Module 13/Week 13

  • Module Overview
  1. We will assume that the theorems, definitions and examples related to diagonalization and complex inner products are well understood, and that you have been able to do, or at least attempted to do, the exercises corresponding to section 6.3 and 6.4.
  1. We will cover the second part of section 6.4 which considersunitary and normal matrices as well as the Schur Decompostition and the Spectral Theorem.
  1. We will cover section 6.6, where we will study quadratic forms and positive definite matrices.
  • Content for Week 12: Hermitian, Unitary and Normal matrices; Schur Decomposition and Spectral Theorem; Quadratic Forms and Positive Definite Matrices.
  1. We will study and prove Schur’s Theorem and then the Spectral Theorem, which relates Hermitian and unitary matrices.
  1. We will define normal matrices, and show how they relate to orthonormal sets of eigenvectors..
  1. We will study quadratic forms and learn a technique for rotating axes that uses eigenvalues and diagonalization.
  • Goals and Objectives:

1) Students will be able to prove Schur’s (Upper Triangularization) Theorem.

2) Students will be able to show that the eigenvalues of a Hermitian matrix H are real and H is unitarily diagonalizable.

3) Students will be able to demonstrate that a matrix is normal if and only if it is unitarily diagonalizable.

4) Students will be able to determine if a real symmetric matrix (or a quadratic form) is positive definite, or positive semidefinite.

5) Students will be able to establish the equivalence of several conditions each of which is equivalent to the matrix A being positive definite.

6) Students will use matrix diagonalization to translate axes when working with conic sections and quadric surfaces.

Assignments and Activities:

  • Assignments and Activities:
  1. Read and Watch

(i)Read Section 6.4, pages 325-334, and Section 6.6, pages 351-363.

(ii)Lesson with tablet

B. Assignments

Assignment #1:Section 6.4:pages334-6. #5,6,7,12,14,15,19, 23 and Section 6.6: pages 35-1: 1,2,3,4,6,8,9,10.

Assignment #2: Do Matlab exercise in Leon:

pages 381-3#19,23,24,25.

C. Forum: Feel free to share your answers and work together. Those who want to post the first answer to a problem, or an alternative answer to a problem already posted, will have it taken into account for participation credit.

Internet Search Assignment: Do a Google search for “Spectral Theorem” or “Quadratic Forms” or “Positive Definite Matrices”and share with the class at least one definition (or explanation), and its source, that complements the material in the book.

Feel free to share your answers and work together. Those who want to post the first answer to a problem, or an alternative answer to a problem already posted, will have it taken into account for participation credit.

.

  1. Recommended supplements for this section. This material is not mandatory. There are some differences in terminology and notation in the different texts. As this is very common, even amongst high school algebra, geometry, precalculus and calculus textbooks, it is very good practice for real life issues in teaching mathematics at any level, or reading articles on mathematics and mathematics education.

(iii)Read Chapters 11 (Quadratic Forms) and 12, and do exercises from:

(iv)Watch MIT videos: Lecture #25 : Symmetric Matrices and Positive Definiteness