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Week 11, Short Day: Diagonalization by Orthogonal Substitution

We will continue the topic from Wednesday by investigating particular properties of symmetric matrices and their possibility of diagonalization. More precisely, what does it mean if a diagonalized matrix is a special-orthogonal matrix. The exercises today will show that a symmetric map can be divided into three steps: a rotation, a scaling and a “backwards” rotation.

Today’s Key Concepts

Symmetric matrices. Orthogonal complement. Special-orthogonal matrix. Positive- and negative-definite and -indefinite. Rotation and scaling.

Prepartion and Syllabus

Again today the topics are from eNote 15 Symmetric Matrices, today until section 15.7. (The rest of this eNote will be covered in the Spring semester.)

Maple Ressources

We do not need new Maple commands for today’s topic.

Today’s Maple demo ScriptDiagonalization provides an advanced look for the interested student into Maple’s programming abilities for creating an automated script for diagonalization. Also, in the final exercise of today you will work with the Maple sheet AnalysisSymMatrix for an advanced analysis of symmetric matrices.

Activity program

  • 13.00 – 14.00: $\,$Lecture (aud. 42, b. 303A) (link to streaming)
  • 14.00 – 16.00: $\,$Group exercises in the study areas (b. 302, bottom floor)
  • 16.00 – 17.00: $\,$Weekly Test.

Group Exercises

  1. The Orthogonal Complement
  2. When there are n Different Eigenvalues
  3. Eigenspaces with gm > 1
  4. Special-Orthogonal Matrix
  5. Special-Orthogonal Matrix as Mapping Matrix
  6. Analysis of Symmetric Maps

Weekly Test

For all Weekly Tests, the following applies:

  • The test is an on-location test, meaning it can only be accessed in the study area.
  • No electronic aids are allowed (except for your own notes on e.g. a tablet).
  • The test can be accessed in the the Möbius quiz system via a link on DTU Learn in the module for 01006 (in the top menu click “Möbius”).
  • The TA will provide a code for Möbius test access.
  • You must be in full-screen mode so the test fills the whole screen.
  • Your solutions to the test questions must be typed into Möbius without in-between calculations or steps. The result is automatically evaluated by Möbius.
  • To ensure a smooth experience use the Firefox or Chrome browser, and disable any add-blocker.
  • Use a DTU network.
  • You may discuss the test questions with fellow students in your study group, but you have your own version of the test with scrambled numbers that you yourself must solve and enter into Möbius.
  • During the final hour on Fridays you have one attempt. Passing this attempt will grant you 1 bonus point. From Friday at 18:00 until Wednesday at 18:00 the test is reopened for repeated attempts. Passing during this phase will grant you ½ bonus point.