Algebra, Week 5

Introduction to Eigenvalues and Eigenvectors

  • A 2x2 matrix A has an eigenvalue if there exists a vector V such that Av = λV.

  • λ is called the eigenvalue of A and V is the eigenvector corresponding to λ.

Finding Eigenvalues

Characteristic Polynomial

  • The eigenvalues can be found using the characteristic polynomial:

    • Formula: x² - (trace A)x + det A = 0

    • Where:

      • trace A = sum of the diagonal elements (a + d)

      • det A = ad - bc

  • Example to find λ: Consider Av = 3V, therefore (A - 2I)V = 0.

Example of Finding Eigenvalues in a 2x2 Matrix

  • For a matrix A = (a c) (b d)

  • Given eigenvalues 21 and λ₁:

    • V₁ = (x₁)

    • V₂ = (y₂) corresponds to λ₂.

  • The eigenvalue equation leads to the equation used to find vectors corresponding to eigenvalues.

Steps to Find Eigenvalues and Eigenvectors

  1. Example: Let A = (1 -3) (2 -1)

  2. Find the eigenvalues of A: Calculate using trace and determinant.

  3. Eigenvalues Calculation:

    • trace A = 1 + (-1) = 0

    • det A = (1)(-1) - (2)(-3) = -1 + 6 = 5

    • Solve: x² - 0x + 5 = 0.

Diagonalization of Matrix A

  • Each eigenvalue corresponds to an eigenvector that can be associated:

    • Set up the equations to get A = EDE^{-1} where D is a diagonal matrix of eigenvalues.

Example Calculation

  1. Find Eigenvalues: Using the characteristic polynomial, derive solutions to the eigenvalue equation.

  2. Find Eigenvectors for each Eigenvalue:

    • Solve (A - λI)V = 0 for V to find corresponding vectors.

Recurrence Relations

  • Solve for recurrence relations of the form Xₙ₊₁ = aXₙ + bXₙ₋₁.

  • Express in matrix form:

    • (Xₙ₊₁)

    • = (a b) (Xₙ) (1 0) (Xₙ₋₁)

  • Initial conditions provided help solve the recurrence sequence.

Conclusion

  • Eigenvalues and eigenvectors play a crucial role in understanding the transformation characteristics of matrices.

  • They can be calculated systematically through characteristic polynomials and recurrence relations.