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# MAI0098 Course information

Prerequisites

Linear Algebra, honours course (TATA53) or equivalent (the following topics should be familiar: complex vector spaces, the spectral theorem for Hermitian and normal operators, the singular value decomposition, the Jordan normal form). The first lecture(s) will review these topics.

Contents

Special matrices: Toeplitz, circulant, Vandermonde, Hankel, and Hessenberg matrices.
Block matrices: inversion formulas, Schur complement.
Real and complex canonical forms.
Vector and matrix norms.
Eigenvalues: location, inequalities, perturbations, Rayleigh quotients, variational characterization. Hadamard´s inequality.
Singular values: inequalities, variational characterization, Schatten and Ky Fan norms. Total least squares.
Matrix products: Kronecker, Hadamard and Khatri-Rao products.
Matrix equations. Stable matrices.
Functions of matrices.
Matrices of functions, matrix calculus and differentiation.
Multilinear algebra, tensor product, decomposition and approximation of tensors.

Examination

The examination consists of two parts, hand-in exercises and oral presentations.

Ph.D. students need 75% of the hand-in exercises well solved plus two oral presentations

Undergraduate/M.Sc. students following TA1004 need for
• grade 5: 75% of the hand-in exercises well solved plus two oral presentations
• grade 4: 50% of the hand-in exercises well solved plus two oral presentations, or, 75% of the hand-in exercises well solved plus one oral presentation
• grade 3: 50% of the hand-in exercises well solved plus one oral presentation

Schedule and plan for the lectures, 2020

Minor modifications of the contents or changes of times are possible (e.g., due to covid-19 situation).

Lecture 1 (2/9, 2020, at 10-12, room A25): review of topics from the Linear Algebra, honours course, e.g.,  the spectral theorem for normal matrices, the singular value decomposition, the Jordan canonical form
Lecture 2 (9/9, 10-12, room A25): Horn-Johnson, Ch. 0, rank properties, partitioned matrices, Schur complement
Lecture 3 (16/9, 10-12, room E236): Horn-Johnson, Ch. 1, spectral properties, simultaneous diagonalization (and triangularization)
Lecture 4 (23/9, 10-12, room R41): Horn-Johnson, Ch. 2 - 3, companion matrices, canonical forms (Jordan, real Jordan, Weyr, rational, rational canonical)
Lecture 5 (30/9, 10-12, room A25): Horn-Johnson, Ch. 4.1 - 4.3, variational characterization and inequalities for eigenvalues of Hermitian matrices
Lecture 6 (7/10, 10-12, room E236): Horn-Johnson, Ch. 5, matrix norms
Lecture 7 (14/10, 10-12, room A25): Horn-Johnson, Ch. 5 + ... , convergence, power series, functions of matrices
Lecture 8 (21/10, 10-12, room S26): Horn-Johnson, Ch. 6.1, 6.3, eigenvalue location, Gershgorin regions
Lecture 9 (5/11, 10-12, room A25): Horn-Johnson, Ch. 7, positive definite matrices, singular values
Lecture 10 (12/11, 10-12, room A25): Horn-Johnson, Ch. 7 + ... , singular value inequalities, Ky-Fan norms, low-rank approximation
Lecture 11 (19/11, 10-12, room E236): Kronecker and Hadamard products, linear matrix equations
Lecture 12 (26/11, 10-12, room A25): Non-linear matrix equations, matrices of functions
Lecture 13 (3/12, 10-12, room A25): Tensors

Sidansvarig: goran.bergqvist@liu.se