## Matrix Analysis

 Unit code: MATH36001 Credit Rating: 10 Unit level: Level 3 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

#### Requisites

Prerequisite

MATH36001 pre-requisites

Students must have taken MATH10202 or MATH10212 (Linar Algebra).  Familiarity with Matlab is helpful but not essential.

#### Aims

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.

#### Overview

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.

#### Learning outcomes

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

• construct some key matrix decompositions such as the Jordan canonical form of a square matrix and the singular value decomposition of a rectangular matrix,
• localize the eigenvalues of matrix using matrix norms and Gershgorin's theorem,
• check the solvability of a linear system and when solvable, provide the set of solutions in terms of the pseudoinverse or the singular value decomposition of the matrix of the linear system,
• solve least squares problems and construct low-rank approximations to a matrix,
• compute 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.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• Mid-semester test: weighting 20%
• Two hours end of semester examination: weighting 80%

#### Syllabus

• 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]

• Roger A. Horn and Charles R. Johnson. Matrix Analysis. Cambridge University Press, 1985.
• Peter Lancaster and Miron Tismenetsky. The Theory of Matrices. Academic Press, London, second edition, 1985.
• Alan J. Laub. Matrix Analysis for Scientists and Engineers. SIAM, Philadelphia, PA, 2005.
• Carl D. Meyer. Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia, PA, 2000.
• James M. Ortega. Matrix Theory: A Second Course. Plenum Press, New York, 1987.

#### Feedback methods

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.

#### Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 67 hours

#### Teaching staff

Francoise Tisseur - Unit coordinator