Course Description
Foundations in the most commonly used transforms in mathematics, science, and engineering. Eigenvector decompositions, Fourier transforms, singular value decompositions, and the Radon transform, with emphasis on mathematical structure and applications.
Additional Requirements for Graduate Students:
Graduate students will be required to have a deeper and more
thorough understanding of the topics of this course. Extra work
will be required, such as more difficult homework, a term
paper, and/or a final project, in addition to undergraduate
homework and exam requirements. A higher level of performance
will be expected.
Athena Title
Matrix and Integral Transforms
Prerequisite
[MATH 3000 or (MATH 3200 and MATH 3300) or MATH 3510 or MATH 3510H] and (MATH 3100 or MATH 3100H)
Grading System
A - F (Traditional)
Course Objectives
Students will learn about the basic mathematical structures underlying some of the most intensively used matrix and integral transforms in applied mathematics and their applications. The student will learn the standard eigenvector decomposition of square matrices and applications to differential equations; discrete and continuous Fourier transforms with applications to asymptotic solutions of differential equations, graph theory, and signal processing; singular value decomposition of non-square matrices with applications to probability, multivariate statistical analysis and dimensional reduction; and Radon transform and related discrete transforms used in computational tomography with applications in medicine and geology.
Topical Outline
1. Eigenvalue problems and eigenvector decompositions. Solutions of linear, homogeneous systems of differential equations. 2. The spectral theorem. Properties of eigenvalues and eigenvectors of self-adjoined matrices. 3. The eigenvector decomposition of shift-invariant matrices: the Fourier transform. 4. Applications of the Fourier transform to graph theory, number theory, and Markov chains. 5. Fast Fourier transforms. 6. Continuous Fourier transforms and applications to partial differential equations. 7. Signal processing: the dangers of discrete sampling of continuous functions and digital signal analysis using Fourier transforms. 8. Singular value decomposition (SVD). 9. Multivariate normal distributions and SVD, applications to multivariate signals. 10. The Radon transform and inverse problem. 11. Discretized inverse problems and computational tomography.