Mathematics Cheat Sheet For Population Biology Page 8
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93.1
Eigenvalues and Eigenvectors
A scalar λ is an eigenvalue of a square matrix A and w = 0 is its associated eigenvector if
Aw = λw.
Eigenvalues of A are calculated as the roots of the characteristic equation,
det(A
λI) = 0,
where I is the identity matrix, a square matrix with ones along the diagonal and zeros elsewhere.
For example, we can calculate the eigenvalues for the matrix,
f
f
1
2
A =
.
p
0
1
Solve the characteristic equation det(A
λI) = 0:
f
f
λ 0
f
λ
f
1
2
1
2
(A
λI) =
=
p
0
0 λ
p
λ
1
1
det(A
λI) =
(f
λ)λ
f
p
1
2
1
2
λ
f
λ
f
p
= 0
1
2
1
Use the quadratic equation to solve for λ:
2
f
f
4f
p
1
2
1
1
2f
1
Numerical Example Define:
1.5 2
A =
(2)
0.5 0
1.5
λ
2
det(A
λI) =
0.5
λ
2
λ
1.5λ
1 = 0
(λ
2)(λ + 0.5) = 0
The roots of this are λ = 2 and λ =
0.5. A k
k matrix will have k eigenvalues. If a
matrix is non-negative, irreducible, and primitive, one of these eigenvalues is guaranteed to be
real, positive, and strictly greater than all the others.
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