Page 121 - Numerical methods for chemical engineering

Eigenvalues/eigenvectors of a 2 × 2 real matrix 107

Eigenvalues/eigenvectors of a 2 × 2 real matrix

Before discussing general eigenvector properties, let us consider the case of a 2×2 real
matrix, for which analytical calculation of eigenvalues is easy,

a 11 − λ a 12
det(A − λI) = = (a 11 − λ)(a 22 − λ) − a 21 a 12
a 21
a 22 − λ
2
= λ − (a 11 + a 22 )λ + (a 11 a 22 − a 21 a 12 ) (3.17)
As the trace (the sum of the diagonal values) and the determinant of A are
(3.18)
tr(A) = a 11 + a 22 det(A) = a 11 a 22 − a 21 a 12
then, using the shorthand T = tr(A), D = det(A),
√ 2
2
det(A − λI) = λ − T λ + D λ 1,2 = T ± T − 4D (3.19)
2
In terms of the elements of A, the eigenvalues are

1 1 2
λ 1,2 = (a 11 + a 22 ) ± (a 11 + a 22 ) − 4(a 11 a 22 − a 21 a 12 ) (3.20)
2 2
For any 2 × 2 real matrix A, the trace equals the sum of the eigenvalues,
λ 1 + λ 2 = a 11 + a 22 = tr(A) (3.21)
Also, we have the following properties for a real 2×2 matrix:
2
1 if T − 4D > 0, then both eigenvalues are real;
2
1
2 if T − 4D = 0,λ 1 = λ 2 = (a 11 + a 22 );
2
2
3 if T − 4D < 0, both eigenvalues are complex and λ 2 = λ 1 .
Also, note that the determinant equals the product of the eigenvalues,
1 1 2 2
2
2
λ 1 λ 2 = (T + T − 4D)(T − T − 4D) = [T − (T − 4D)] = D (3.22)
4 4
Thus, A is singular if it has one or more zero eigenvalues.
We now consider some special cases of 2 × 2 real matrices.
A is triangular

a 11 a 12
A = det(A) = a 11 a 12 (3.23)
0 a 22
The eigenvalues are

2
(a 11 + a 22 ) ± (a 11 + a 22 ) − 4a 11 a 22
λ 1,2 = ={a 11 , a 22 } (3.24)
2
For any upper triangular matrix, the eigenvalues are found on the principal diagonal. This
also is true for lower triangular and diagonal matrices,

a 11 0 a 11 0
A = A = (3.25)
a 21 a 22 0 a 22