Engineering Math Lecture 8.12: Diagonalization

Introduction

• Last section of Chapter 8: Diagonalization
• Definitions:
  • Diagonalizable Matrix: An n x n square matrix A is diagonalizable if there exists an invertible n x n matrix P and a diagonal matrix D such that AP = PD.
  • Similar Matrices: If A and D have this relationship, they are similar matrices.

Theorems

• Theorem 1: If matrix A has n distinct eigenvalues, then A is diagonalizable.
  • Diagonalizable if and only if A has n linearly independent eigenvectors.
• Visualizing Theorems:
  • A is diagonalizable ↔ A has n linearly independent eigenvectors.
  • n distinct eigenvalues → A is diagonalizable.
  • n distinct eigenvalues → n linearly independent eigenvectors.

Process of Diagonalizing a Matrix

1. Find Eigenvalues
   • Include multiple roots (e.g., lambda - 1 squared).
   • Arrange eigenvalues as diagonal entries of D.
2. Find Eigenvectors
   • For each eigenvalue, find corresponding eigenvectors.
   • Use these eigenvectors as columns in matrix P.
   • Ensure Vi is an eigenvector corresponding to lambda i.
   • Ensure enough eigenvectors are present; otherwise, A is not diagonalizable.

Examples

Example 1

• Given eigenvalues of A are 0 and 6.
• Steps:
  1. Note: Two distinct eigenvalues imply A is diagonalizable.
  2. Matrix D: Diagonal matrix with eigenvalues 0 and 6.
  3. Matrix P: Construct using eigenvectors for 0 and 6.
• Calculations:
  • Solve A - 0I2 and A - 6I2 for eigenvectors.
  • Construct P using eigenvectors.
  • Verify AP = PD.

Example 2

• No given eigenvalues.
• Steps:
  1. Compute determinant of A - lambda I2 for eigenvalues.
  2. Check eigenvalues: lambda = -3 with multiplicity 2.
  3. Find eigenvectors for lambda = -3.
• Result:
  • Only one independent eigenvector found → A is not diagonalizable.

Example 3 (3x3 Matrix)

• Eigenvalues: lambda = 1 (multiplicity 2), lambda = 2 (multiplicity 1).
• Steps:
  1. Solve for eigenvectors corresponding to each eigenvalue.
  2. Two independent vectors for lambda = 1, one for lambda = 2.
  3. Construct matrices D and P from eigenvectors.
  4. Verify AP = PD.

Remarks

• Diagonalizing matrices with complex eigenvalues works similarly.
• Purpose of diagonalization: Simplifies computations, especially for A^k.
• Matrix Powers: If A is diagonalizable, A^k = PD^kP^-1 simplifies computations.

Example: Compute A^10

• Given matrices P, D, and P^-1.
• Compute D^10 (diagonal elements powered individually).
• Multiply P, D^10, and P^-1 to find A^10.
• Provides large numerical results but simplifies computational process.

Conclusion

• Last section of Chapter 8.
• Students advised to practice odd-numbered problems.
• Upcoming topics in the next chapter.