Matrix Exponentials

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In this session we will learn the basic linear theory for systems. We will also see how we can write the solutions to both homogeneous and inhomogeneous systems efficiently by using a matrix form, called the fundamental matrix, and then matrix-vector algebra.

Session Activities

Read the course notes:

• Matrix Exponentials: Introduction (PDF)

Watch the lecture video clip:

• Theory of Systems (00:11:22)

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  Theory of Systems (00:11:22)

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• Transcript (PDF)

Read the course notes:

• General Linear ODE Systems and Independent Solutions (PDF)
• The Existence and Uniqueness Theorem for Linear Systems (PDF)
• The Wronskian (PDF)
• Existence and Uniqueness and Superposition in the General Case (PDF)

Watch the lecture video clip:

• Fundamental Matrices (00:08:18)

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  Fundamental Matrices (00:08:18)

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• Transcript (PDF)

Read the course notes:

• Fundamental Matrices (PDF)
• The Normalized Fundamental Matrix (PDF)

Watch the lecture video clips:

• The Matrix Exponential I (00:26:26)

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  The Matrix Exponential I (00:26:26)

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• Transcript (PDF)
• The Matrix Exponential II (00:04:26)

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  The Matrix Exponential II (00:04:26)

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• Transcript (PDF)

Read the course notes:

• The Exponential Matrix (PDF)

Watch the lecture video clips:

• Example: Solving a System with the Matrix Exponential (00:08:17)

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  Example: Solving a System with the Matrix Exponential (00:08:17)

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• Calculating the Exponential Matrix (00:09:46)

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  Calculating the Exponential Matrix (00:09:46)

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• Variation of Parameters (00:13:19)

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  Variation of Parameters (00:13:19)

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Read the course notes:

• Inhomogeneous Case: Variation of Parameters Formula (PDF)

Watch the problem solving video:

• Matrix Exponentials (00:11:54)

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  Matrix Exponentials

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• Transcript (PDF)

Complete the practice problems:

• Practice Problems 35 (PDF)
• Practice Problems 35 Solutions (PDF)

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)

Session Activities

Watch the lecture video clip:

• Decoupling (00:47:06)

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  Decoupling (00:47:06)

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