Applied Matrix Algebra

Learning Outcomes

• Students who completely master the course material will:
  
  Be knowledgeable of
  
• The Frobenius and other matrix norms.  Orthogonal (including Householder and Givens) and unitary matrices.  The QR factorization.  Projection and orthogonal projection matrices;
  
• The singular value decomposition, low rank approximations, total least squares, and the Moore-Penrose pseudo-inverse;
  
• Linear programming and applications, revised simplex, artificial variables, shadow prices, duality, degeneracy, Bland’s anti-cycling rules, the integer solution property of integer transportation problems, integer and logical variables;
  
• Positive definiteness, congruence, the modified Cholesky factorization.  Quadratic forms, equality constrained quadratic programming.
  
  Be able to use techniques from the course (including MATLAB where appropriate) to
  
• Calculate the QR factorization of a matrix via Givens and via Householder matrices;
  
• Find least squares / shortest solutions via the QR factorization and the SVD;
  
• Find orthogonal bases for the range and null spaces of a matrix;
  
• Find the total least squares fit to a set of data points via the SVD;
  
• Form linear programming models of data fitting (one and infinity norm), blending, transportation, rostering, resource allocation, stock cutting, and similar problems.
  
• Put a linear program into standard form, obtain an initial basic feasible solution via artificial variables, and then solve the original linear program via revised simplex;
  
• Solve an integer linear or mixed integer linear program via branch and bound;
  
• Identify a positive definite matrix by calculating its’ modified Cholesky factorization;
  
• Categorize stationary points of multivariable functions via the Hessian;
  
• Find the minimizer of a real valued complex quadratic;
  
• Solve equality constrained quadratic programs.

Prerequisites

Restrictions

For further information see Mathematics and Statistics .