Lecture Notes: Eigenvalues and Eigenvectors of a 3x3 Matrix

Introduction

• Focus on computation of eigenvalues and eigenvectors for a 3x3 matrix.
• No explanation on the concept of eigenvalues and eigenvectors; links provided for detailed explanations in the video description.

Finding Eigenvalues

• Use the characteristic equation: ( \det(A - \lambda I) = 0 ).
• Subtract ( \lambda ) from the main diagonal of the matrix.
• Calculate the determinant:
  • Example matrix:
    • [\begin{bmatrix} 2-\lambda & 0 & 1 \ -1 & 4-\lambda & -1 \ -1 & 2 & -\lambda \end{bmatrix}].
• Formula for determinant of 3x3 matrix: Use cofactor expansion on the chosen row (often row or column with zeros).
• Simplify the resulting polynomial equation to find eigenvalues:
  • In this case, eigenvalues (( \lambda )) are found to be 1, 2, and 3.

Finding Eigenvectors

• Focus on computation for ( \lambda = 1 ).
• Use the matrix ( A - \lambda I ) and solve ((A - \lambda I)\mathbf{v} = \mathbf{0}) to find the corresponding eigenvector.
  • Example matrix after substitution:
    • [\begin{bmatrix} 1 & 0 & 1 \ -1 & 3 & -1 \ -1 & 2 & -1 \end{bmatrix}].
• Solve the system of equations:
  • Example: From the first row, ( x_1 + x_3 = 0 ) implies ( x_3 = -x_1 ).
  • Second row simplifies to ( 3x_2 = 0 ), leading to ( x_2 = 0 ).
• Resulting eigenvector for ( \lambda = 1 ) can be represented as: [\begin{bmatrix} 1 \ 0 \ -1 \end{bmatrix}] (choosing ( x_1 = 1 ), ( x_3 = -1 )).

Exercise

• Compute eigenvectors for ( \lambda = 2 ) and ( \lambda = 3 ).
• Answers available in the video description for verification.

Conclusion

• Reiterate the importance of eigenvalues and eigenvectors.
• Encourage continuous learning.
• Mention that several resources are available for further understanding.