MAS 801 It's a Discreetly Discrete World – Mathematics in Real-life ...

Lecture 9: 22 October 2008
Eigenvalues & Eigenvectors
• How can we find the eigenvalues and eigenvectors of a given n x n
matrix A
• We want to find λ and x such that Ax = λx
• This can be rewritten as Ax - λx = 0 or (A - λI)x = 0
• Turns out that λ is an eigenvalue of A if and only if
det(A - λI) = 0
The above equation is called the characteristic equation of A
Example
• If A = 3 1 , then det(A - λI) = 3-λ 1 = (3 - λ) 2 - 1 = λ 2 - 6λ + 8
1 3
1 3-λ
• Solving the quadratic equation λ 2 - 6λ + 8 = 0 gives λ = 2 and λ = 4
• The eigenvalues of A are therefore 2 and 4