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MATH 426 : Advanced Linear Algebra, Applications and Numerical Methods

Credits: 3

Course Description: This course is a continuation of linear algebra, towards topics relevant to applications as well as theoretical concepts. Topics to be discussed are algebraic systems, the singular value decomposition (SVD) of a matrix and some of its modern applications. We will discuss Principal component analysis (PCA) and its applications to data analysis. We will study linear transformations and change of basis. We will discuss complex vector spaces and Jordan canonical form of Matrices. We will discuss non-negative matrices and Perron-Frobenius Theory. We will explain multiple matrix factorisations, such as LU, QR, NMF. Finally we will discuss other applications such as the Fast Discrete Fourier Transform. For each of these topics we will discuss numerical computer algorithms and their implementations. In particular we will discuss in detail eigenvalue estimation, including iterative and direct methods, such as Hausholder methods, tri-diagonalzation, power methods, and power method with shifts. We will explain concepts of numerical analysis that are important to consider when we talk about the implementation of algorithms, such as stability and convergence. We will discuss iterative methods as well as direct ones, their advantages and disadvantages. The methods are their applications will be illustrated using a common programming language such as python and/or R.

Pre-Requisites: MATH 260

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