Programma Mathematics II

Onderwerpen

• Eigen value Decomposition: Nature of eigen values. Computation of eigen values and eigenvectors: characteristic polynomial. Geometric interpretation. Distinct real, repeated real and complex eigen values. Number of independent eigenvectors, modal matrix. Gram-Schmidt process to obtain orthonormal basis. 
• Computation of eigen values and eigenvectors using MATLAB and Maple. 
• Matrix eigen values theorems. Numerical computation of eigen values: QR decomposition, uniqueness of the QR decomposition. MATLAB QR decomposition and eigen values. 
• Matrix Calculus: Integration and differentiation, polynomials and powers, exponentials, MATLAB matrix functions. Cayley-Hamilton theorem.Similar and diagonalizable matrices. Similar transformations. Special matrices and their eigen values, real symmetric matrices, orthogonal matrices. 
• Application to systems of differential equations and state/space representation. 
• Singular Value Decomposition: motivation for SVD. Orthogonal matrices revisited. Singular values of matrix A. Subspaces of A. SVD of a square matrix. Input/output viewpoint of SVD, geometric representation of SVD in two and three dimensions. Picture of the gain as a function of input/ouput directions. 
• Computation of singular values and singular vectors. Relation to the eigen values decomposition. Computation of the SVD using MATLAB and Maple. Full and reduced SVD’s, thin and compact SVD’s. Condition number of a matrix. 
• Applications of SVD: computing the rank and range of a matrix, computing the (pseudo-) inverse of a matrix, least squares solutions of m x n systems, curve fitting, linear regression. 
• Examples of commercial applications of SVD: Principal Component Analysis, face recognition and image compression (on-line SVD for computer vision), Latent Semantic Indexing for Google’s page ranking.

Competenties

• To create sufficient background necessary to understand and use eigen values and eigenvectors (Eigen value Decomposition, ED) and Singular Value Decomposition (SVD) in modern control engineering.
• To practice the usage of the computer tools MATLAB/Simulink and Maple to analyze and solve problems using ED and SVD.

Doelen

After this course participants know:

• the nature of eigen values and eigenvectors and how to compute these for simple matrices using pencil-and-paper.
• how to compute eigen values and eigenvectors of a matrix using MATLAB and Maple. 
• how to compute an orthonormal basis from independent eigenvectors using the Gram-Schmidt process. 
• how eigen values values are computed numerically with the QR decomposition and the numerical errors that can occur. 
• how to integrate, differentiate a matrix, form matrix polynomials and exponentials by hand and using MATLAB or Maple. 
• when a matrix is (orthogonal) diagonalizable and how to do it. 
• how to compute the solution of a differential equations system with matrices. 
• the need for the Singular Value Decomposition and the nature of it. 
• how to compute the SVD for a simple 2 x 2 matrix by hand and using MATLAB and/or Maple for more complicated ones. 
• how to compute a reduced SVD using MATLAB and/or Maple. 
• how to use the SVD to solve m x n systems of equations using the pseudo-inverse, the normal equations or QR decomposition. 
• that SVD is also used in various commercial applications.