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.
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.
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.
Example: Let A = (1 -3) (2 -1)
Find the eigenvalues of A: Calculate using trace and determinant.
Eigenvalues Calculation:
trace A = 1 + (-1) = 0
det A = (1)(-1) - (2)(-3) = -1 + 6 = 5
Solve: x² - 0x + 5 = 0.
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.
Find Eigenvalues: Using the characteristic polynomial, derive solutions to the eigenvalue equation.
Find Eigenvectors for each Eigenvalue:
Solve (A - λI)V = 0 for V to find corresponding vectors.
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.
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.