Eigenvalues and Eigenvectors
45 min 4 Examples
- Overview and Definition of Eigenvalues and Eigenvectors
- Find the Eigenvalue for the given Eigenvector (Example #1)
- Find the Eigenvector given its corresponding Eigenvalue (Example #2)
- Find a basis for the corresponding Eigenspace (Example #3 & #4)
- Exploring the general pattern for Eigenvalues
- Theorems involving Eigenvalues and Eigenvectors
The Characteristic Equation
53 min 4 Examples
- The Characteristic Equation: Finding Eigenvalues and Eigenvectors
- Find all Eigenvalues and Eigenvectors for the given matrix (Example #1 & #2)
- Invertible Matrix Theorem Continued and Similarity Theorem
- Find all Eigenvalues and Eigenvectors for the given matrix (Example #3 & #4)
- Application to Markov Chains
Diagonalization
35 min 3 Examples
- Understand the Importance of a Diagonal Matrix
- Procedure for Diagonalizing a Matrix
- Diagonalize the Matrix (Example #1 & #2)
- Determine if the Matrix is Diagonalizable by Inspection (Example #3)
- Definition and Theorem about Diagonalization and Distinct Eigenvalues
Eigenvectors and Linear Transformations
33 min 6 Examples
- Overview of Matrix of a Linear Transformation
- Algorithm for Finding a B-Matrix
- Using the Algorithm for Finding the matrix T relative to B (Example #1 & #2)
- Facts regarding matrix for T relative to B
- Mapping a polynomial and finding matrix T relative to a base (Example)
- Diagonal Matrix Representation Theorem and Overview of Similarity
- Find the B-Matrix for the Transformation (Example #1, #2 & #3)
Complex Eigenvalues
19 min 3 Examples
- Overview of Complex Eigenvalue and Complex Eigenvectors and their Applications
- Find the Eigenvalues for the given matrix (Example #1)
- Find all Eigenvalues and Eigenvectors for the given matrix (Example #2)
- Rotation due to a Complex Eigenvalue
- Find an invertible matrix P for a Complex Eigenvalue (Example #3)
Chapter Test
1 hr 30 min 9 Problems
- Is lambda an eigenvalue of the matrix? (Problem #1)
- Find the eigenvalues and multiplicities for the characteristic polynomial (Problem #2)
- Use diagonalization to compute A^10 (Problem #3)
- Find the eigenvalues of A and a basis for each eigenspace (Problem #4)
- Suppose the characteristic polynomial is given what can you say about the size of the eigenspace (Problem #5a-c)
- Find a basis for each eigenspace and determine if there is a P matrix that diagonalizes A (Problem #6)
- Find a basis B for the linear transformation such that [T]B is diagonal (Problem #7)
- Find the image of the transformation, show that T is a linear transformation and a matrix relative to the bases (Problem #8a-c)
- True or False (Problem #9a-d)
