FINAL REVIEW SHEET
The final exam may address material from any of the sections of the book we have covered,
meaning chapters 1 − 3, 5.1 − 5.4, 6, 7.1 − 7.4, 8.1 and 8.3, but will primarily focus on material
from chapters 7 and 8.
Chapter 7 – Eigenvalues and Eigenvectors
• What is an eigenvalue? What is an eigenvector? What is the characteristic polynomial?
• How do we compute the characteristic polynomial?
• How do we find eigenvalues and eigenvectors?
• How do we use eigenvalues and eigenvectors to approximate An⃗v for n large?
• How do we use eigenvalues and eigenvectors to analyze dynamical systems?
• What does it mean to diagonalize a matrix? How do you do it? When can you do it? Why?
• What does it mean for two matrices to be similar? What does this have to do with coordinate
systems? What properties are preserved under similarity?
Chapter 8 – Symmetric Matrices and Singular Value Decomposition
• What can you say about the eigenvalues of a symmetric matrix? The eigenvectors?
• How do you compute an orthonormal eigenbasis for a symmetric matrix?
• How do you compute the singular values of a matrix?
• What do singular values say about the geometry of a linear map?
Remark. Cofactor expansion and Sarrus’ rule were excluded from the second midterm only.
They are part of the material covered and may be addressed and useful in the final exam.