householder transformation, QR decomposition, Cholesky decomposition, singular value decomposition, Simultaneous diagonalization
Reference Books:
• Linear Algebra and its Applications by Gilbert Strang
• Matrix Methods: An Introduction by R. Bronson
• Matrix Analysis and Applied Linear Algebra by C. D. Meyer
Assignments: Lectures:
• Lecture -0 (17-18/7/17): Conducted by Prof. Nanduri for both Sec-1 and Sec-2 students, vector space, basis, row-reduced echelon form, linear transformation
• Lecture -1 (24/7/17): Introduction to Algebra of Matrices, Examples (e.g. control systems, network systems, data analysis) where the tools of Matrix Algebra can be employed to solve real life problems, Finally I introduced a notion called 'personality of a matrix'
• Lecture -2 (25/7/17): Column space, Row space, Rank, Classification of rank one matrices, an attempt to characterize matrices of any fixed rank for an algebra of matrices, Rank Normal Form (RNF) of a matrix (with proof), Proof of row rank=column rank using RNF
• Lecture -3 (31/7/17): Rank decomposition/factorization (RF) of a matrix from RNF, Computing basis of Column space, Row Space and Null space of a matrix using its RNF/RF, concept of inner product of vectors in real/complex finite dimensional vector spaces, inner product space (IPS), concept of norm/length induced by inner product, cosine of angle between two vectors in an IPS, orthogonality of vectors in an IPS
• Lecture -4 (01/8/17): Orthogonal/Orthonormal set and basis of IPS over the filed of real/complex numbers, example of non-standard inner products in euclidean space, orthogonal projection of a vector on a given vector in xy-plane
• Lecture -5 (07/8/17): Projection matrix and its personality, The Gram-Schmidt orhonormalization process, Fundamental theorem of Linear Algebra, orthogonal/unitary matrix, eigenvalues and eigenvectors of a square matrix, eigenvectors of a rotation matrix
• Lecture -6 (08/8/17): Applications of ONB in an IPS, eigenvalues of a matrix and its transpose/conjugate transpose, eigenvalues of product of two matrices, left and right eigenvectors and their orthogonality, eigenvalues and eigenvectors of triangular matrices, some speculation on eigenvalues and eigenvectors of symmetric/Hermitian matrices
• Lecture -7 (14/8/17): Spectral factorization/decomposition theorem for real symmetric/complex Hermitian matrices (with proof), Some speculation on personality of real symmetric/complex Hermitian matrices derived from its spectral decomposition, unitarily diagonalization, Schur's triangularization theorem (with proof), normal matrix, classification of all matrices that are unitarily diagonalizable (with proof)
• Lecture -8 (21/8/17): Importance of the theorems dicussed last day, matrix polynomial, eigenvalues of a matrix polynomial, annihilating polynomials and its usefulness, an annihilating polynomial of upper-triangular matrices, characteristic polynomial of a matrix, Cayley-Hamilton theorem (with proof), minimal/minimum polynomial of a matrix and its roots
• Lecture -9 (22/8/17): Minimal polynomial of a diagonal matrix, minimal polynomials of similar matrices, minimal polynomial of a normal matrix, exponential of a matrix and its properties (and its applications in control systems), exponential of a normal matrix, sine/cosine of a matrix
• Lecture -10 (28/8/17): Discussions on problems in the Assignment-1, companion matrix, generalized eigenvectors associated with an eigenvalue, chain/Jordan chain corresponding to an eigenvalue and examples
• Lecture -11 (29/8/17): Canonical/Jordan canonical basis of a matrix, Jordan block, Jordan Canonical Form of a matrix (with proof)
• Lecture -12 (04/9/17): Coomputations of Jordan Canonical Form of certain matrices, introduction to quadratic forms
• Lecture -13 (05/9/17): Doubt clearing session
• Lecture -14 (11/9/17): Doubt clearing session
• Lecture -15 (12/9/17): Doubt clearing session
• Lecture -16 (03/10/17): Quadratic forms (QF), equivalent QF, Diagonal QF, Orthogonal and Lagrange reduction methods for determining a diagonal form of a QF (examples)
• Lecture -17 & 18 (09-10/10/17): Classes are conducted by Prof. Nanduri: Bilinear forms-symmetric, conjugate, examples, matrix representation of bilinear forms, norm-def, examples-usual norm, p-norm, Cauchy-Schwartz inequality
• Lecture -19 (16/10/17): (combined with Section-2) Application and need of vector norms (p-norm) in real/complex vector space, linear isomorphism between any real/complex linear space of dimension n with R^n/C^n respectively, Holder's inequality (with proof), Minkowski's inequality (with proof)
• Lecture -20 (17/10/17): Behavior of p-norm as a function of p, equivalence of p-norms (norm inequalities) (with proof), definition of matrix norm
• Lecture -21 (23/10/17): Vectorization of a matrix and matrix norm defined by a corresponding vector norm, Subordinate/induced matrix norms, formula for induced norm with respect to infinity/max norm and 1-norm
• Lecture -22 (24/10/17): Classification of quadratic forms (positive definite, positive semi-definite, negative definite, negative semi-definite, indefinite)
• Lecture -23 (30/10/17): QR factorization of a matrix using Gram-Schmidt orthogonalization process, Householder reflector, QR factorization using Householder reflector
• Lecture -24 (31/10/17): Class Test
• Lecture -25 (06/11/17): Cholesky factorization of a positive definite matrix and its uniqueness, introduction to svd
• Lecture -26 (07/11/17): Householder reflector revisited, why should svd exist for any matrix
• Lecture -27 (13/11/17): Existence and computation of svd
• Lecture -28 (14/11/17): Doubt clearing session