Syllabus Fall 2000

Mon. Wed. 4:30 - 5:45, Rm 106, Tyler Hall

Office: Tyler 222

phone: 874-4439, Office Hours: Tues 3:30-4:00, Wed 2:00-3:00, Thurs 1:00-2:00

or by appointment

eaton@math.uri.edu

A basic knowlege of linear algebra such as is covered in an undergraduate course is required as background for this course. You must be familiar with such concepts as matrix algebra, solving systems of equations, finding inverses of matrices, vectors, vector spaces, linear transformations. This material is included in both texts and can be used for reference.

We will cover the following sections from *Matrix
Analysis*.

Chapter | Topics |

1 | All Eigenvalues and Eigenvectors |

2 | Unitary Matrices |

Normal Matrices | |

Shur's Theorem | |

Cayley-Hamilton Theorem | |

QR-factorization | |

3 | Jordan Canonical Form |

The minimal polynomial | |

4 | Hermitian Matrices |

The Reyleigh-Ritz Theorem and the Courant-Fischer Theorem | |

7 | Single Value Decomposition |

Determining the structure of the eigensystems
of a matrix **A** is very important and motivates much of what is covered
in the course. We see that it is equivalent to questions concerning decompositions
of **A** and changes of basis. When the matrix **A** is *normal*,
we may use *unitary matices* in such decompositions. We will study
several such decompositions, which are not only useful in obtaining knowlege
of of the structure of the eigensystems but for other applications as well.
Other techniques apply to general matrices. The *Jordan form* is useful
in analyzing *defective matrices*, those which are not *diagonalizable*.

Finally, we study *quadratic forms* which
arise in diverse areas of applications and are useful in studying matrices.

Each paper should contain the following elements:

- Give a brief overview of the area of application, enough so that the problem can be understood by a reader who is unfamiliar with the area.
- State the problem.
- Give the mathematics that are needed in solving the problem. State all theorems, even if we covered them in class. You don't have to provide the proofs of the theorems.
- Show how the math is used in solving the general problem.
- Illustrate with an example using maple.
- Give a list of references.

- clarity,
- thoroughness,
- accuracy, and
- interest of the application.

Assignment | Problem Numbers | Due Date |

Handout | 1,2,3 | Sept 27 |

Section 1.1 | 1,5 | Sept 27 |

Section 1.2 | 3,4 | Oct 4 |

Section 1.3 | 2, 6, 10 | Oct 11 |

Section 1.4 | 4,10 | Oct 23 |

Section 2.1 | 1,2,3,12 | Nov 8 |

Section 2.2 | none | |

Section 2.3 | 6,8 | Nov 8 |

Section 2.4 | 1,9 | Nov 15 |

Project - Application | Nov 27 | |

Section 2.5 | 2,3,15 | Dec 11 |

Section 2.6 | 5 | Dec 11 |

Section 3.3 | 5,6 | Dec 11 |

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