Unit 2: Spectral Theory In this unit, you will study Spectral Theory, which refers to the study of eigenvalues and eigenvectors of a matrix. The name Spectral Theory is due to David Hilbert, who coined the phrase in his study of Hilbert space theory. Hilbert's original work was in the setting of quadratic forms, and only later was it discovered that Spectral Theory had applications to quantum mechanics, where it could be used to describe the behavior of atomic spectra. Eigenvalues and eigenvectors of a linear operator are two of the most important concepts in Linear Algebra with applications to many fields, such as computer science (Google's PageRank algorithm), physics (quantum mechanics, vibration analysis), and economics (equilibrium states of Markov models). You will then learn about trace and determinants, two important numbers associated to a matrix. There are several operations that can be applied to a square matrix, and the determinant is a very important operation of this type. The determinant is a number that is calculated from a square matrix and is used to check for many different properties of that matrix, including invertibility. We will learn to compute the determinant and study properties of determinants and the effects of row operations on them. The trace of a matrix is related to the characteristic polynomial of the matrix and can be used to detect nilpotency. You will then learn about Schur's Theorem, which describes how every matrix is related to an upper triangular matrix. Finally, you will learn about quadratic forms, the second derivative test, and some advanced theorems.
Unit 2 Time Advisory
This unit will take you approximately 18 hours to complete.
☐ Subunit 2.1: 8.5 hours ☐ Sub-subunit 2.1.1: 2.5 hours
☐ Sub-subunit 2.1.2: 6 hours
☐ Subunit 2.2: 2 hours
☐ Subunit 2.3: 3.5 hours
☐
Subunit 2.4: 4 hours
☐
Reading: 1 hour
☐ Activity: 3 hours
Unit2 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Define and compute eigenvectors and eigenvalues.
- Define and compute invariant subspaces.
- Define and compute an eigenspace of a linear operator.
- Prove properties of invariant subspaces.
- State Schur's Theorem.
- Define and identify normal matrices.
- Explain the composition and the inversion of permutations.
- Define and compute the determinant.
- Explain when eigenvalues exist for a given operator.
- Determine the normal form of a nilpotent operator.
- Explain the idea of Jordan blocks, Jordan matrices, and the Jordan
form of a matrix.
- Define quadratic forms.
- State the second derivative test.
2.1 Eigenvalues, Eigenvectors, and Applications 2.1.1 Eigenvalues and Eigenvectors - Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 7: Eigenvalues and Eigenvectors”
Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne
Schilling’s *Linear Algebra: As an Introduction to Abstract
Mathematics*: “[Chapter 7: Eigenvalues and
Eigenvectors”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/10/mat67_course_notes.pdf)
(PDF)
Instructions: Please read Sections 7.1–7.5. The eigenvectors and
eigenvalues of a matrix help to describe the behavior of the
associated linear transformation.
Studying this reading should take approximately 1.5 hours to
complete.
Terms of Use: These materials have been reproduced for educational
and non-commercial purposes and can be viewed in their original
format
[here](https://mail.whittier.edu/owa/redir.aspx?C=fa77263df66842f9a74d8d1b871cd15b&URL=http%3A%2F%2Fwww.math.ucdavis.edu%2F~anne%2Flinear_algebra%2Fmat67_course_notes.pdf).
Any reproduction or redistribution for commercial use is strictly
prohibited.
Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory”
Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory” (PDF)
Instructions: Please read Section 7.1 (pages 157–164) in its entirety. Work through the eigenvalue/eigenvector examples on your own and check your work with that in the text. Being able to accurately and efficiently perform these computations is essential.Studying this reading should take approximately 1 hour to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
2.1.2 Eigenvalues and Eigenvectors - Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory”
Link: Brigham Young University: Kenneth Kuttler’s *An Introduction
to Linear Algebra:* [“Chapter 7: Spectral
Theory”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/02/Linear-Algebra-Kuttler-1-30-11-OTC.pdf)
(PDF)
Instructions: Please read Section 7.2 (pages 164–167) in its
entirety. Here, you will learn about applications of eigenvalues
and eigenvectors.
Studying this reading should take approximately 45 minutes to
complete.
Terms of Use: *An Introduction to Linear Algebra* was written by
Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the
Saylor Foundation’s Open Textbook Challenge.
Activity: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory”
Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory” (PDF)
Instructions: Please work through the odd-numbered problems for 19–33 in Section 7.3 on pages 168–170. When you are done, check your solutions with the answers on page 492.
Completing this activity should take approximately 2.5 hours to complete.Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 7: Eigenvalues and Eigenvectors”
Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 7: Eigenvalues and Eigenvectors” (PDF)
Instructions: Please read Section 7.6. Note the relationship between the eigenvectors for a rotation matrix and the angle of rotation.Studying this reading should take approximately 45 minutes to complete.
Terms of Use: These materials have been reproduced for educational and non-commercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 7”
Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 7” (PDF)
Instructions: Please complete calculational exercises 3 and 7 and proof-writing exercises 10, 11, and 12 (pages 95, 96, and 98).Completing this activity should take approximately 2 hours to complete.
Terms of Use: These materials have been reproduced for educational and non-commercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
2.2 Schur’s Theorem - Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory”
Link: Brigham Young University: Kenneth Kuttler’s *An Introduction
to Linear Algebra:* [“Chapter 7: Spectral
Theory”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/02/Linear-Algebra-Kuttler-1-30-11-OTC.pdf)
(PDF)
Instructions: Please read Section 7.4 (pages 173–180) in its
entirety. Schur's Theorem relates any matrix to an associated upper
triangular matrix in which the eigenvalues for the original matrix
appear on the diagonal. Read through the proof of this theorem and
the accompanying lemmas and corollaries.
Studying this reading should take approximately 2 hours to
complete.
Terms of Use: *An Introduction to Linear Algebra* was written by
Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the
Saylor Foundation’s Open Textbook Challenge.
2.3 Trace and Determinant - Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory”
Link: Brigham Young University: Kenneth Kuttler’s *An Introduction
to Linear Algebra:* [“Chapter 7: Spectral
Theory”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/02/Linear-Algebra-Kuttler-1-30-11-OTC.pdf)
(PDF)
Instructions: Please read Section 7.5 (page 180) in its entirety.
Pay particular attention to the algebraic properties of the trace.
Studying this reading should take approximately 30 minutes to
complete.
Terms of Use: *An Introduction to Linear Algebra* was written by
Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the
Saylor Foundation’s Open Textbook Challenge.
Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 8: Permutations and the Determinant of a Square Matrix”
Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 8: Permutations and the Determinant of a Square Matrix” (PDF)
Instructions: Please read Chapter 8 in its entirety. Pay particular attention to the algebraic properties of the determinant.Studying this reading should take approximately 1 hour to complete.
Terms of Use: These materials have been reproduced for educational and non-commercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 8”
Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 8” (PDF)
Instructions: Please complete calculational exercises 1, 4, and 5 and the proof-writing exercises 1, 2, and 3 (pages 115 and 116).This activity should take approximately 2 hours complete.
Terms of Use: These materials have been reproduced for educational and non-commercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
2.4 Quadratic Forms, Second Derivative Test, and Advanced Theorems - Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory”
Link: Brigham Young University: Kenneth Kuttler’s *An Introduction
to Linear Algebra:* [“Chapter 7: Spectral
Theory”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/02/Linear-Algebra-Kuttler-1-30-11-OTC.pdf)
(PDF)
Instructions: Please read Sections 7.6–7.9 (pages 181–190) in their
entirety. Note how the matrix algebra you have been studying is
applied to prove a familiar theorem from multivariable calculus.
Studying this reading should take approximately 1 hour to
complete.
Terms of Use: *An Introduction to Linear Algebra* was written by
Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the
Saylor Foundation’s Open Textbook Challenge.
Activity: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory”
Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory” (PDF)
Instructions: Please work through problems 1, 4, 15, and the odd-numbered problems for 19–31 for Section 7.10. When you are done, check your solutions with the answers on page 493.
This activity should take approximately 3 hours to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.