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6FMAI14 Course information
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
Plan for the lectures 2023
Minor modifications of the contents or changes of times or room are possible.
Lecture 1 (10.15-12.00, 30/8, room E236): 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 (10.15-12.00, 6/9, E236): Horn-Johnson, Ch. 0, rank properties, partitioned matrices, Schur complement
Lecture 3 (10.15-12.00, 8/9, E236): Horn-Johnson, Ch. 1, spectral properties, simultaneous diagonalization (and triangularization)
Lecture 4 (10.15-12.00, 13/9, E236): Horn-Johnson, Ch. 2 - 3, companion matrices, canonical forms (Jordan, real Jordan, Weyr, rational, rational canonical)
Lecture 5 (10.15-12.00, 20/9, E236): Horn-Johnson, Ch. 4.1 - 4.3, variational characterization and inequalities for eigenvalues of Hermitian matrices
Lecture 6 (10.15-12.00, 22/9, E236): Horn-Johnson, Ch. 5, matrix norms
Lecture 7 (10.15-12.00, 27/9, E236): Horn-Johnson, Ch. 5 + ... , convergence, power series, functions of matrices
Lecture 8 (10.15-12.00, 4/10, E236): Horn-Johnson, Ch. 6.1, 6.3, eigenvalue location, Gershgorin regions
Lecture 9 (10.15-12.00, 11/10, E236): Horn-Johnson, Ch. 7, positive definite matrices, singular values
Lecture 10 (10.15-12.00, 18/10, E236): Horn-Johnson, Ch. 7 + ... , singular value inequalities, Ky-Fan norms, low-rank approximation
Lecture 11 (10.15-12.00, 2/11, E236): Kronecker and Hadamard products, linear matrix equations
Lecture 12 (10.15-12.00, 3/11, E236): Non-linear matrix equations, matrices of functions
Lecture 13 (10.15-12.00, 9/11, E236): Tensors
Sidansvarig: goran.bergqvist@liu.se
Senast uppdaterad: 2023-11-24