Differential Equations

Matrix Methods: Eigenvalues and Normal Modes

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In this session we learn matrix methods for solving constant coefficient linear systems of DE’s. This method will supersede the method of elimination used in the last session. In order to use matrix methods we will need to learn about eigenvalues and eigenvectors of matrices.

Session Activities

Read the course notes:

• Matrix Methods: Eigenvalues and Normal Modes: Introduction (PDF)
• Vectors and Matrices: Homogeneous Systems (PDF)

Complete the practice problem:

• Exercise: Linear Algebra (PDF)
• Answer (PDF)

Watch the lecture video clip:

• Matrix Methods Introduction, First Example

Read the course notes:

• Motivation and Derivation: Worked Example (PDF)

Watch the lecture video clip:

• Matrix Systems: General Terminology

Read the course notes:

• General Case: Eigenvalues and Eigenvectors (PDF)
• Worked Example: Distinct Real Roots (PDF)

Learn from the Mathlet materials:

• Read about how to work with the Matrix/Vector Applet (PDF)
• Work with the Matrix/Vector Applet

Watch the lecture video clip:

• Complex Eigenvalues: Example

Read the course notes:

• Complex Eigenvalues (PDF)

Watch the lecture video clip:

• Three Chamber Mixing Example: Repeated Roots

Read the course notes:

• Repeated Eigenvalues (PDF)

Watch the problem solving videos:

• Linear Systems: Matrix Methods
• Linear Systems: Complex Roots

Check Yourself

Complete the problem sets:

• Problem Set Part I Problems (PDF)
• Problem Set Part I Solutions (PDF)
• Problem Set Part II Problems (PDF)
• Problem Set Part II Solutions (PDF)

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