MMath Mathematics and Statistics

This course unit is an introduction to matrix analysis, covering both classical and more recent results that are useful in applying matrix algebra to practical problems. In particular it treats eigenvalues and singular values, matrix factorizations, function of matrices, and structured matrices. It builds on the first year linear algebra course. Apart from being used in many areas of mathematics, Matrix Analysis has broad applications in fields such as engineering, physics, statistics, econometrics and in modern application areas such as data mining and pattern recognition. Examples from some of these areas will be used to illustrate and motivate some of the theorems developed in the course.

MATH36001 Pre-Requisites: MATH11022

Familiarity with Matlab is helpful but not essential.

To introduce students to matrix analysis through the development of essential tools such as the Jordan canonical form, Perron-Frobenius theory, the singular value decomposition, and matrix functions.

On successful completion of this course unit students will be able to: 

• Use the Cayley-Hamilton Theorem to manipulate matrix polynomials and use Gershgorin’s Theorem to localize the eigenvalues of a matrix.
• Construct and use properties of the Schur, Spectral, Jordan and Singular Value Decompositions of matrices.
• Solve least squares problems and construct low-rank approximations to a matrix using the Singular Value Decomposition.
• Check the solvability of a linear system, and when solvable, provide the set of solutions in terms of a generalized inverse.
• Compute and prove basic properties of the matrix exponential and use it for solving differential and algebraic equations.
• Apply Perron’s Theorem to positive matrices and the Perron-Frobenius Theorem to nonnegative matrices.

• Define, prove, and use properties of vector norms and matrix norms.

   

• Basics: Summary/recap of basic concepts from linear algebra including matrices and vectors, determinants, singularity of matrices, rank. [2 lectures]
• Theory of eigensystems: Eigenvalues, eigenvectors, and invariant subspaces; reduction of square matrices to simpler form including the Schur decomposition, spectral decomposition for normal matrices and the Jordan canonical decomposition; minimal and characteristic polynomials, Cayley-Hamilton Theorem; Sylvester's inertia theorem. [6]
• Norms: Vector norms and matrix norms, bounds for eigenvalues, Gershgorin theorem. [2]
• Singular value decomposition (SVD): Projectors; pseudo-inverses; application to linear least squares; polar decomposition. [3]
• Nonnegative matrices and related results: Irreducible matrices; Perron-Frobenius theorem; diagonally dominant matrices. [2]
• Matrix functions: Definitions; the matrix exponential function and application to the solution of differential equations and higher order equations; difference equations and matrix powers. [5]
• Kronecker product. Definition, properties and application to the solution of Sylvester's equation (if time). [2]

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
 

• Matrix analysis    Horn, Roger A., author.    Cambridge University Press    1985    ISBN: 9780511810817
• Matrix analysis and applied linear algebra    Meyer, C. D. (Carl Dean)    Society for Industrial and Applied Mathematics    2000    ISBN: 0898714540
• Matrix theory : a second course    Ortega, James M. (James McDonough), 1932-Plenum    1987    ISBN: 0306424339
• Matrix analysis for scientists & engineers    Laub, Alan J., 1948-    Society for Industrial and Applied Mathematics    2005    ISBN: 0898715768