MATH 304, Sections 501 and 503
Handout 02
How to Diagonalize a Matrix: Let A ∈ Mn,n (F).
1. Compute the characteristic polynomials χA (x) = det(xIn − A).
2. Use χA to compute the eigenvalues of A and the algebraic multiplicity of each eigenvalue.
3. If
X
amA (λ) < n
λ an eigenvalue of A
the matrix is not diagonalizable over F (this will not occur if F = C).
4. For each eigenvalue λ, find a basis for EA (λ).
5. If dim(EA (λ)) < amA (λ) for any eigenvalue λ, the matrix A is not diagonalizable.
6. Otherwise A is diagonalizable. Combine the bases for each EA (λ) to obtain a basis {~v1 , . . . , ~vn } of
eigenvectors with corresponding eigenvalues {λ1 , . . . , λn }.
7. Let Q be the n × n matrix with k th -column ~vk for all 1 ≤ k ≤ n and let D = diag(λ1 , . . . , λn ). Then
Q is invertible,
A = QDQ−1 ,
and
D = Q−1 AQ.
Matrix Limit Problem:
In a wildlife preserve, bunnies and wolves want to live in harmony. Each year the number of bunnies
increases to 54 times the previous year’s bunny population minus 1 times the previous year’s wolf population.
Similarly, each year the number of wolves increases to 38 times the previous year’s bunny population
minus half of the previous year’s wolf population. If the reserve starts with 100 bunnies and 7 wolves,
what values do the populations tend to over time?